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In mathematics, Vincent's theorem, named after Alexandre Joseph Hidulph Vincent, is a little-known theorem that was (almost) totally forgotten, having been overshadowed by Sturm's theorem. Even though Vincent's theorem is of great interest because it can be used to isolate the real roots of polynomials with rational coefficients, it cannot be found in any of the classical books on Theory of Equations (of the 20th century), except for Uspensky's book. Two variants of this theorem are presented along with several (continued fractions as well as bisection) real root isolation methods that are derived from it.

Sign variation

Let c₀, c₁, c₂, ... be a finite or infinite sequence of real numbers. Suppose l < r and the following conditions hold:

If r = l+1 the numbers c_l and c_r have opposite signs.
If r ≥ l+2 the numbers c_l+1, ..., c_r−1 are all zero and the numbers c_l and c_r have opposite signs.

This is called a sign variation or sign change between the numbers c_l and c_r.

When dealing with the polynomial p(x) in one variable, one defines the number of sign variations of p(x) as the number of sign variations in the sequence of its coefficients.

Two versions of this theorem are presented: the continued fractions version due to Vincent,^[1]^[2] and the bisection version due to Alesina and Galuzzi.^[3]^[4]

This statement of the continued fractions version can be found also in the Wikipedia article Budan's theorem.

Vincent's theorem: Continued fractions version (1834 and 1836)

If in a polynomial equation with rational coefficients and without multiple roots, one makes successive transformations of the form

x = a_{1} + \frac{1}{x^{'}}, x^{'} = a_{2} + \frac{1}{x^{″}}, x^{″} = a_{3} + \frac{1}{x^{‴}}, \dots

where $a_{1}, a_{2}, a_{3}, \dots$ are any positive numbers greater than or equal to one, then after a number of such transformations, the resulting transformed equation either has zero sign variations or it has a single sign variation. In the first case there is no root, whereas in the second case there is a single positive real root. Furthermore, the corresponding root of the proposed equation is approximated by the finite continued fraction:^[1]^[2]^[5]

a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3} + \frac{1}{⋱}}}

Moreover, if infinitely many numbers $a_{1}, a_{2}, a_{3}, \dots$ satisfying this property can be found, then the root is represented by the (infinite) corresponding continued fraction.

The above statement is an exact translation of the theorem found in Vincent's original papers;^[1]^[2]^[5] however, the following remarks are needed for a clearer understanding:

If $f_{n} (x)$ denotes the polynomial obtained after n substitutions (and after removing the denominator), then there exists N such that for all $n \geq N$ either $f_{n} (x)$ has no sign variation or it has one sign variation. In the latter case $f_{n} (x)$ has a single positive real root for all $n \geq N$ .
The continued fraction represents a positive root of the original equation, and the original equation may have more than one positive root. Moreover, assuming $a_{1} \geq 1$ , we can only obtain a root of the original equation which is > 1. To obtain an arbitrary positive root we need to assume that $a_{1} \geq 0$ .
Negative roots are obtained by replacing x by −x, in which case the negative roots become positive.

Vincent's theorem: Bisection version (Alesina and Galuzzi 2000)

Let p(x) be a real polynomial of degree deg(p) which has only simple roots. It is possible to determine a positive quantity δ so that for every pair of positive real numbers a, b with $| b - a | < δ$ , every transformed polynomial of the form

Template:NumBlk

has exactly 0 or 1 sign variations. The second case is possible if and only if p(x) has a single root within (a, b).

The Alesina-Galuzzi "a_b roots test"

From equation (Template:EquationNote) the following criterion is obtained for determining whether a polynomial has any roots in the interval (a, b):

Perform on p(x) the substitution

x \leftarrow \frac{a + b x}{1 + x}

and count the number of sign variations in the sequence of coefficients of the transformed polynomial; this number gives an upper bound on the number of real roots p(x) has inside the open interval (a, b). More precisely, the number ρ_ab(p) of real roots in the open interval (a, b) — multiplicities counted — of the polynomial p(x) in R[x], of degree deg(p), is bounded above by the number of sign variations var_ab(p), where

v a r_{a b} (p) = v a r ((1 + x)^{\deg (p)} p (\frac{a + b x}{1 + x})),

v a r_{a b} (p) = v a r_{b a} (p) \geq ρ_{a b} (p) .

As in the case of Descartes' rule of signs if var_ab(p) = 0 it follows that ρ_ab(p) = 0 and if var_ab(p) = 1 it follows that ρ_ab(p) = 1.

A special case of the Alesina-Galuzzi "a_b roots test" is Budan's "0_1 roots test".

Sketch of a Proof

A detailed discussion of Vincent's theorem, its extension, the geometrical interpretation of the transformations involved and three different proofs can be found in the work by Alesina and Galuzzi.^[3]^[4] A fourth proof is due to Ostrowski^[6] who rediscovered a special case of a theorem stated by Obreschkoff,^[7] p. 81, back in 1920-1923.

To prove (both versions of) Vincent's theorem Alesina and Galuzzi show that after a series of transformations mentioned in the theorem, a polynomial with one positive root will eventually have one sign variation. To show this they use the following corollary to the theorem by Obreschkoff of 1920-1923 mentioned earlier; that is, the following corollary gives the necessary conditions under which a polynomial with one positive root has exactly one sign variation in the sequence of its coefficients; see also the corresponding figure.

Corollary (to Obreschkoff's cone or sector theorem, 1920-1923^[7] p. 81): If a real polynomial has one simple root x₀, and all other (possibly multiple) roots lie in the sector
$S_{\sqrt{3}} = {x = - α + i β | α > 0 and | β | \leq \sqrt{3} | α |}$
then the sequence of its coefficients has exactly one sign variation.

Consider now the Möbius transformation

M (x) = \frac{a x + b}{c x + d}, a, b, c, d \in ℤ_{> 0}

and the three circles shown in the corresponding figure; assume that $\frac{a}{c} < \frac{b}{d}$ .

The (yellow) circle

| x - \frac{1}{2} (\frac{a}{c} + \frac{b}{d}) | = \frac{1}{2} (\frac{b}{d} - \frac{a}{c})

whose diameter lies on the real axis, with endpoints

\frac{a}{c}

and

\frac{b}{d}

, is mapped by the inverse Möbius transformation

M^{- 1} (x) = \frac{d x - b}{- c x + a}

onto the imaginary axis. For example the point

\frac{1}{2} (\frac{a}{c} + \frac{b}{d}) + \frac{i}{2} (\frac{b}{d} - \frac{a}{c})

gets mapped onto the point

- i \frac{d}{c}

. The exterior points get mapped onto the half-plane with Re(x) < 0.

The two circles (only their blue crescents are visible) with center

\frac{1}{2} (\frac{a}{c} + \frac{b}{d}) \pm \frac{i}{2 \sqrt{3}} (\frac{b}{d} - \frac{a}{c})

and radius

\frac{(b / d - a / c)}{\sqrt{3}}

are mapped by the inverse Möbius transformation

M^{- 1} (x) = \frac{d x - b}{- c x + a}

onto the lines

I m (x) = \pm \sqrt{3} R e (x)

. For example the point

\frac{1}{2} (\frac{a}{c} + \frac{b}{d}) - \frac{3 i}{2 \sqrt{3}} (\frac{b}{d} - \frac{a}{c})

gets mapped to the point

\frac{- d}{2 c} (1 - i \sqrt{3}) .

The exterior points (those outside the eight-shaped figure) get mapped onto the

S_{\sqrt{3}}

sector.

From the above it becomes obvious that if a polynomial has a single positive root inside the eight-shaped figure and all other roots are outside of it, it will present one sign variation in the sequence of its coefficients. This also guarantees the termination of the process.

Historical background

Early applications of Vincent's theorem

Vincent presented in both of his papers^[1]^[2] several examples showing precisely how his theorem is to be used in order to isolate the real roots of polynomials with continued fractions. However the resulting method had exponential computing time, a fact that must have been realized then, as was also realized by Uspensky^[8] p. 136, a century later.

The exponential nature of Vincent's algorithm is due to the way the partial quotients a_i (in Vincent's theorem) are computed. That is, to compute each partial quotient a_i (that is, to locate where the roots lie on the x-axis) Vincent uses Budan's theorem as a "no roots test"; in other words, to find the integer part of a root Vincent performs successive substitutions of the form x ← x+1 and stops only when the polynomials p(x) and p(x+1) differ in the number of sign variations in the sequence of their coefficients (i.e. when the number of sign variations of p(x+1) is decreased).^[1]^[2]

See the corresponding diagram where the root lies in the interval (5, 6). It can be easily inferred that, if the root is far away from the origin, it will take a lot of time to find its integer part this way; hence the exponential nature of Vincent's method. Below there is an explanation of how this drawback is overcome.

Disappearance of Vincent's theorem

Vincent was the last author in the 19th century to use his theorem for the isolation of the real roots of a polynomial.

The reason for that was the appearance of Sturm's theorem in 1827 which solved the real root isolation problem in polynomial time, by defining the precise number of real roots a polynomial has in a real open interval (a, b). The resulting (Sturm's) method for computing the real roots of polynomials has been the only one widely known and used ever since – up to about 1980, when it was replaced (in almost all computer algebra systems) by methods derived from Vincent's theorem, the fastest one being the Vincent–Akritas–Strzeboński (VAS) method.^[9]

Serret included in his Algebra,^[10] pp 363–368, Vincent's theorem along with its proof and directed all interested readers to Vincent's papers for examples on how it is used. Serret was the last author to mention Vincent's theorem in the 19th century.

Comeback of Vincent's theorem

In the 20th century Vincent's theorem cannot be found in any of the theory of equations books; the only exceptions are the books by Uspensky^[8] and Obreschkoff,^[7] where in the second there is just the statement of the theorem.

It was in Uspensky's book^[8] that Akritas found Vincent's theorem and made it the topic of his Ph.D. Thesis "Vincent's Theorem in Algebraic Manipulation", North Carolina State University, USA, 1978. A major achievement at the time was getting hold of Vincent's original paper of 1836, something that had eluded Uspensky — resulting thus in a great misunderstanding. Vincent's original paper of 1836 was made available to Akritas through the commendable efforts (interlibrary loan) of a librarian in the Library of the University of Wisconsin–Madison, USA.

Real root isolation methods derived from Vincent's theorem

Isolation of the real roots of a polynomial is the process of finding open disjoint intervals such that each contains exactly one real root and every real root is contained in some interval. According to the French school of mathematics of the 19th century, this is the first step in computing the real roots, the second being their approximation to any degree of accuracy; moreover, the focus is on the positive roots, because to isolate the negative roots of the polynomial p(x) replace x by −x (x ← −x) and repeat the process.

The continued fractions version of Vincent's theorem can be used to isolate the positive roots of a given polynomial p(x) of degree deg(p). To see this, represent by the Möbius transformation

M (x) = \frac{a x + b}{c x + d}, a, b, c, d \in ℕ

the continued fraction that leads to a transformed polynomial Template:NumBlk with one sign variation in the sequence of its coefficients. Then, the single positive root of f(x) (in the interval (0, ∞)) corresponds to that positive root of p(x) which is located in the open interval with endpoints $\frac{b}{d}$ and $\frac{a}{c}$ . These endpoints are not ordered and correspond to M(0) and M(∞) respectively.

Therefore, to isolate the positive roots of a polynomial, all that has to be done is to compute — for each root — the variables $a, b, c, d$ of the corresponding Möbius transformation

M (x) = \frac{a x + b}{c x + d}

that leads to a transformed polynomial as in equation (Template:EquationNote), with one sign variation in the sequence of its coefficients.

Crucial Observation: The variables $a, b, c, d$ of a Möbius transformation

M (x) = \frac{a x + b}{c x + d}

(in Vincent's theorem) leading to a transformed polynomial — as in equation (Template:EquationNote) — with one sign variation in the sequence of its coefficients can be computed:

either by continued fractions, leading to the Vincent-Akritas-Strzebonski (VAS) continued fractions method,^[9]
or by bisection, leading to (among others) the Vincent-Collins-Akritas (VCA) bisection method.^[11]

The "bisection part" of this all important observation appeared as a special theorem in the papers by Alesina and Galuzzi.^[3]^[4]

All methods described below (see the article on Budan's theorem for their historical background) need to compute (once) an upper bound, ub, on the values of the positive roots of the polynomial under consideration. Exception is the VAS method where additionally lower bounds, lb, need to be computed at almost every cycle of the main loop. To compute the lower bound lb of the polynomial p(x) compute the upper bound ub of the polynomial $x^{\deg (p)} p (\frac{1}{x})$ and set $l b = \frac{1}{u b}$ .

Excellent (upper and lower) bounds on the values of just the positive roots of polynomials have been developed by Akritas, Strzeboński and Vigklas based on previous work by Doru Stefanescu. They are described in P. S. Vigklas' Ph.D. Thesis^[12] and elsewhere.^[13] These bounds have already been implemented in the computer algebra systems Mathematica, Sage, SymPy, Xcas etc.

All three methods described below follow the excellent presentation of François Boulier,^[14] p. 24.

Continued fractions method

There is only one continued fractions method derived from Vincent's theorem. As has been stated above it started in the 1830s when Vincent presented in both of his papers^[1]^[2] several examples showing precisely how his theorem is to be used in order to isolate the real roots of polynomials with continued fractions. However the resulting method had exponential computing time. Below is an explanation of how this method evolved.

Vincent–Akritas–Strzeboński (VAS, 2005)

This is the second method (after VCA) developed to handle the exponential behavior of Vincent's method.

The VAS continued fractions method is a direct implementation of Vincent's theorem. It was originally presented by Vincent in his 1834^[1] and 1836^[2] papers in an exponential form; namely, Vincent computed each partial quotient a_i by a series of unit increments a_i ← a_i + 1, which are equivalent to substitutions of the form x ← x+1.

Vincent's method was converted into its polynomial complexity form by Akritas, who in his 1978 Ph.D. Thesis ("Vincent's theorem in algebraic manipulation", North Carolina State University, USA) computed each partial quotient a_i as the lower bound, lb, on the values of the positive roots of a polynomial; this is called the ideal positive lower root bound which computes the integer part of the smallest positive root (see the corresponding figure). To wit, now set $a_{i} \leftarrow l b$ or, equivalently, perform the substitution x ← x+lb, which takes about the same time as the substitution x ← x+1.

Finally, since the ideal positive lower root bound does not exist, Strzeboński^[15] introduced in 2005 the substitution $x \leftarrow l b_{c o m p u t e d} * x$ , whenever $l b_{c o m p u t e d} > 16$ ; in general $l b > l b_{c o m p u t e d}$ and the value 16 was determined experimentally. Moreover, it has been shown^[15] that the VAS (continued fractions) method is faster than the fastest implementation of the VCA (bisection) method,^[16] a fact that was confirmed^[17] independently; more precisely, for the Mignotte polynomials of high degree VAS is about 50,000 times faster than the fastest implementation of VCA.

In 2007, Sharma^[18] removed the hypothesis of the ideal positive lower bound and proved that VAS is still polynomial in time.

VAS is the default algorithm for root isolation in Mathematica, Sage, SymPy, Xcas.

For a comparison between Sturm's method and VAS use the functions realroot(poly) and time(realroot(poly)) of Xcas. By default, to isolate the real roots of poly realroot uses the VAS method; to use Sturm's method write realroot(sturm, poly). See also the External links for two applications that do the same thing: one for Android devices by A. Berkakis and another one for the Apple devices iPhone/iPod/iPad by S. Kehagias.

Here is how VAS(p, M) works, where for simplicity Strzeboński's contribution is not included:

Let p(x) be a polynomial of degree deg(p) such that p(0) ≠ 0. To isolate its positive roots, associate with p(x) the Möbius transformation M(x) = x and repeat the following steps while there are pairs {p(x), M(x)} to be processed.
Use Descartes' rule of signs on p(x) to compute, if possible, (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (0, ∞). If there are no roots return the empty set, ∅ whereas if there is one root return the interval (a, b), where a = min(M(0), M(∞)), and b = max(M(0), M(∞)); if b = ∞ set b = ub, where ub is an upper bound on the values of the positive roots of p(x).^[12]^[13]
If there are two or more sign variations Descartes' rule of signs implies that there may be zero, two or an even number of real roots inside the interval (0, ∞); in this case consider separately the roots of p(x) which lie inside the interval (0, 1) from those which lie inside the interval (1, ∞). A special test has to be made for 1.
- To guarantee that there will be roots inside the interval (0, 1) the ideal lower bound, lb is used; that is the integer part of the smallest positive root is computed with the help of the lower bound,^[12]^[13] $l b_{c o m p u t e d}$ , on the values of the positive roots of p(x). If $l b_{c o m p u t e d} > 1$ , the substitution $x \leftarrow x + l b_{c o m p u t e d}$ is performed to p(x) and M(x), whereas if $l b_{c o m p u t e d} \leq 1$ use substitution(s) x ← x+1 to find the integer part of the root(s).
- To compute the roots inside the interval (0, 1) perform the substitution $x \leftarrow \frac{1}{1 + x}$ to p(x) and M(x) and process the pair

{(1 + x)^{\deg (p)} p (\frac{1}{1 + x}), M (\frac{1}{1 + x})},

whereas to compute the roots in the interval (1, ∞) perform the substitution x ← x+1 to p(x) and M(x) and process the pair

{p (1 + x), M (1 + x)}

. It may well turn out that 1 is a root of p(x), in which case M(1) is a root of the original polynomial and the isolation interval reduces to a point.

Below is a recursive presentation of VAS(p, M).

VAS(p, M):

Input: A univariate, square-free polynomial $p (x) \in ℤ [x], p (0) \neq 0$ and of degree deg(p), and the Möbius transformation

$M (x) = \frac{a x + b}{c x + d} = x, a, b, c, d \in ℕ .$

Output: A list of isolating intervals of the positive roots of p(x).

1 var ← the number of sign variations of p(x) // Descartes' rule of signs; 2 if var = 0 then RETURN ∅; 3 if var = 1 then RETURN {(a, b)} // a = min(M(0), M(∞)), b = max(M(0), M(∞)), but if b = ∞ set b = ub, where ub is an upper bound on the values of the positive roots of p(x); 4 lb ← the ideal lower bound on the positive roots of p(x); 5 if $l b \geq 1$ then $p \leftarrow p (x + l b), M \leftarrow M (x + l b)$ ; 6 $p_{01} \leftarrow (x + 1)^{\deg (p)} p (\frac{1}{x + 1}), M_{01} \leftarrow M (\frac{1}{x + 1})$ // Look for real roots in (0, 1); 7 m ← M(1) // Is 1 a root? ; 8 $p_{1 \infty} \leftarrow p (x + 1), M_{1 \infty} \leftarrow M (x + 1)$ // Look for real roots in (1, ∞); 9 if p(1) ≠ 0 then 10 RETURN $V A S (p_{01}, M_{01}) \cup V A S (p_{1 \infty}, M_{1 \infty})$ 11 else 12 RETURN $V A S (p_{01}, M_{01}) \cup {[m, m]} \cup V A S (p_{1 \infty}, M_{1 \infty})$ 13 end

Remarks

For simplicity Strzeboński's contribution is not included.
In the above algorithm with each polynomial there is associated a Möbius transformation M(x).
In line 1 Descartes' rule of signs is applied.
If lines 4 and 5 are removed from VAS(p, M) the resulting algorithm is Vincent's exponential one.
Any substitution performed on the polynomial p(x) is also performed on the associated Möbius transformation M(x) (lines 5 6 and 8).
The isolating intervals are computed from the Möbius transformation in line 3, except for integer roots computed in line 7 (also 12).

Example of VAS(p, M)

Given the polynomial $p (x) = x^{3} - 7 x + 7$ the arguments of the VAS method are: $p (x) = x^{3} - 7 x + 7$ and M(x) = x.

1 var ← 2 // the number of sign variations in the sequence of coefficients of $p (x) = x^{3} - 7 x + 7$

4 lb ← 1 // the ideal lower bound — found using $l b_{c o m p u t e d}$ and substitution(s) x ← x+1

5 $p \leftarrow x^{3} + 3 x^{2} - 4 x + 1, M \leftarrow x + 1$

6 $p_{01} \leftarrow x^{3} - x^{2} - 2 x + 1, M_{01} \leftarrow \frac{x + 2}{x + 1}$

7 m ← 1

8 $p_{1 \infty} \leftarrow x^{3} + 6 x^{2} + 5 x + 1, M_{1 \infty} \leftarrow x + 2$

10 RETURN $V A S (x^{3} - x^{2} - 2 x + 1, \frac{x + 2}{x + 1}) \cup V A S (x^{3} + 6 x^{2} + 5 x + 1, x + 2)$

List of isolation intervals: {}. List of pairs ${p o l y, M o e b i u s t r a n s f o r m a t i o n}$ to be processed: ${{x^{3} - x^{2} - 2 x + 1, \frac{x + 2}{x + 1}}, {x^{3} + 6 x^{2} + 5 x + 1, x + 2}}$ . Remove the first and process it.

$V A S (x^{3} - x^{2} - 2 x + 1, \frac{x + 2}{x + 1})$

1 var ← 2 // the number of sign variations in the sequence of coefficients of $p (x) = x^{3} - x^{2} - 2 x + 1$

4 lb ← 0 // the ideal lower bound — found using $l b_{c o m p u t e d}$ and substitution(s) x ← x+1

6 $p_{01} \leftarrow x^{3} + x^{2} - 2 x - 1, M_{01} \leftarrow \frac{2 x + 3}{x + 2}$

7 $m \leftarrow \frac{3}{2}$

8 $p_{1 \infty} \leftarrow x^{3} + 2 x^{2} - x - 1, M_{1 \infty} \leftarrow \frac{x + 3}{x + 2}$

10 RETURN $V A S (x^{3} + x^{2} - 2 x - 1, \frac{2 x + 3}{x + 2}) \cup V A S (x^{3} + 2 x^{2} - x - 1, \frac{x + 3}{x + 2})$

List of isolation intervals: {}. List of pairs ${p o l y, M o e b i u s t r a n s f o r m a t i o n}$ to be processed: ${{x^{3} + x^{2} - 2 x - 1, \frac{2 x + 3}{x + 2}}, {x^{3} + 2 x^{2} - x - 1, \frac{x + 3}{x + 2}}, {x^{3} + 6 x^{2} + 5 x + 1, x + 2}}$ . Remove the first and process it.

$V A S (x^{3} + x^{2} - 2 x - 1, \frac{2 x + 3}{x + 2})$

1 var ← 1 // the number of sign variations in the sequence of coefficients of $p (x) = x^{3} + x^{2} - 2 x - 1$

3 RETURN ${(\frac{3}{2}, 2)}$

List of isolation intervals: ${(\frac{3}{2}, 2)}$ . List of pairs ${p o l y, M o e b i u s t r a n s f o r m a t i o n}$ to be processed: ${{x^{3} + 2 x^{2} - x - 1, \frac{x + 3}{x + 2}}, {x^{3} + 6 x^{2} + 5 x + 1, x + 2}}$ . Remove the first and process it.

$V A S (x^{3} + 2 x^{2} - x - 1, \frac{x + 3}{x + 2})$

1 var ← 1 // the number of sign variations in the sequence of coefficients of $p (x) = x^{3} + 2 x^{2} - x - 1$

3 RETURN ${(1, \frac{3}{2})}$

List of isolation intervals: ${(1, \frac{3}{2}), (\frac{3}{2}, 2)}$ . List of pairs ${p o l y, M o e b i u s t r a n s f o r m a t i o n}$ to be processed: ${{x^{3} + 6 x^{2} + 5 x + 1, x + 2}}$ . Remove the first and process it.

$V A S (x^{3} + 6 x^{2} + 5 x + 1, x + 2)$

1 var ← 0 // the number of sign variations in the sequence of coefficients of $p (x) = x^{3} + 6 x^{2} + 5 x + 1$

2 RETURN ∅

List of isolation intervals: ${(1, \frac{3}{2}), (\frac{3}{2}, 2)}$ . List of pairs ${p o l y, M o e b i u s t r a n s f o r m a t i o n}$ to be processed: ∅. Finished.

Therefore, the two positive roots of the polynomial $p (x) = x^{3} - 7 x + 7$ lie inside the isolation intervals $(1, \frac{3}{2})$ and $(\frac{3}{2}, 2)$ . Each root can be approximated by (for example) bisecting the isolation interval — inside which it lies — until the difference of the endpoints is smaller than $1 0^{- 6}$ ; following this approach, the roots turn out to be $ρ_{1} = 1.3569$ and $ρ_{2} = 1.69202$ .

Bisection methods

There are various bisection methods derived from Vincent's theorem; they are all presented and compared elsewhere.^[19] Here the two most important of them are described, namely, the Vincent-Collins-Akritas (VCA) method and the Vincent-Alesina-Galuzzi (VAG) method.

The Vincent-Alesina-Galuzzi (VAG) method is the simplest of all methods derived from Vincent's theorem but has the most time consuming test (in line 1) to determine if a polynomial has roots in the interval of interest; this makes it the slowest of the methods presented in this article.

By contrast, the Vincent-Collins-Akritas (VCA) method is more complex but uses a simpler test (in line 1) than VAG. This along with certain improvements^[16] have made VCA the fastest bisection method.

Vincent–Collins–Akritas (VCA, 1976)

This was the first method developed to overcome the exponential nature of Vincent's original approach, and has had quite an interesting history as far as its name is concerned. This method, which isolates the real roots, using Descartes' rule of signs and Vincent's theorem, had been originally called modified Uspensky's algorithm by its inventors Collins and Akritas.^[11] After going through names like "Collins-Akritas method" and "Descartes' method" (too confusing if ones considers Fourier's article^[20]), it was finally François Boulier, of Lille University, who gave it the name Vincent-Collins-Akritas (VCA) method,^[14] p. 24, based on the fact that "Uspensky's method" does not exist^[21] and neither does "Descartes' method".^[22] The best implementation of this method is due to Rouillier and Zimmerman,^[16] and to this date, it is the fastest bisection method. It has the same worst case complexity as Sturm's algorithm, but is almost always much faster. It has been implemented in Maple's RootFinding package.

Here is how VCA(p, (a, b)) works:

Given a polynomial $p_{o r i g} (x)$ of degree deg(p), such that $p_{o r i g} (0) \neq 0$ , whose positive roots need to be isolated, first compute an upper bound,^[12]^[13] ub on the values of these positive roots and set $p (x) = p_{o r i g} (u b * x)$ and (a, b) = (0, ub). The positive roots of p(x) all lie in the interval (0, 1) and there is a bijection between them and the roots of $p_{o r i g} (x)$ , which all lie in the interval (a, b) = (0, ub) (see the corresponding figure); this bijection is expressed by $α_{(a, b)} = a + α_{(0, 1)} (b - a)$ . Likewise, there is a bijection between the intervals (0, 1) and (0, ub).

Bijection between the roots of $p_{o r i g} (x)$ and p(x).

Repeat the following steps while there are pairs {p(x), (a, b)} to be processed.
Use Budan's "0_1 roots test" on p(x) to compute (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (0, 1). If there are no roots return the empty set, ∅ and if there is one root return the interval (a, b).
If there are two or more sign variations Budan's "0_1 roots test" implies that there may be two or more real roots inside the interval (0, 1). In this case cut it in half and consider separately the roots of p(x) which lie inside the interval $(0, \frac{1}{2})$ — and which correspond to the roots of $p_{o r i g} (x)$ inside the interval $(a, \frac{1}{2} (a + b))$ — from those which lie inside the interval $(\frac{1}{2}, 1)$ — and which correspond to the roots of $p_{o r i g} (x)$ inside the interval $(\frac{1}{2} (a + b), b)$ ; that is, process, respectively, the pairs

{2^{\deg (p)} p (\frac{x}{2}), (a, \frac{1}{2} (a + b))}, {2^{\deg (p)} p (\frac{1}{2} (x + 1)), (\frac{1}{2} (a + b), b)}

(see the corresponding figure).It may well turn out that $\frac{1}{2}$ is a root of p(x), in which case \tfrac{1}{2}(a+b) is a root of $p_{o r i g} (x)$ and the isolation interval reduces to a point.

Bijections between the roots of p(x) and those of $p (\frac{x}{2})$ and $p (\frac{x + 1}{2})$ .

Below is a recursive presentation of the original algorithm VCA(p, (a, b)).

VCA(p, (a, b))

Input: A univariate, square-free polynomial $p (u b * x) \in ℤ [x], p (0) \neq 0$ of degree deg(p), and the open interval (a, b) = (0, ub), where ub is an upper bound on the values of the positive roots of p(x). (The positive roots of $p (u b * x)$ are all in the open interval (0, 1)).
Output: A list of isolating intervals of the positive roots of p(x)

1 var ← the number of sign variations of $(x + 1)^{\deg (p)} p (\frac{1}{x + 1})$ // Budan's "0_1 roots test";
2 if var = 0 then RETURN ∅;
3 if var = 1 then RETURN {(a, b)};
4 $p_{0 \frac{1}{2}} \leftarrow 2^{\deg (p)} p (\frac{x}{2})$ // Look for real roots in $(0, \frac{1}{2})$ ;
5 $m \leftarrow \frac{1}{2} (a + b)$ // Is $\frac{1}{2}$ a root? ;
6 $p_{\frac{1}{2} 1} \leftarrow 2^{\deg (p)} p (\frac{x + 1}{2})$ // Look for real roots in $(\frac{1}{2}, 1)$ ;
7 if $p (\frac{1}{2}) \neq 0$ then
8 RETURN $V C A (p_{0 \frac{1}{2}}, (a, m)) \cup V C A (p_{\frac{1}{2} 1}, (m, b))$
9 else
10 RETURN $V C A (p_{0 \frac{1}{2}}, (a, m)) \cup {[m, m]} \cup V C A (p_{\frac{1}{2} 1}, (m, b))$
11 end

Remark

In the above algorithm with each polynomial there is associated an interval (a, b). As shown elsewhere,^[22] p. 11, a Möbius transformation can also be associated with each polynomial in which case VCA looks more like VAS.
In line 1 Budan's "0_1 roots test" is applied.

Example of VCA(p,(a,b))

Given the polynomial $p_{o r i g} (x) = x^{3} - 7 x + 7$ and considering as an upper bound^[12]^[13] on the values of the positive roots ub = 4 the arguments of the VCA method are: $p (x) = 64 x^{3} - 28 x + 7$ and (a, b) = (0, 4).

1 var ← 2 // the number of sign variations in the sequence of coefficients of $(x + 1)^{3} p (\frac{1}{x + 1}) = 7 x^{3} - 7 x^{2} - 35 x + 43$

4 $p_{0 \frac{1}{2}} \leftarrow 64 x^{3} - 112 x + 56$

5 m ← 2

6 $p_{\frac{1}{2} 1} \leftarrow 64 x^{3} + 192 x^{2} + 80 x + 8$

7 $p (\frac{1}{2}) = 1$

8 RETURN $V C A (64 x^{3} - 112 x + 56, (0, 2)) \cup V C A (64 x^{3} + 192 x^{2} + 80 x + 8, (2, 4))$

List of isolation intervals: {}. List of pairs ${p o l y, i n t e r v a l}$ to be processed: ${{64 x^{3} - 112 x + 56, (0, 2)}, {64 x^{3} + 192 x^{2} + 80 x + 8, (2, 4)}}$ . Remove the first and process it.

$V C A (64 x^{3} - 112 x + 56, (0, 2))$

1 var ← 2 // the number of sign variations in the sequence of coefficients of $(x + 1)^{3} p (\frac{1}{x + 1}) = 56 x^{3} + 56 x^{2} - 56 x + 8$

4 $p_{0 \frac{1}{2}} \leftarrow 64 x^{3} - 448 x + 448$

5 m ← 1

6 $p_{\frac{1}{2} 1} \leftarrow 64 x^{3} + 192 x^{2} - 256 x + 64$

7 $p (\frac{1}{2}) = 8$

8 RETURN $V C A (64 x^{3} - 448 x + 448, (0, 1)) \cup V C A (64 x^{3} + 192 x^{2} - 256 x + 64, (1, 2))$

List of isolation intervals: {}. List of pairs ${p o l y, i n t e r v a l}$ to be processed: ${{64 x^{3} - 448 x + 448, (0, 1)}, {64 x^{3} + 192 x^{2} - 256 x + 64, (1, 2)}, {64 x^{3} + 192 x^{2} + 80 x + 8, (2, 4)}}$ . Remove the first and process it.

$V C A (64 x^{3} - 448 x + 448, (0, 1))$

1 var ← 0 // the number of sign variations in the sequence of coefficients of $(x + 1)^{3} p (\frac{1}{x + 1}) = 448 x^{3} + 896 x^{2} + 448 x + 64$

2 RETURN ∅

List of isolation intervals: {}. List of pairs ${p o l y, i n t e r v a l}$ to be processed: ${{64 x^{3} + 192 x^{2} - 256 x + 64, (1, 2)}, {64 x^{3} + 192 x^{2} + 80 x + 8, (2, 4)}}$ . Remove the first and process it.

$V C A (64 x^{3} + 192 x^{2} - 256 x + 64, (1, 2))$

1 var ← 2 // the number of sign variations in the sequence of coefficients of $(x + 1)^{3} p (\frac{1}{x + 1}) = 64 x^{3} - 64 x^{2} - 128 x + 64$

4 $p_{0 \frac{1}{2}} \leftarrow 64 x^{3} + 384 x^{2} - 1024 x + 512$

5 $m \leftarrow \frac{3}{2}$

6 $p_{\frac{1}{2} 1} \leftarrow 64 x^{3} + 576 x^{2} - 64 x - 64$

7 $p (\frac{1}{2}) = - 8$

8 RETURN $V C A (64 x^{3} + 384 x^{2} - 1024 x + 512, (1, \frac{3}{2})) \cup V C A (64 x^{3} + 576 x^{2} - 64 x - 64, (\frac{3}{2}, 2))$

List of isolation intervals: {}. List of pairs ${p o l y, i n t e r v a l}$ to be processed: ${{64 x^{3} + 384 x^{2} - 1024 x + 512, (1, \frac{3}{2})}, {64 x^{3} + 576 x^{2} - 64 x - 64, (\frac{3}{2}, 2)}, {64 x^{3} + 192 x^{2} + 80 x + 8, (2, 4)}}$ . Remove the first and process it.

$V C A (64 x^{3} + 384 x^{2} - 1024 x + 512, (1, \frac{3}{2}))$

1 var ← 1 // the number of sign variations in the sequence of coefficients of $(x + 1)^{3} p (\frac{1}{x + 1}) = 512 x^{3} + 512 x^{2} - 128 x - 64$

3 RETURN ${(1, \frac{3}{2})}$

List of isolation intervals: ${(1, \frac{3}{2})}$ . List of pairs ${p o l y, i n t e r v a l}$ to be processed: ${{64 x^{3} + 576 x^{2} - 64 x - 64, (\frac{3}{2}, 2)}, {64 x^{3} + 192 x^{2} + 80 x + 8, (2, 4)}}$ . Remove the first and process it.

$V C A (64 x^{3} + 576 x^{2} - 64 x - 64, (\frac{3}{2}, 2))$

1 var ← 1 // the number of sign variations in the sequence of coefficients of $(x + 1)^{3} p (\frac{1}{x + 1}) = - 64 x^{3} - 256 x^{2} + 256 x + 512$

3 RETURN ${(\frac{3}{2}, 2)}$

List of isolation intervals: ${(1, \frac{3}{2}), (\frac{3}{2}, 2)}$ . List of pairs ${p o l y, i n t e r v a l}$ to be processed: ${{64 x^{3} + 192 x^{2} + 80 x + 8, (2, 4)}}$ . Remove the first and process it.

$V C A (64 x^{3} + 192 x^{2} + 80 x + 8, (2, 4))$

1 var ← 0 // the number of sign variations in the sequence of coefficients of $(x + 1)^{3} p (\frac{1}{x + 1}) = 8 x^{3} + 104 x^{2} + 376 x + 344$

2 RETURN ∅

List of isolation intervals: ${(1, \frac{3}{2}), (\frac{3}{2}, 2)}$ . List of pairs ${p o l y, i n t e r v a l}$ to be processed: ∅. Finished.

Therefore, the two positive roots of the polynomial $p (x) = x^{3} - 7 x + 7$ lie inside the isolation intervals $(1, \frac{3}{2})$ and $(\frac{3}{2}, 2)$ . Each root can be approximated by (for example) bisecting the isolation interval — inside which it lies — until the difference of the endpoints is smaller than $1 0^{- 6}$ ; following this approach, the roots turn out to be $ρ_{1} = 1.3569$ and $ρ_{2} = 1.69202$ .

Vincent–Alesina–Galuzzi (VAG, 2000)

This was developed last and is the simplest real root isolation method derived from Vincent's theorem.

Here is how VAG(p, (a, b)) works:

Given a polynomial p(x) of degree deg(p), such that p(0) ≠ 0, whose positive roots need to be isolated, first compute an upper bound,^[12]^[13] ub on the values of these positive roots and set (a, b) = (0, ub). The positive roots of p(x) all lie in the interval (a, b).
Repeat the following steps while there are intervals (a, b) to be processed; in this case the polynomial p(x) stays the same.
Use the Alesina-Galuzzi "a_b roots test" on p(x) to compute (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (a, b). If there are no roots return the empty set, ∅ and if there is one root return the interval (a, b).
If there are two or more sign variations the Alesina-Galuzzi "a_b roots test" implies that there may be two or more real roots inside the interval (a, b). In this case cut it in half and consider separately the roots of p(x) which lie inside the interval $(a, \frac{1}{2} (a + b))$ from those which lie inside the interval $(\frac{1}{2} (a + b), b)$ ; that is, process, respectively, the intervals $(a, \frac{1}{2} (a + b))$ and $(\frac{1}{2} (a + b), b)$ . It may well turn out that $\frac{1}{2} (a + b)$ is a root of p(x), in which case the isolation interval reduces to a point.

Below is a recursive presentation of VAG(p, (a, b)).

VAG(p, (a, b))

Input: A univariate, square-free polynomial $p (x) \in ℤ [x], p (0) \neq 0$ of degree deg(p) and the open interval (a, b) = (0, ub), where ub is an upper bound on the values of the positive roots of p(x).
Output: A list of isolating intervals of the positive roots of p(x).

1 var ← the number of sign variations of $(x + 1)^{\deg (p)} p (\frac{a + b x}{1 + x})$ // The Alesina-Galuzzi "a_b roots test";
2 if var = 0 then RETURN ∅;
3 if var = 1 then RETURN {(a, b)};
4 $m \leftarrow \frac{1}{2} (a + b)$ // Subdivide the interval (a,b) in two equal parts;
5 if p(m) ≠ 0 then
6 RETURN $V A G (p, (a, m)) \cup V A G (p, (m, b))$
7 else
8 RETURN $V A G (p, (a, m)) \cup {[m, m]} \cup V A G (p, (m, b))$
9 end

Remarks

Compared to VCA the above algorithm is extremely simple; by contrast, VAG uses the time consuming "a_b roots test" and that makes it much slower than VCA.^[19]
As Alesina and Galuzzi point out,^[4] p. 189, there is a variant of this algorithm due to Donato Saeli. Saeli suggested that the mediant of the endpoints be used instead of their midpoint $\frac{1}{2} (a + b)$ . However, it has been shown^[19] that using the mediant of the endpoints is in general much slower than the "mid-point" version.

Example of VAG(p,(a,b))

Given the polynomial $p (x) = x^{3} - 7 x + 7$ and considering as an upper bound^[12]^[13] on the values of the positive roots ub = 4 the arguments of VAG are: $p (x) = x^{3} - 7 x + 7$ and (a, b) = (0, 4).

1 var ← 2 // the number of sign variations in the sequence of coefficients of $(1 + x)^{3} p (\frac{4 x}{1 + x}) = 43 x^{3} - 35 x^{2} - 7 x + 7$

4 $m \leftarrow \frac{a + b}{2} = \frac{0 + 4}{2} = 2$

5 p(m) = 1

8 RETURN $V A G (x^{3} - 7 x + 7, (0, 2)) \cup V A G (x^{3} - 7 x + 7, (2, 4)$

List of isolation intervals: {}. List of intervals to be processed: {(0, 2), (2, 4)}. Remove the first and process it.

$V A G (x^{3} - 7 x + 7, (0, 2))$

1 var ← 2 // the number of sign variations in the sequence of coefficients of $(1 + x)^{3} p (\frac{2 x}{1 + x}) = x^{3} - 7 x^{2} + 7 x + 7$

4 $m \leftarrow \frac{a + b}{2} = \frac{0 + 2}{2} = 1$

5 p(m) = 1

8 RETURN $V A G (x^{3} - 7 x + 7, (0, 1)) \cup V A G (x^{3} - 7 x + 7, (1, 2)$

List of isolation intervals: {}. List of intervals to be processed: ${(0, 1), (1, 2), (2, 4)}$ . Remove the first and process it.

$V A G (x^{3} - 7 x + 7, (0, 1))$

1 var ← 0 // the number of sign variations in the sequence of coefficients of $(1 + x)^{3} p (\frac{x}{1 + x}) = x^{3} + 7 x^{2} + 14 x + 7$

2 RETURN ∅

List of isolation intervals: {}. List of intervals to be processed: ${(1, 2), (2, 4)}$ . Remove the first and process it.

$V A G (x^{3} - 7 x + 7, (1, 2))$

1 var ← 2 // the number of sign variations in the sequence of coefficients of $(1 + x)^{3} p (\frac{1 + 2 x}{1 + x}) = x^{3} - 2 x^{2} - x + 1$

4 $m \leftarrow \frac{a + b}{2} = \frac{1 + 2}{2} = \frac{3}{2}$

5 $p (m) = - \frac{1}{8}$

8 RETURN $V A G (x^{3} - 7 x + 7, (1, \frac{3}{2})) \cup V A G (x^{3} - 7 x + 7, (\frac{3}{2}, 2)$

List of isolation intervals: {}. List of intervals to be processed: ${(1, \frac{3}{2}), (\frac{3}{2}, 2), (2, 4)}$ . Remove the first and process it.

$V A G (x^{3} - 7 x + 7, (1, \frac{3}{2}))$

1 var ← 1 // the number of sign variations in the sequence of coefficients of $2^{3} (1 + x)^{3} p (\frac{1 + \frac{3}{2} x}{1 + x}) = x^{3} + 2 x^{2} - 8 x - 8$

3 RETURN ${(1, \frac{3}{2})}$

List of isolation intervals: ${(1, \frac{3}{2})}$ . List of intervals to be processed: ${(\frac{3}{2}, 2), (2, 4)}$ . Remove the first and process it.

$V A G (x^{3} - 7 x + 7, (\frac{3}{2}, 2))$

1 var ← 1 // the number of sign variations in the sequence of coefficients of $2^{3} (1 + x)^{3} p (\frac{\frac{3}{2} + 2 x}{1 + x}) = 8 x^{3} + 4 x^{2} - 4 x - 1$

3 RETURN ${(\frac{3}{2}, 2)}$

List of isolation intervals: ${(1, \frac{3}{2}), (\frac{3}{2}, 2)}$ . List of intervals to be processed: {(2, 4)}. Remove the first and process it.

$V A G (x^{3} - 7 x + 7, (2, 4))$

1 var ← 0 // the number of sign variations in the sequence of coefficients of $(1 + x)^{3} p (\frac{2 + 4 x}{1 + x}) = 344 x^{3} + 376 x^{2} + 104 x + 8$

2 RETURN ∅

List of isolation intervals: ${(1, \frac{3}{2}), (\frac{3}{2}, 2)}$ . List of intervals to be processed: ∅. Finished.

Therefore, the two positive roots of the polynomial $p (x) = x^{3} - 7 x + 7$ lie inside the isolation intervals $(1, \frac{3}{2})$ and $(\frac{3}{2}, 2)$ . Each root can be approximated by (for example) bisecting the isolation interval — inside which it lies — until the difference of the endpoints is smaller than $1 0^{- 6}$ ; following this approach, the roots turn out to be $ρ_{1} = 1.3569$ and $ρ_{2} = 1.69202$ .

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Berkakis, Antonis: RealRoots, a free App for Android devices to compare Sturm's method and VAS
https://play.google.com/store/apps/details?id=org.kde.necessitas.berkakis.realroots

Encyclopedia of Mathematics http://www.encyclopediaofmath.org/index.php

Kehagias, Spyros: RealRoots, a free App for iPhone, iPod Touch and iPad to compare Sturm's method and VAS http://itunes.apple.com/gr/app/realroots/id483609988?mt=8

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

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Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^3.0 ^3.1 ^3.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^4.0 ^4.1 ^4.2 ^4.3 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^5.0 ^5.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^7.0 ^7.1 ^7.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ ^8.0 ^8.1 ^8.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ ^9.0 ^9.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ ^11.0 ^11.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ ^12.0 ^12.1 ^12.2 ^12.3 ^12.4 ^12.5 ^12.6 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ ^13.0 ^13.1 ^13.2 ^13.3 ^13.4 ^13.5 ^13.6 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^14.0 ^14.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ ^15.0 ^15.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ ^16.0 ^16.1 ^16.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ ^19.0 ^19.1 ^19.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ ^22.0 ^22.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[paper_1834-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 ^1.5 ^1.6 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[paper_1836-2] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 ^2.5 ^2.6 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[AG_1998-3] 3.0 ^3.1 ^3.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[AG_2000-4] 4.0 ^4.1 ^4.2 ^4.3 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[paper_1838-5] 5.0 ^5.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[O_1950-6] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[Obr_1920-7] 7.0 ^7.1 ^7.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[Uspensky-8] 8.0 ^8.1 ^8.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[VAS-9] 9.0 ^9.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[Serret-10] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[CA-11] 11.0 ^11.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[Panos-12] 12.0 ^12.1 ^12.2 ^12.3 ^12.4 ^12.5 ^12.6 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[bounds-13] 13.0 ^13.1 ^13.2 ^13.3 ^13.4 ^13.5 ^13.6 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[FB-14] 14.0 ^14.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[AS-15] 15.0 ^15.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[RZ-16] 16.0 ^16.1 ^16.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[TE-17] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[VS-18] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[ASV_2008-19] 19.0 ^19.1 ^19.2 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[Fourier-20] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[akritas-21] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[noDec-22] 22.0 ^22.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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Carlitz exponential: Difference between revisions

Latest revision as of 01:17, 30 July 2013

Contents

Sign variation

Vincent's theorem: Continued fractions version (1834 and 1836)

Vincent's theorem: Bisection version (Alesina and Galuzzi 2000)

The Alesina-Galuzzi "a_b roots test"

Sketch of a Proof

Historical background

Early applications of Vincent's theorem

Disappearance of Vincent's theorem

Comeback of Vincent's theorem

Real root isolation methods derived from Vincent's theorem

Continued fractions method

Vincent–Akritas–Strzeboński (VAS, 2005)

Example of VAS(p, M)

Bisection methods

Vincent–Collins–Akritas (VCA, 1976)

Example of VCA(p,(a,b))

Vincent–Alesina–Galuzzi (VAG, 2000)

Example of VAG(p,(a,b))

See also

References

External links

Navigation menu

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+In mathematics, '''Vincent's theorem''', named after Alexandre Joseph Hidulph Vincent, is a little-known theorem that was (almost) totally forgotten, having been overshadowed by [[Sturm's theorem]]. Even though Vincent's theorem is of great interest because it can be used to isolate the real roots of polynomials with rational coefficients, it cannot be found in any of the classical books on ''Theory of Equations'' (of the 20th century), except for [[J. V. Uspensky|Uspensky]]'s book. Two variants of this theorem are presented along with several (continued fractions as well as bisection) real root isolation methods that are derived from it.
+==Sign variation==
+:Let ''c''<sub>0</sub>, ''c''<sub>1</sub>, ''c''<sub>2</sub>, ... be a finite or infinite sequence of real numbers. Suppose ''l'' < ''r'' and the following conditions hold:
+# If ''r'' = ''l''+1 the numbers ''c<sub>l</sub>'' and ''c<sub>r</sub>'' have opposite signs.
+# If ''r'' ≥ ''l''+2 the numbers ''c<sub>l+1</sub>'', ..., ''c<sub>r−1</sub>'' are all zero and the numbers ''c<sub>l</sub>'' and ''c<sub>r</sub>'' have opposite signs.
+: This is called a ''sign variation'' or ''sign change'' between the numbers ''c<sub>l</sub>'' and ''c<sub>r</sub>''.
+: When dealing with the polynomial ''p''(''x'') in one variable, one defines the number of '''sign variations of ''p''(''x'')''' as the number of sign variations in the sequence of its coefficients.
+Two versions of this theorem are presented: the ''[[continued fractions]]'' version due to Vincent,<ref name=paper_1834>{{cite journal|last=Vincent|first=Alexandre Joseph Hidulph|title=Mémoire sur la résolution des équations numériques|url=http://gallica.bnf.fr/ark:/12148/bpt6k57787134/f4.image.r=Agence%20Rol.langEN|journal=Mémoires de la Société Royale des Sciences, de l' Agriculture et des Arts, de Lille|year=1834|pages=1–34}}</ref><ref name=paper_1836>{{cite journal|last=Vincent|first=Alexandre Joseph Hidulph|title=Sur la résolution des équations numériques|url=http://www-mathdoc.ujf-grenoble.fr/JMPA/PDF/JMPA_1836_1_1_A28_0.pdf|journal=Journal de Mathématiques Pures et Appliquées|volume=1|year=1836|pages=341–372}}</ref> and the ''[[Bisection method|bisection]]'' version due to Alesina and Galuzzi.<ref name=AG_1998>{{cite journal|last=Alesina|first=Alberto|coauthor=Massimo Galuzzi|title=A new proof of Vincent's theorem|url=http://retro.seals.ch/cntmng?type=pdf&rid=ensmat-001:1998:44::149&subp=hires|journal=L'Enseignement Mathématique|year=1998|volume=44|number=3-4|pages=219–256}}</ref><ref name=AG_2000>{{cite journal|last=Alesina|first=Alberto|coauthor=Massimo Galuzzi|title=Vincent's Theorem from a Modern Point of View|url=http://inf-server.inf.uth.gr/~akritas/Alessina_Galuzzi_b.pdf|journal=Categorical Studies in Italy 2000, Rendiconti del Circolo Matematico di Palermo, Serie II, n. 64|year=2000|pages=179–191}}</ref>
+This statement of the ''[[continued fractions]]'' version can be found also in the Wikipedia article [[Budan's theorem#Vincent's theorem (1834 and 1836)|Budan's theorem]].
+===Vincent's theorem: Continued fractions version (1834 and 1836)===
+If in a polynomial equation with rational coefficients and without multiple roots, one makes successive transformations of the form
+: <math>x = a_1 + \frac{1}{x'},\quad x' = a_2 + \frac{1}{x''},\quad x'' = a_3 + \frac{1}{x'''}, \ldots</math>
+where <math>a_1, a_2, a_3,\ldots </math> are any positive numbers greater than or equal to one, then after a number of such transformations, the resulting transformed equation either has zero [[Vincent's theorem#Sign variation|sign variation]]s or it has a single [[Vincent's theorem#Sign variation|sign variation]]. In the first case there is no root, whereas in the second case there is a single positive real root. Furthermore, the corresponding root of the proposed equation is approximated by the finite continued fraction:<ref name="paper_1834"/><ref name="paper_1836"/><ref name=paper_1838>{{cite journal|last=Vincent|first=Alexandre Joseph Hidulph|title=Addition à une précédente note relative à la résolution des équations numériques|url=http://math-doc.ujf-grenoble.fr/JMPA/PDF/JMPA_1838_1_3_A19_0.pdf|journal=Journal de Mathématiques Pures et Appliquées|volume=3|year=1838|pages=235–243}}</ref>
+: <math>a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots}}} </math>
+Moreover, if infinitely many numbers <math>a_1, a_2, a_3,\ldots </math> satisfying this property can be found, then the root is represented by the (infinite) corresponding continued fraction.
+The above statement is an exact translation of the theorem found in Vincent's original papers;<ref name="paper_1834"/><ref name="paper_1836"/><ref name="paper_1838"/> however, the following remarks are needed for a clearer understanding:
+*If <math>f_n(x)</math> denotes the polynomial obtained after ''n'' substitutions (and after removing the denominator), then there exists ''N'' such that for all <math>n\ge N</math> either <math>f_n(x)</math> has no sign variation or it has one sign variation. In the latter case <math>f_n(x)</math> has a single positive real root for all <math>n\ge N</math>.
+* The continued fraction represents a positive root of the original equation, and the original equation may have more than one positive root. Moreover, assuming <math>a_1 \ge 1</math>, we can only obtain a root of the original equation which is > 1. To obtain an arbitrary positive root we need to assume that <math>a_1 \ge 0</math>.
+* Negative roots are obtained by replacing ''x'' by −''x'', in which case the negative roots become positive.
+===Vincent's theorem: Bisection version (Alesina and Galuzzi 2000)===
+Let ''p''(''x'') be a real polynomial of degree deg(''p'') which has only simple roots. It is possible to determine a positive quantity δ so that for every pair of positive real numbers ''a'', ''b'' with <math>|b-a| < \delta</math>, every transformed polynomial of the form
+{{NumBlk|:|<math>f(x) = (1+x)^{\deg(p)}p \left (\frac{a+bx}{1+x} \right )</math>|{{EquationRef|1}}}}
+has exactly 0 or 1 [[Vincent's theorem#Sign variation|sign variation]]s. The second case is possible if and only if  ''p''(''x'') has a single root within (''a'', ''b'').
+====The Alesina-Galuzzi "a_b roots test"====
+From equation ({{EquationNote|1}}) the following criterion is obtained for determining whether a polynomial has any roots in the interval (''a'', ''b''):
+Perform on ''p''(''x'') the substitution
+:<math>x \leftarrow \frac{a+bx}{1+x} </math>
+and count the number of [[Vincent's theorem#Sign variation|sign variations]] in the sequence of coefficients of the transformed polynomial; this number gives an ''upper bound'' on the number of real roots ''p''(''x'') has inside the open interval (''a'', ''b''). More precisely, the number ρ<sub>''ab''</sub>(''p'') of real roots in the open interval (''a'', ''b'') — multiplicities counted — of the polynomial ''p''(''x'') in '''R'''[''x''], of degree deg(''p''), is bounded above by the number of [[Vincent's theorem#Sign variation|sign variations]] ''var''<sub>''ab''</sub>(''p''), where
+:<math>var_{ab}(p) = var \left ((1+x)^{\deg(p)}p\left (\frac{a+bx}{1+x} \right ) \right ),</math>
+:<math>var_{ab}(p) = var_{ba}(p) \ge \rho_{ab}(p).</math>
+As in the case of [[Descartes' rule of signs]] if ''var''<sub>''ab''</sub>(''p'') = 0 it follows that ρ<sub>''ab''</sub>(''p'') = 0 and if ''var''<sub>''ab''</sub>(''p'') = 1 it follows that ρ<sub>''ab''</sub>(''p'') = 1.
+A special case of the Alesina-Galuzzi '''"a_b roots test"''' is [[Budan's theorem#Early applications of Budan's theorem|Budan's '''"0_1 roots test"''']].
+===Sketch of a Proof===
+A detailed discussion of Vincent's theorem, its extension, the geometrical interpretation of the transformations involved and three different proofs can be found in the work by Alesina and Galuzzi.<ref name="AG_1998"/><ref name="AG_2000"/> A fourth proof is due to [[Alexander Ostrowski|Ostrowski]]<ref name=O_1950>{{cite journal |last=Ostrowski|first=A. M.|title=Note on Vincent's theorem|url=http://www.jstor.org/stable/10.2307/1969443|journal=Annals of Mathematics | series = Second Series |year=1950 |volume=52 |number=3|pages=702–707}}</ref> who rediscovered a special case of a theorem stated by [[Nikola Obreshkov|Obreschkoff]],<ref name=Obr_1920>{{cite book |last=Obreschkoff |first=Nikola|title=Verteilung und Berechnung der Nullstellen reeller Polynome|publisher=VEB Deutscher Verlag der Wissenschaften |year=1963 |location=Berlin}}</ref> p.&nbsp;81, back in 1920-1923.
+To prove (both versions of) Vincent's theorem Alesina and Galuzzi show that after a series of transformations mentioned in the theorem, a polynomial with one positive root will eventually have one sign variation. To show this they use the following corollary to the theorem by [[Nikola Obreshkov|Obreschkoff]] of 1920-1923 mentioned earlier; that is, the following corollary gives the necessary conditions under which a polynomial with one positive root has exactly one sign variation in the sequence of its coefficients; see also the corresponding figure.
+<blockquote>'''Corollary''' (to [[Nikola Obreshkov|Obreschkoff]]'s cone or sector theorem, 1920-1923<ref name="Obr_1920"/> p.&nbsp;81): If a real polynomial has one simple root ''x''<sub>0</sub>, and all other (possibly multiple) roots lie in the sector
+:<math>S_{\sqrt{3}}= \{x = -\alpha+i\beta \ \  |\ \   \alpha>0  \ \text{   and   }\   |\beta| \le \sqrt{3}\,|\alpha|\}</math>
+then the sequence of its coefficients has exactly one sign variation.</blockquote>
+[[File:Sketch of proof.jpg|thumb|x220px|center|[[Nikola Obreshkov|Obreschkoff]]'s <math>S_{\sqrt{3}}</math> sector and his famous eight-shaped figure (of circles).]]
+Consider now the [[Möbius transformation]]
+:<math>M(x)=\frac{ax+b}{cx+d}, \qquad a,b,c,d \in \mathbb{Z}_{>0}</math>
+and the three circles shown in the corresponding figure; assume that <math>\frac{a}{c} < \frac{b}{d}</math>.
+*The (yellow) circle
+::<math>\left |x-\tfrac{1}{2}\left(\tfrac{a}{c} + \tfrac{b}{d} \right ) \right |=\tfrac{1}{2}\left (\tfrac{b}{d} - \tfrac{a}{c} \right )</math>
+:whose diameter lies on the real axis, with endpoints <math>\frac{a}{c}</math> and <math> \frac{b}{d}</math>, is mapped by the inverse Möbius transformation
+::<math>M^{-1}(x)=\frac{dx-b}{-cx+a}</math>
+:onto the imaginary axis. For example the point
+::<math>\tfrac{1}{2}\left(\tfrac{a}{c} + \tfrac{b}{d} \right )+\tfrac{i}{2}\left (\tfrac{b}{d} - \tfrac{a}{c} \right )</math>
+:gets mapped onto the point <math>-i\frac{d}{c}</math>. The exterior points get mapped onto the half-plane with Re(''x'') < 0.
+*The two circles (only their blue crescents are visible) with center
+::<math>\tfrac{1}{2}\left(\tfrac{a}{c} + \tfrac{b}{d} \right ) \pm \tfrac{i}{2\sqrt{3}} \left (\tfrac{b}{d} - \tfrac{a}{c} \right )</math>
+:and radius <math>\frac{(b/d-a/c)}{\sqrt{3}}</math> are mapped by the inverse Möbius transformation
+::<math>M^{-1}(x)=\frac{dx-b}{-cx+a}</math>
+:onto the lines <math>Im(x) = \pm \sqrt{3}Re(x)</math>. For example the point
+::<math>\tfrac{1}{2}\left(\tfrac{a}{c} + \tfrac{b}{d} \right ) -\tfrac{3 i}{2\sqrt{3}} \left (\tfrac{b}{d} - \tfrac{a}{c} \right )</math>
+:gets mapped to the point
+::<math>\frac{-d}{2c}(1-i\sqrt{3}).</math>
+:The exterior points (those outside the eight-shaped figure) get mapped onto the <math>S_{\sqrt{3}}</math> sector.
+From the above it becomes obvious that if a polynomial has a single positive root inside the eight-shaped figure and all other roots are outside of it, it will present one sign variation in the sequence of its coefficients. This also guarantees the termination of the process.
+==Historical background==
+===Early applications of Vincent's theorem===
+Vincent presented in both of his papers<ref name="paper_1834"/><ref name="paper_1836"/> several examples showing precisely how his theorem is to be used in order to [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|isolate the real roots]] of polynomials with [[continued fractions]]. However the resulting method had [[Exponential time#Exponential time|exponential]] computing time, a fact that must have been realized then, as was also realized by [[J. V. Uspensky|Uspensky]]<ref name=Uspensky>{{cite book|last=Uspensky|first=James Victor|title=Theory of Equations|year=1948|publisher=McGraw–Hill Book Company|location=New York|url=http://www.google.com/search?q=uspensky+theory+of+equations&btnG=Search+Books&tbm=bks&tbo=1}}</ref> p.&nbsp;136, a century later.
+[[File:Vincent method.jpg|thumb|Vincent's search for a root (applying Budan's theorem)|right|450px]]
+The [[Exponential time#Exponential time|exponential]] nature of Vincent's algorithm is due to the way the partial [[Continued fractions#Notations for continued fractions|quotient]]s ''a<sub>i</sub>'' (in [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]]) are computed. That is, to compute each partial [[Continued fractions#Notations for continued fractions|quotient]] ''a<sub>i</sub>'' (that is,  to locate where the roots lie on the ''x''-axis) Vincent uses [[Budan's theorem]] as a [[Budan's theorem#Early applications of Budan's theorem|"no roots test"]]; in other words, to find the integer part of a root Vincent performs successive substitutions of the form ''x'' ← ''x''+1 and stops only when the polynomials ''p''(''x'') and ''p''(''x''+1) differ in the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of their coefficients (i.e. when the number of [[Vincent's theorem#Sign variation|sign variation]]s of ''p''(''x''+1) is decreased).<ref name="paper_1834"/><ref name="paper_1836"/>
+See the corresponding diagram where the root lies in the interval (5, 6). It can be easily inferred that, if the root is far away from the origin, it will take a lot of time to find its integer part this way; hence the [[Exponential time#Exponential time|exponential]] nature of Vincent's method. [[Vincent's theorem#Vincent–Akritas–Strzeboński (VAS, 2005)|Below]] there is an explanation of how this drawback is overcome.
+===Disappearance of Vincent's theorem===
+Vincent was the last author in the 19th century to use his theorem for the [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|isolation of the real roots]] of a polynomial.
+The reason for that was the appearance of [[Budan's theorem#Disappearance of Budan's theorem|Sturm's theorem]] in 1827 which solved the [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|real root isolation problem]] in [[Exponential time#Polynomial time|polynomial]] time, by defining the precise number of real roots a polynomial has in a  real open interval (''a'', ''b''). The resulting (Sturm's) method for computing the real roots of polynomials has been the only one widely known and used ever since – up to about 1980, when it was replaced (in almost all [[computer algebra system]]s) by [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|methods derived from Vincent's theorem]], the fastest one being the [[Vincent's theorem#Vincent–Akritas–Strzeboński (VAS, 2005)|Vincent–Akritas–Strzeboński]] (VAS) method.<ref name=VAS>{{cite journal|last=Akritas|first=Alkiviadis G.|coauthors=A.W. Strzeboński, P.S. Vigklas|title=Improving the performance of the continued fractions method using new bounds of positive roots|journal=Nonlinear Analysis: Modelling and Control|year=2008|volume=13|pages=265–279|url=http://www.lana.lt/journal/30/Akritas.pdf}}</ref>
+Serret included in his Algebra,<ref name=Serret>{{cite book|last=Serret|first=Joseph A.|title=Cours d'algèbre supérieure. Tome I|year=1877|publisher=Gauthier-Villars|url=http://archive.org/details/coursdalgbresu01serruoft}}</ref> pp 363–368, [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]] along with its proof and directed all interested readers to Vincent's papers for examples on how it is used. Serret was the last author to mention [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]] in the 19th century.
+===Comeback of Vincent's theorem===
+In the 20th century [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]] cannot be found in any of the theory of equations books; the only exceptions are the books by [[J. V. Uspensky|Uspensky]]<ref name="Uspensky"/> and [[Nikola Obreshkov|Obreschkoff]],<ref name="Obr_1920"/> where in the second there is just the statement of the theorem.
+It was in [[J. V. Uspensky|Uspensky]]'s book<ref name="Uspensky"/> that Akritas found [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]] and made it the topic of his Ph.D. Thesis "Vincent's Theorem in Algebraic Manipulation", [[North Carolina State University|North Carolina State University, USA]], 1978.  A major achievement at the time was getting hold of Vincent's original paper of 1836, something that had eluded [[J. V. Uspensky|Uspensky]] — resulting thus in a [[Budan's theorem#Uspensky's implementation of Vincent's theorem|great misunderstanding]].  Vincent's original paper of 1836 was made available to Akritas through the commendable efforts (interlibrary loan) of a librarian in the Library of the [[University of Wisconsin–Madison|University of Wisconsin–Madison, USA]].
+==Real root isolation methods derived from Vincent's theorem==
+'''Isolation of the real roots''' of a polynomial is the process of finding open disjoint intervals such that each contains exactly one real root and every real root is contained in some interval. According to the French school of mathematics of the 19th century, this is the first step in computing the real roots, the second being their '''approximation''' to any degree of accuracy; moreover, the focus is on the '''positive''' roots, because to isolate the '''negative''' roots of the polynomial ''p''(''x'') replace ''x'' by −''x'' (''x'' ← −''x'') and repeat the process.
+The [[continued fractions]] version of [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]] can be used to isolate the positive roots of a given polynomial ''p''(''x'') of degree deg(''p''). To see this, represent by the [[Möbius transformation]]
+:<math>M(x)=\frac{ax+b}{cx+d}, a,b,c,d \in \mathbb{N}</math>
+the continued fraction that leads to a transformed polynomial {{NumBlk|:|<math>f(x) = (cx+d)^{\deg(p)}p \left (\frac{ax+b}{cx+d} \right )</math>|{{EquationRef|2}}}} with one [[Vincent's theorem#Sign variation|sign variation]]  in the sequence of its coefficients. Then, the single positive root of ''f''(''x'') (in the interval (0, ∞)) corresponds to ''that'' positive root of ''p''(''x'') which is located in the open interval with endpoints <math>\frac{b}{d}</math> and <math>\frac{a}{c}</math>. These endpoints are ''not'' ordered and correspond to ''M''(0) and ''M''(∞) respectively.
+Therefore, to isolate the positive roots of a polynomial, all that has to be done is to compute — for ''each'' root — the variables <math>a, b, c, d</math> of the corresponding [[Möbius transformation]]
+:<math>M(x)=\frac{ax+b}{cx+d}</math>
+that leads to a transformed polynomial as in equation ({{EquationNote|2}}), with one [[Vincent's theorem#Sign variation|sign variation]] in the sequence of its coefficients.
+'''Crucial Observation:''' The variables <math>a, b, c, d</math> of a [[Möbius transformation]]
+:<math>M(x)=\frac{ax+b}{cx+d}</math>
+(in [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]]) leading to a transformed polynomial — as in equation ({{EquationNote|2}}) — with one [[Vincent's theorem#Sign variation|sign variation]] in the sequence of its coefficients can be computed:
+*either by ''[[continued fractions]]'', leading to the ''[[Vincent's theorem#Vincent–Akritas–Strzeboński (VAS, 2005)|Vincent-Akritas-Strzebonski (VAS)]]'' continued fractions method,<ref name="VAS"/>
+*or by ''[[Bisection method|bisection]]'', leading to (among others) the ''[[Vincent's theorem#Vincent–Collins–Akritas (VCA, 1976)|Vincent-Collins-Akritas (VCA)]]'' bisection method.<ref name=CA>{{cite book|last=Collins|first=George E.|coauthor=Alkiviadis G. Akritas|title =Polynomial Real Root Isolation Using Descartes' Rule of Signs|year = 1976|pages=272–275|series = SYMSAC '76, Proceedings of the third ACM symposium on Symbolic and algebraic computation|publisher = ACM|location = Yorktown Heights, NY, USA|url=http://doi.acm.org/10.1145/800205.806346}}</ref>
+The "bisection part" of this all important observation appeared as a special [[Vincent's theorem#Vincent's theorem: Bisection version (Alesina and Galuzzi 2000)|theorem]] in the papers by Alesina and Galuzzi.<ref name="AG_1998"/><ref name="AG_2000"/>
+All methods described below (see the article on [[Budan's theorem]] for their historical background) need to compute (once) an upper bound, ''ub'', on the values of the positive roots of the polynomial under consideration. Exception is the [[Vincent's theorem#Vincent–Akritas–Strzeboński (VAS, 2005)|VAS]] method where additionally lower bounds, ''lb'', need to be computed at almost every cycle of the main loop. To compute the lower bound ''lb'' of the polynomial ''p''(''x'') compute the upper bound ''ub'' of the polynomial <math>x^{\deg(p)}p\left (\frac{1}{x} \right )</math> and set <math>lb = \frac{1}{ub}</math>.
+Excellent (upper and lower) bounds on the values of just the positive roots of polynomials have been developed by Akritas, Strzeboński and Vigklas based on previous work by Doru Stefanescu. They are described in P. S. Vigklas' Ph.D. Thesis<ref name=Panos>{{cite book|last=Vigklas|first=Panagiotis, S.|title=Upper bounds on the values of the positive roots of polynomials|year=2010|publisher=Ph. D. Thesis, University of Thessaly, Greece|url=http://www.inf.uth.gr/images/PHDTheses/phd_thesis_vigklas.pdf}}</ref> and elsewhere.<ref name=bounds>{{cite journal|last=Akritas|first=Alkiviadis, G.|title=Linear and Quadratic Complexity Bounds on the Values of the Positive Roots of Polynomials|journal=Journal of Universal Computer Science|year=2009|volume=15|number=3 |pages=523–537 | url=http://www.jucs.org/jucs_15_3/linear_and_quadratic_complexity}}</ref> These bounds have already been implemented in the [[computer algebra system]]s  [[Mathematica]], [[Sage (mathematics software)|Sage]], [[SymPy]], [[Xcas]] etc.
+All three methods described below follow the excellent presentation of François Boulier,<ref name=FB>{{cite book|last=Boulier|first=François|title=Systèmes polynomiaux : que signifie " résoudre " ?|url=http://www.lifl.fr/~boulier/polycopies/resoudre.pdf|year=2010|publisher=Université Lille 1}}</ref> p.&nbsp;24.
+===Continued fractions method===
+There is only one [[continued fractions]] method derived from [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]]. As has been stated [[Vincent's theorem#Early applications of Vincent's theorem|above]] it started in the 1830s when Vincent presented in both of his papers<ref name="paper_1834"/><ref name="paper_1836"/> several examples showing precisely how his theorem is to be used in order to [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|isolate the real roots]] of polynomials with [[continued fractions]]. However the resulting method had [[Exponential time#Exponential time|exponential]] computing time. Below is an explanation of how this method evolved.
+====Vincent–Akritas–Strzeboński (VAS, 2005)====
+This is the second method (after [[Vincent's theorem#Vincent–Collins–Akritas (VCA, 1976)|VCA]]) developed to handle the [[Exponential time#Exponential time|exponential]] behavior of Vincent's method.
+The VAS continued fractions method is a ''direct'' implementation of Vincent's theorem. It was originally presented by Vincent in his 1834<ref name="paper_1834"/> and 1836<ref name="paper_1836"/> papers in an [[Vincent's theorem#Early applications of Vincent's theorem|exponential form]]; namely, Vincent computed each partial [[Continued fractions#Notations for continued fractions|quotient]] ''a<sub>i</sub>'' by a series of ''unit'' increments ''a<sub>i</sub>'' ← ''a<sub>i</sub>'' + 1, which are equivalent to substitutions of the form ''x'' ← ''x''+1.
+Vincent's method was converted into its [[Exponential time#Polynomial time|polynomial]] complexity form by Akritas, who in his 1978 Ph.D. Thesis ("Vincent's theorem in algebraic manipulation", North Carolina State University, USA) computed each partial [[Continued fractions#Notations for continued fractions|quotient]] ''a<sub>i</sub>'' as the lower bound, ''lb'', on the values of the positive roots of a polynomial; this is called the ''ideal'' positive lower root bound which computes the integer part of the smallest positive root (see the corresponding figure). To wit, now set <math>a_i \leftarrow lb</math> or, equivalently, perform the substitution ''x'' ← ''x''+''lb'', which takes about the same time as the substitution ''x'' ← ''x''+1.
+[[File:VAS method example.jpg|x220px|thumb|center|VAS searching for a root: The ''ideal'' lower bound is 5, hence ''x'' ← ''x''+5.]]
+Finally, since the ideal positive lower root bound does not exist, Strzeboński<ref name=AS>{{cite journal|last=Akritas|first=Alkiviadis G.|coauthors=Adam W. Strzeboński|title=A Comparative Study of Two Real Root Isolation Methods|journal=Nonlinear Analysis: Modelling and Control|year=2005|volume=10|number=4|pages=297–304|url=http://www.lana.lt/journal/19/Akritas.pdf}}</ref> introduced in 2005 the substitution <math>x \leftarrow lb_{computed}*x</math>, whenever <math>lb_{computed}>16</math>; in general <math>lb > lb_{computed}</math> and the value 16 was determined experimentally. Moreover, it has been shown<ref name="AS"/> that the VAS ([[continued fractions]]) method is faster than the fastest implementation of the VCA (bisection) method,<ref name=RZ>{{cite journal |last=Rouillier |first=F.|coauthor= P. Zimmerman|title=Efficient isolation of polynomial's real roots|journal=Journal of Computational and Applied Mathematics|volume=162|pages=33–50|year=2004|url=http://dl.acm.org/citation.cfm?id=972166}}</ref> a fact that was confirmed<ref name=TE>{{cite journal|last=Tsigaridas, P.E.|coauthors=I.Z. Emiris,|title=Univariate polynomial real root isolation: Continued fractions revisited |journal=LNCS |year=2006 |volume=4168 |pages=817–828| url=http://www.springerlink.com/content/c70468755x403481/}}</ref> independently; more precisely, for the Mignotte polynomials of high degree VAS is about 50,000 times faster than the fastest implementation of VCA.
+In 2007, Sharma<ref name=VS>{{cite book|last=Sharma|first=Vikram|title=Complexity Analysis of Algorithms in Algebraic Computation|year=2007|publisher=Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University,USA|url=http://www.cs.nyu.edu/web/Research/Theses/sharma_vikram.pdf}}</ref> removed the hypothesis of the ideal positive lower bound and proved that VAS is still [[Exponential time#Polynomial time|polynomial]] in time.
+VAS is the default algorithm for root isolation in [[Mathematica]], [[Sage (mathematics software)|Sage]], [[SymPy]], [[Xcas]].
+For a comparison between Sturm's method and VAS use the functions realroot(poly) and time(realroot(poly)) of [[Xcas]]. By default, to isolate the real roots of poly realroot uses the VAS method; to use Sturm's method write realroot(sturm, poly). See also the [[Vincent's theorem#External links|External links]] for two applications that do the same thing: one for Android devices by A. Berkakis and another one for the Apple devices iPhone/iPod/iPad by S. Kehagias.
+Here is how VAS(''p'', ''M'') works, where for simplicity Strzeboński's contribution is not included:
+*Let ''p''(''x'') be  a polynomial of degree deg(''p'') such that ''p''(0) ≠ 0. To isolate its positive roots, associate with ''p''(''x'') the [[Möbius transformation]] ''M''(''x'') = ''x'' and repeat the following steps while there are pairs {''p''(''x''), ''M''(''x'')} to be processed.
+*Use [[Descartes' rule of signs]] on ''p''(''x'') to compute, if possible, (using the number ''var'' of [[Vincent's theorem#Sign variation|sign variations]] in the sequence of its coefficients) the number of its roots inside the interval (0, ∞). If there are no roots return the empty set, ∅ whereas if there is one root return the interval (''a'', ''b''), where ''a'' = min(''M''(0), ''M''(∞)), and ''b'' = max(''M''(0), ''M''(∞)); if ''b'' = ∞ set ''b'' = ''ub'', where ''ub'' is an upper bound on the values of the positive roots of ''p''(''x'').<ref name="Panos"/><ref name="bounds"/>
+*If there are two or more sign variations [[Descartes' rule of signs]] implies that there may be zero, two or an even number of real roots inside the interval (0, ∞); in this case consider separately the roots of ''p''(''x'') which lie inside the interval (0, 1) from those which lie inside the interval (1, ∞). A special test has to be made for 1.
+**To guarantee that there will be roots inside the interval (0, 1) the ideal lower bound, ''lb'' is used; that is the integer part of the smallest positive root is computed with the help of the lower bound,<ref name="Panos"/><ref name="bounds"/> <math>lb_{computed} </math>, on the values of the positive roots of ''p''(''x''). If <math>lb_{computed}>1 </math>, the substitution <math>x \leftarrow x+lb_{computed}</math> is performed to ''p''(''x'') and ''M''(''x''), whereas if <math>lb_{computed} \le 1</math> use substitution(s) ''x'' ← ''x''+1 to find the integer part of the root(s).
+**To compute the roots inside the interval (0, 1) perform the substitution <math>x \leftarrow \frac{1}{1+x}</math> to ''p''(''x'') and ''M''(''x'') and process the pair
+:::<math>\left \{(1+x)^{\deg(p)}p\left (\tfrac{1}{1+x} \right ),M(\tfrac{1}{1+x}) \right\},</math>
+::whereas to compute the roots in the interval (1, ∞) perform the substitution ''x'' ← ''x''+1 to ''p''(''x'') and ''M''(''x'') and process the pair <math>\{p(1+x),M(1+x)\}</math>. It may well turn out that 1 is a root of ''p''(''x''), in which case ''M''(1) is a root of the original polynomial and the isolation interval reduces to a point.
+Below is a [[Recursion#Recursion in computer science|recursive]] presentation of VAS(''p'', ''M'').
+<blockquote>'''VAS'''('''''p''''', '''''M'''''):<br>
+'''Input''': A univariate, square-free polynomial <math>p(x) \in \mathbb{Z}[x], p(0) \neq 0</math> and of degree deg(''p''), and the [[Möbius transformation]]
+:<math>M(x)= \frac{ax+b}{cx+d}=x, \qquad a, b, c, d \in \mathbb{N}.</math>
+'''Output''': A list of isolating intervals of the positive roots of ''p''(''x'').<br>
+<code>
+''var'' ← the number of [[Vincent's_theorem#Sign_variation|sign variations]] of ''p''(''x'') // [[Descartes' rule of signs]];<br>
+'''if''' ''var'' = 0 then '''RETURN''' ∅;<br>
+'''if''' ''var'' = 1 then '''RETURN''' {(''a'', ''b'')} // ''a'' = min(''M''(0), ''M''(∞)), ''b'' = max(''M''(0), ''M''(∞)), but if ''b'' = ∞ set ''b'' = ''ub'', where ''ub'' is an upper bound on the values of the positive roots of ''p''(''x'');<br>
+''lb'' ← the ''ideal'' lower bound on the positive roots of ''p''(''x'');<br>
+'''if''' <math>lb \ge 1</math> '''then''' <math>p \leftarrow p(x + lb), M \leftarrow M(x + lb)</math>;<br>
+<math>p_{01} \leftarrow (x+1)^{\deg(p)}p(\tfrac{1}{x + 1}), M_{01} \leftarrow M(\tfrac{1}{x + 1})</math>  // Look for real roots in (0, 1);<br>
+''m'' ← ''M''(1) // Is 1 a root? ;<br>
+<math>p_{1\infty} \leftarrow p(x + 1), M_{1\infty}\leftarrow M(x + 1)</math> // Look for real roots in (1, ∞);<br>
+'''if''' ''p''(1) ≠ 0  '''then'''<br>
+'''RETURN''' <math>VAS(p_{01},M_{01}) \cup VAS(p_{1\infty},M_{1\infty})</math><br>
+'''else''' <br>
+'''RETURN''' <math>VAS(p_{01},M_{01}) \cup \{[m,m]\} \cup VAS(p_{1\infty},M_{1\infty})</math><br>
+'''end'''<br>
+</code></blockquote>
+'''Remarks'''
+*For simplicity Strzeboński's contribution is not included.
+*In the above algorithm with each polynomial there is associated a [[Möbius transformation]] ''M''(''x'').
+*In line 1 [[Descartes' rule of signs]] is applied.
+*If lines 4 and 5 are removed from VAS(''p'', ''M'') the resulting algorithm is Vincent's exponential one.
+*Any substitution performed on the polynomial ''p''(''x'') is also performed on the associated [[Möbius transformation]] ''M''(''x'') (lines 5 6 and 8).
+*The isolating intervals are computed from the [[Möbius transformation]] in line 3, except for integer roots computed in line 7 (also 12).
+====Example of VAS(''p'', ''M'')====
+Given the polynomial <math>p(x)=x^3 -7x + 7 </math> the arguments of the VAS method are:<math>p(x) = x^3 - 7x +7</math> and ''M''(''x'') = ''x''.
+<code>
+''var'' ← 2 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>p(x) = x^3 - 7x +7</math>
+''lb'' ← 1 // the ideal lower bound — found using  <math>lb_{computed}</math> and substitution(s) ''x'' ← ''x''+1
+<math> p \leftarrow  x^3+3x^2-4x+1, M\leftarrow  x+1</math>
+<math>p_{01} \leftarrow x^3-x^2-2x+1, M_{01}\leftarrow \frac{x+2}{x+1}</math>
+''m'' ← 1
+<math>p_{1\infty} \leftarrow x^3+6x^2+5x+1, M_{1\infty}\leftarrow x+2</math>
+          '''RETURN''' <math>VAS(x^3-x^2-2x+1,\frac{x+2}{x+1})  \cup VAS(x^3+6x^2+5x+1,x+2)</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of pairs <math>\{poly,Moebius\  transformation\}</math> to be processed: <math>\{\{x^3-x^2-2x+1,\frac{x+2}{x+1}\},\{x^3+6x^2+5x+1,x+2\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VAS(x^3-x^2-2x+1,\frac{x+2}{x+1}) </math>
+<hr />
+<br>
+<code>
+''var'' ← 2 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>p(x) = x^3-x^2-2x+1</math>
+''lb'' ← 0 // the ideal lower bound — found using  <math>lb_{computed}</math> and substitution(s) ''x'' ← ''x''+1
+<math>p_{01} \leftarrow x^3+x^2-2x-1, M_{01}\leftarrow \frac{2x+3}{x+2}</math>
+<math>m \leftarrow \tfrac{3}{2}</math>
+<math>p_{1\infty} \leftarrow x^3+2x^2-x-1, M_{1\infty}\leftarrow \frac{x+3}{x+2}</math>
+          '''RETURN''' <math>VAS(x^3+x^2-2x-1,\frac{2x+3}{x+2})  \cup  VAS(x^3+2x^2-x-1,\frac{x+3}{x+2})</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of pairs <math>\{poly,Moebius\  transformation\}</math> to be processed: <math>\{\{x^3+x^2-2x-1,\frac{2x+3}{x+2}\},\{x^3+2x^2-x-1,\frac{x+3}{x+2}\},\{x^3+6x^2+5x+1,x+2\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VAS(x^3+x^2-2x-1,\frac{2x+3}{x+2}) </math>
+<hr />
+<br>
+<code>
+''var'' ← 1 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>p(x) = x^3+x^2-2x-1</math>
+          '''RETURN''' <math>\{(\tfrac{3}{2},2)\}</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: <math>\{(\tfrac{3}{2},2)\}</math>. List of pairs <math>\{poly,Moebius\  transformation\}</math> to be processed: <math>\{\{x^3+2x^2-x-1,\frac{x+3}{x+2}\},\{x^3+6x^2+5x+1,x+2\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VAS(x^3+2x^2-x-1,\frac{x+3}{x+2}) </math>
+<hr />
+<br>
+<code>
+''var'' ← 1 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>p(x) = x^3+2x^2-x-1</math>
+          '''RETURN''' <math>\{(1,\tfrac{3}{2})\}</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: <math>\{(1,\tfrac{3}{2}),(\tfrac{3}{2},2)\}</math>. List of pairs <math>\{poly, Moebius\  transformation\}</math> to be processed: <math>\{\{x^3+6x^2+5x+1,x+2\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VAS(x^3+6x^2+5x+1,x+2) </math>
+<hr />
+<br>
+<code>
+''var'' ← 0 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>p(x) = x^3+6x^2+5x+1</math>
+          '''RETURN''' ∅
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: <math>\{(1,\tfrac{3}{2}),(\tfrac{3}{2},2)\}</math>. List of pairs <math>\{poly,Moebius\  transformation\}</math> to be processed: ∅. Finished.
+<hr />
+<br>
+Therefore, the two positive roots of the polynomial <math>p(x)=x^3 -7x + 7</math> lie inside the isolation intervals <math>{(1,\tfrac{3}{2})}</math> and <math>{(\tfrac{3}{2},2)}</math>. Each root can be approximated by (for example) bisecting the isolation interval — inside which it lies — until the difference of the endpoints is smaller than <math>10^{-6}</math>; following this approach, the roots turn out to be <math>\rho_{1} = 1.3569</math> and <math>\rho_{2} = 1.69202 </math>.
+===Bisection methods===
+There are various [[bisection method]]s derived from [[Vincent's theorem#Vincent's theorem: Bisection version (Alesina and Galuzzi 2000)|Vincent's theorem]]; they are all presented and compared elsewhere.<ref name=ASV_2008>{{cite journal|last=Akritas|first=Alkiviadis G.|coauthor=Adam W. Strzeboński, Panagiotis S. Vigklas|title=On the Various Bisection Methods Derived from Vincent's Theorem|url=http://sci-gems.math.bas.bg:8080/jspui/handle/10525/376|journal=Serdica Journal of Computing|year=2008|volume=2|number=1|pages=89–104}}</ref> Here the two most important of them are described, namely, the [[Vincent's theorem#Vincent–Collins–Akritas (VCA, 1976)|Vincent-Collins-Akritas (VCA)]] method and the [[Vincent's theorem#Vincent–Alesina–Galuzzi (VAG, 2000)|Vincent-Alesina-Galuzzi (VAG)]] method.
+The [[Vincent's theorem#Vincent–Alesina–Galuzzi (VAG, 2000)|Vincent-Alesina-Galuzzi (VAG)]] method is the simplest of all methods derived from Vincent's theorem but has the most time consuming test (in line 1) to determine if a polynomial has roots in the interval of interest; this makes it the slowest of the methods presented in this article.
+By contrast, the [[Vincent's theorem#Vincent–Collins–Akritas (VCA, 1976)|Vincent-Collins-Akritas (VCA)]] method is more complex but uses a simpler test (in line 1) than [[Vincent's theorem#Vincent–Alesina–Galuzzi (VAG, 2000)|VAG]]. This along with certain improvements<ref name="RZ"/> have made [[Vincent's theorem#Vincent–Collins–Akritas (VCA, 1976)|VCA]] the fastest bisection method.
+====Vincent–Collins–Akritas (VCA, 1976)====
+This was the first method developed to overcome the [[Exponential time#Exponential time|exponential]] nature of Vincent's [[Vincent's theorem#Continued fractions method|original approach]], and has had quite an interesting history as far as its name is concerned. This method, which isolates the real roots, using Descartes' rule of signs and [[Vincent's theorem#Vincent's theorem: Continued fractions version (1834 and 1836)|Vincent's theorem]], had been originally called ''modified Uspensky's algorithm'' by its inventors Collins and Akritas.<ref name="CA"/> After going through names like "Collins-Akritas method" and "Descartes' method" (too confusing if ones considers Fourier's article<ref name=Fourier>{{cite journal|last=Fourier|first=Jean Baptiste Joseph|title=Sur l'usage du théorème de Descartes dans la recherche des limites des racines|year=1820|journal=Bulletin des Sciences, par la Société Philomatique de Paris|pages=156–165 | url=http://ia600309.us.archive.org/22/items/bulletindesscien20soci/bulletindesscien20soci.pdf}}</ref>), it was finally François Boulier, of Lille University, who gave it the name ''Vincent-Collins-Akritas'' (VCA) method,<ref name="FB"/> p.&nbsp;24, based on the fact that "Uspensky's method" does not exist<ref name="akritas">{{cite book|last=Akritas|first=Alkiviadis G.|title=There's no "Uspensky's Method"|url=http://dl.acm.org/citation.cfm?id=32457|year=1986|publisher=In: Proceedings of the fifth ACM Symposium on Symbolic and Algebraic Computation (SYMSAC '86, Waterloo, Ontario, Canada), pp. 88–90}}</ref> and neither does "Descartes' method".<ref name=noDec>{{cite book|last=Akritas|first=Alkiviadis G.|title=There is no "Descartes' method"|url=http://books.google.com/books?id=SJR2ybQdZFgC&lpg=PR1&pg=PR1#v=onepage&q&f=false |year=2008|publisher=In: M.J.Wester and M. Beaudin (Eds), Computer Algebra in Education, AullonaPress, USA, pp. 19-35}}</ref> The best implementation of this method is due to Rouillier and Zimmerman,<ref name="RZ"/> and  to this date, it is the fastest bisection method. It has the same worst case [[Computational complexity theory|complexity]] as Sturm's algorithm, but is almost always much faster. It has been implemented in [[Maple (software)|Maple]]'s RootFinding package.
+Here is how VCA(''p'', (''a'', ''b'')) works:
+*Given a polynomial <math>p_{orig}(x)</math> of degree deg(''p''), such that <math>p_{orig}(0)\ne 0</math>, whose positive roots need to be isolated, first compute an upper bound,<ref name="Panos"/><ref name="bounds"/> ''ub'' on the values of these positive roots and set <math>p(x) = p_{orig}(ub*x)</math> and (''a'', ''b'') = (0, ''ub''). The positive roots of ''p''(''x'') all lie in the interval (0, 1) and there is a [[bijection]] between them and the roots of <math>p_{orig}(x)</math>, which all lie in the interval (''a'', ''b'') = (0, ''ub'') (see the corresponding figure); this [[bijection]] is expressed by <math>\alpha_{(a,b)}=a+\alpha_{(0,1)}(b-a)</math>. Likewise, there is a [[bijection]] between the intervals (0, 1) and (0, ''ub'').
+[[File:VCA Algorithm.jpg|x220px|thumb|center|[[Bijection]] between the roots of <math>p_{orig}(x)</math> and ''p''(''x'').]]
+*Repeat the following steps while there are pairs {''p''(''x''), (''a'', ''b'')} to be processed.
+*Use Budan's [[Budan's theorem#Early applications of Budan's theorem|"'''0_1 roots test'''"]] on ''p''(''x'') to compute (using the number ''var'' of [[Budan's theorem#Sign variation|sign variations]] in the sequence of its coefficients) the number of its roots inside the interval (0, 1). If there are no roots return the empty set, ∅ and if there is one root return the interval (''a'', ''b'').
+*If there are two or more sign variations Budan's [[Budan's theorem#Early applications of Budan's theorem|"'''0_1 roots test'''"]] implies that there may be two or more real roots inside the interval (0, 1). In this case cut it in half and consider separately the roots of ''p''(''x'') which lie inside the interval <math>(0,\tfrac{1}{2})</math> — and which correspond to the roots of <math>p_{orig}(x)</math> inside the interval <math>(a,\tfrac{1}{2}(a+b))</math> — from those which lie inside the interval <math>(\tfrac{1}{2},1)</math> — and which correspond to the roots of <math>p_{orig}(x)</math> inside the interval <math>(\tfrac{1}{2}(a+b),b)</math>; that is, process, respectively, the pairs
+:<math>\left \{2^{\deg(p)}p(\tfrac{x}{2}), (a, \tfrac{1}{2}(a+b)) \right \}, \quad \left \{2^{\deg(p)}p(\tfrac{1}{2} (x+1)), (\tfrac{1}{2}(a+b), b) \right \}</math>
+(see the corresponding figure).It may well turn out that <math>\tfrac{1}{2}</math> is a root of ''p''(''x''), in which case \tfrac{1}{2}(a+b) is a root of <math>p_{orig}(x)</math> and the isolation interval reduces to a point.
+[[File:VCA Example.jpg|x220px|thumb|center|[[Bijection]]s between the roots of ''p''(''x'') and those of <math>p(\tfrac{x}{2})</math> and <math>p(\tfrac{x+1}{2})</math>.]]
+Below is a [[Recursion#Recursion in computer science|recursive]] presentation of the original algorithm VCA(''p'', (''a'', ''b'')).
+<blockquote>'''VCA'''('''''p''''', ('''''a''''', '''''b'''''))<br>
+'''Input''': A univariate, square-free polynomial <math>p(ub * x) \in \mathbb{Z}[x], p(0) \neq 0</math> of degree deg(''p''), and the open
+interval (''a'', ''b'') = (0, ''ub''), where ''ub'' is an upper bound on the values of the positive
+roots of ''p''(''x''). (The positive roots of <math>p(ub * x)</math> are all in the open interval (0, 1)).<br>
+'''Output''': A list of isolating intervals of the positive roots of ''p''(''x'')<br>
+<code>
+''var'' ← the number of [[Vincent's theorem#Sign variation|sign variation]]s of <math>(x + 1)^{\deg(p)}p(\tfrac{1}{x+1})</math> // Budan's [[Budan's theorem#Early applications of Budan's theorem|"'''0_1 roots test'''"]];<br>
+'''if''' ''var'' = 0 '''then RETURN''' ∅;<br>
+'''if''' ''var'' = 1 '''then RETURN''' {(''a'', ''b'')};<br>
+<math>p_{0 \tfrac{1}{2}} \leftarrow 2^{\deg(p)}p(\tfrac{x}{2})</math> // Look for real roots in <math>(0, \tfrac{1}{2})</math>;<br>
+<math> m \leftarrow \tfrac{1}{2} (a+b) </math> // Is <math>\tfrac{1}{2}</math> a root? ;<br>
+<math>p_{\tfrac{1}{2}1} \leftarrow 2^{\deg(p)}p(\tfrac{x+1}{2})</math> // Look for real roots in <math>(\tfrac{1}{2}, 1)</math>;<br>
+'''if''' <math>p(\tfrac{1}{2}) \neq 0</math> '''then'''<br>
+          '''RETURN''' <math>VCA (p_{0 \tfrac{1}{2}}, (a, m))  \cup VCA (p_{\tfrac{1}{2}1},(m, b))</math><br>
+'''else'''<br>
+        '''RETURN''' <math>VCA (p_{0 \frac{1}{2}}, (a, m))  \cup   \{[m, m]\} \cup VCA (p_{\frac{1}{2}1},(m, b))</math><br>
+'''end'''
+</code></blockquote>
+'''Remark'''
+*In the above algorithm with each polynomial there is associated an interval (''a'', ''b''). As shown elsewhere,<ref name="noDec"/> p.&nbsp;11, a [[Möbius transformation]] can also be associated with each polynomial in which case VCA looks more like [[Vincent's theorem#VAS(p,M)|VAS]].
+*In line 1 Budan's [[Budan's theorem#Early applications of Budan's theorem|"'''0_1 roots test'''"]] is applied.
+====Example of VCA(p,(a,b))====
+Given the polynomial <math>p_{orig}(x)=x^3 -7x + 7 </math> and considering as an upper bound<ref name="Panos"/><ref name="bounds"/> on the values of the positive roots ''ub'' = 4 the arguments of the VCA method are:<math>p(x) = 64x^3 - 28x +7</math> and (''a'', ''b'') = (0, 4).
+<code>
+''var'' ← 2 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(x + 1)^{3}p(\frac{1}{x+1}) = 7x^3-7x^2-35x+43</math>
+<math>p_{0 \tfrac{1}{2}} \leftarrow 64x^3-112x+56</math>
+''m'' ← 2
+<math>p_{\tfrac{1}{2}1} \leftarrow 64x^3+192x^2+80x+8</math>
+<math>p({\tfrac{1}{2}}) = 1</math>
+          '''RETURN''' <math>VCA(64x^3-112x+56,(0,2)) \cup VCA(64x^3+192x^2+80x+8,(2,4))</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of pairs <math>\{poly,interval\}</math> to be processed: <math>\{\{64x^3-112x+56,(0,2)\},\{64x^3+192x^2+80x+8,(2,4)\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VCA(64x^3-112x+56,(0,2)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 2 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(x + 1)^{3}p(\tfrac{1}{x+1}) = 56x^3+56x^2-56x+8</math>
+<math>p_{0 \frac{1}{2}} \leftarrow 64x^3-448x+448</math>
+''m'' ← 1
+<math>p_{\tfrac{1}{2}1} \leftarrow 64x^3+192x^2-256x+64</math>
+<math>p({\tfrac{1}{2}}) = 8</math>
+          '''RETURN''' <math>VCA(64x^3-448x+448,(0,1)) \cup  VCA(64x^3+192x^2-256x+64,(1,2))</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of pairs <math>\{poly,interval\}</math> to be processed: <math>\{\{64x^3-448x+448,(0,1)\},\{64x^3+192x^2-256x+64,(1,2)\},\{64x^3+192x^2+80x+8,(2,4)\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VCA(64x^3-448x+448,(0,1)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 0 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(x + 1)^{3}p(\tfrac{1}{x+1}) = 448x^3+896x^2+448x+64</math>
+          '''RETURN''' ∅
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of pairs <math>\{poly,interval\}</math> to be processed: <math>\{\{64x^3+192x^2-256x+64,(1,2)\},\{64x^3+192x^2+80x+8,(2,4)\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VCA(64x^3+192x^2-256x+64,(1,2)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 2 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(x + 1)^{3}p(\frac{1}{x+1}) = 64x^3-64x^2-128x+64</math>
+<math>p_{0 \tfrac{1}{2}} \leftarrow 64x^3+384x^2-1024x+512</math>
+<math> m \leftarrow \tfrac{3}{2} </math>
+<math>p_{\tfrac{1}{2}1} \leftarrow 64x^3+576x^2-64x-64</math>
+<math>p({\tfrac{1}{2}}) = -8</math>
+'''RETURN''' <math>VCA(64x^3+384x^2-1024x+512,(1,\tfrac{3}{2})) \cup VCA(64x^3+576x^2-64x-64,(\tfrac{3}{2},2))</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of pairs <math>\{poly,interval\}</math> to be processed: <math>\{\{64x^3+384x^2-1024x+512,(1,\tfrac{3}{2})\},\{64x^3+576x^2-64x-64,(\tfrac{3}{2},2)\},\{64x^3+192x^2+80x+8,(2,4)\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VCA(64x^3+384x^2-1024x+512,(1,\tfrac{3}{2})) </math>
+<hr />
+<br>
+<code>
+''var'' ← 1 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(x + 1)^{3}p(\frac{1}{x+1}) = 512x^3+512x^2-128x-64</math>
+'''RETURN''' <math>\{(1,\tfrac{3}{2})\}</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: <math>\{(1,\tfrac{3}{2})\}</math>. List of pairs <math>\{poly,interval\}</math> to be processed: <math>\{\{64x^3+576x^2-64x-64,(\frac{3}{2},2)\},\{64x^3+192x^2+80x+8,(2,4)\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VCA(64x^3+576x^2-64x-64,(\frac{3}{2},2)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 1 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(x + 1)^{3}p(\tfrac{1}{x+1}) = -64x^3-256x^2+256x+512</math>
+'''RETURN''' <math>\{(\tfrac{3}{2},2)\}</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: <math>\{(1,\tfrac{3}{2}),(\tfrac{3}{2},2)\}</math>. List of pairs <math>\{poly,interval\}</math> to be processed: <math>\{\{64x^3+192x^2+80x+8,(2,4)\}\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VCA(64x^3+192x^2+80x+8,(2,4)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 0 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(x + 1)^{3}p(\tfrac{1}{x+1}) = 8x^3+104x^2+376x+344</math>
+'''RETURN''' ∅
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: <math>\{(1,\tfrac{3}{2}),(\tfrac{3}{2},2)\}</math>. List of pairs <math>\{poly,interval\}</math> to be processed: ∅. Finished.
+<hr />
+<br>
+Therefore, the two positive roots of the polynomial <math>p(x)=x^3 -7x + 7</math> lie inside the isolation intervals <math>{(1,\tfrac{3}{2})}</math> and <math>{(\tfrac{3}{2},2)}</math>. Each root can be approximated by (for example) bisecting the isolation interval — inside which it lies — until the difference of the endpoints is smaller than <math>10^{-6}</math>; following this approach, the roots turn out to be <math>\rho_{1} = 1.3569</math> and <math>  \rho_{2} = 1.69202 </math>.
+====Vincent–Alesina–Galuzzi (VAG, 2000)====
+This was developed last and is the simplest [[Vincent's theorem#Real root isolation methods derived from Vincent's theorem|real root isolation method]] derived from [[Vincent's theorem#Vincent's theorem: Bisection version (Alesina and Galuzzi 2000)|Vincent's theorem]].
+Here is how VAG(''p'', (''a'', ''b'')) works:
+*Given a polynomial ''p''(''x'') of degree deg(''p''), such that ''p''(0) ≠ 0, whose positive roots need to be isolated, first compute an upper bound,<ref name="Panos"/><ref name="bounds"/> ''ub'' on the values of these positive roots and set (''a'', ''b'') = (0, ''ub''). The positive roots of ''p''(''x'') all lie in the interval (''a'', ''b'').
+*Repeat the following steps while there are intervals (''a'', ''b'') to be processed; in this case the polynomial ''p''(''x'') stays the same.
+*Use the Alesina-Galuzzi [[Vincent's theorem#The Alesina-Galuzzi "a b roots test"|"'''a_b roots test'''"]]  on ''p''(''x'') to compute (using the number ''var'' of [[Vincent's theorem#Sign variation|sign variations]] in the sequence of its coefficients) the number of its roots inside the interval (''a'', ''b''). If there are no roots return the empty set, ∅ and if there is one root return the interval (''a'', ''b'').
+*If there are two or more sign variations the Alesina-Galuzzi [[Vincent's theorem#The Alesina-Galuzzi "a b roots test"|"'''a_b roots test'''"]] implies that there may be two or more real roots inside the interval (''a'', ''b''). In this case cut it in half and consider separately the roots of ''p''(''x'') which lie inside the interval <math>(a, \tfrac{1}{2}(a+b))</math> from those which lie inside the interval <math>(\tfrac{1}{2}(a+b),b)</math>; that is, process, respectively, the intervals <math>(a, \tfrac{1}{2}(a+b))</math> and <math>(\tfrac{1}{2}(a+b),b)</math>. It may well turn out that <math>\tfrac{1}{2}(a+b)</math> is a root of ''p''(''x''), in which case the isolation interval reduces to a point.
+Below is a [[Recursion#Recursion in computer science|recursive]] presentation of VAG(''p'', (''a'', ''b'')).
+<blockquote>'''VAG'''('''''p''''', ('''''a''''', '''''b'''''))<br>
+'''Input''': A univariate, square-free polynomial <math>p(x) \in \mathbb{Z}[x], p(0) \neq 0</math> of degree deg(''p'') and the open interval (''a'', ''b'') = (0, ''ub''), where ''ub'' is an upper bound on the values of the positive roots of ''p''(''x''). <br>
+'''Output''': A list of isolating intervals of the positive roots of ''p''(''x'').<br>
+<code>
+''var'' ← the number of [[Vincent's theorem#Sign variation|sign variation]]s of <math>(x + 1)^{\deg(p)}p\left(\frac{a+bx}{1+x} \right)</math> // The Alesina-Galuzzi [[Vincent's theorem#The Alesina-Galuzzi "a_b roots test"|"'''a_b roots test'''"]];<br>
+'''if''' ''var'' = 0 '''then RETURN''' ∅;<br>
+'''if''' ''var'' = 1 '''then RETURN''' {(''a'', ''b'')};<br>
+<math> m \leftarrow \tfrac{1}{2}(a+b) </math> // Subdivide the interval (a,b) in two equal parts;<br>
+'''if''' ''p''(''m'') ≠ 0 '''then'''<br>
+          '''RETURN''' <math>VAG (p, (a, m))\cup VAG (p,(m, b))</math><br>
+'''else'''<br>
+        '''RETURN''' <math>VAG (p, (a, m)) \cup  \{[m, m]\} \cup VAG (p,(m, b))</math><br>
+'''end'''<br>
+</code></blockquote>
+'''Remarks'''
+*Compared to [[Vincent's theorem#VCA(p,(a,b))|VCA]] the above algorithm is extremely simple; by contrast, VAG uses the time consuming '''[[Vincent's theorem#The Alesina-Galuzzi "a b roots test"|"a_b roots test"]]''' and that makes it much slower than [[Vincent's theorem#VCA(p,(a,b))|VCA]].<ref name="ASV_2008"/>
+*As Alesina and Galuzzi point out,<ref name="AG_2000"/> p.&nbsp;189, there is a variant of this algorithm due to Donato Saeli. Saeli suggested that the ''[[Mediant (mathematics)|mediant]]'' of the endpoints be used instead of their midpoint <math> \tfrac{1}{2}(a+b)</math>. However, it has been shown<ref name="ASV_2008"/> that using the [[Mediant (mathematics)|mediant]] of the endpoints is in general much slower than the "mid-point" version.
+====Example of VAG(p,(a,b))====
+Given the polynomial  <math>p(x)=x^3 -7x + 7 </math> and considering as an upper bound<ref name="Panos"/><ref name="bounds"/> on the values of the positive roots ''ub'' = 4 the arguments of VAG are:<math>p(x) = x^3 -7x + 7</math> and (''a'', ''b'') = (0, 4).
+<code>
+''var'' ← 2 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(1+x)^{3}p(\tfrac{4x}{1+x}) = 43x^3-35x^2-7x+7</math>
+<math>m \leftarrow \frac{a+b}{2} = \frac{0+4}{2} = 2</math>
+''p''(''m'') = 1
+          '''RETURN''' <math>VAG(x^3 -7x + 7,(0,2)) \cup VAG(x^3 -7x + 7,(2,4)</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of intervals to be processed: {(0, 2), (2, 4)}. Remove the first and process it.
+<hr />
+<br>
+<math>VAG(x^3 -7x + 7,(0,2)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 2 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(1+x)^{3}p(\frac{2x}{1+x}) = x^3-7x^2+7x+7</math>
+<math>m \leftarrow \frac{a+b}{2} = \frac{0+2}{2} = 1</math>
+''p''(''m'') = 1
+          '''RETURN''' <math>VAG(x^3 -7x + 7,(0,1)) \cup VAG(x^3 -7x + 7,(1,2)</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of intervals to be processed: <math>\{(0,1),(1,2),(2,4)\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VAG(x^3 -7x + 7,(0,1)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 0 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(1+x)^{3}p(\frac{x}{1+x}) = x^3+7x^2+14x+7</math>
+          '''RETURN''' ∅
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of intervals to be processed: <math>\{(1,2),(2,4)\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VAG(x^3 -7x + 7,(1,2)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 2 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(1+x)^{3}p(\frac{1+2x}{1+x}) = x^3-2x^2-x+1</math>
+<math>m \leftarrow \frac{a+b}{2} = \frac{1+2}{2} = \frac{3}{2}</math>
+<math>p(m) = -\tfrac{1}{8}</math>
+          '''RETURN''' <math>VAG(x^3 -7x + 7,(1,\frac{3}{2})) \cup VAG(x^3 -7x + 7,(\frac{3}{2},2)</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: {}. List of intervals to be processed: <math>\{(1,\tfrac{3}{2}),(\tfrac{3}{2},2),(2,4)\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VAG(x^3 -7x + 7,(1,\tfrac{3}{2})) </math>
+<hr />
+<br>
+<code>
+''var'' ← 1 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>2^3(1+x)^{3}p(\frac{1+\frac{3}{2}x}{1+x}) = x^3+2x^2-8x-8</math>
+          '''RETURN''' <math>\{(1,\tfrac{3}{2})\}</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: <math>\{(1,\tfrac{3}{2})\}</math>. List of intervals to be processed: <math>\{(\tfrac{3}{2},2),(2,4)\}</math>. Remove the first and process it.
+<hr />
+<br>
+<math>VAG(x^3 -7x + 7,(\frac{3}{2},2)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 1 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>2^3(1+x)^{3}p(\frac{\frac{3}{2}+2x}{1+x}) = 8x^3+4x^2-4x-1</math>
+          '''RETURN''' <math>\{(\tfrac{3}{2},2)\}</math>
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: <math>\{(1,\tfrac{3}{2}),(\tfrac{3}{2},2)\}</math>. List of intervals to be processed: {(2, 4)}. Remove the first and process it.
+<hr />
+<br>
+<math>VAG(x^3 -7x + 7,(2,4)) </math>
+<hr />
+<br>
+<code>
+''var'' ← 0 // the number of [[Vincent's theorem#Sign variation|sign variation]]s in the sequence of coefficients of <math>(1+x)^3 p(\frac{2+4x}{1+x}) = 344x^3+376x^2+104x+8</math>
+          '''RETURN''' ∅
+</code>
+<br>
+<hr />
+<br>
+List of isolation intervals: <math>\{(1,\tfrac{3}{2}),(\tfrac{3}{2},2)\}</math>. List of intervals to be processed: ∅.  Finished.
+<hr />
+<br>
+Therefore, the two positive roots of the polynomial <math>p(x)=x^3 -7x + 7</math> lie inside the isolation intervals <math>{(1,\tfrac{3}{2})}</math> and <math>{(\tfrac{3}{2},2)}</math>. Each root can be approximated by (for example) bisecting the isolation interval — inside which it lies — until the difference of the endpoints is smaller than <math>10^{-6}</math>; following this approach, the roots turn out to be <math>\rho_{1} = 1.3569</math> and <math>  \rho_{2} = 1.69202 </math>.
+==See also==
+*[[Properties of polynomial roots]]
+*[[Root-finding algorithm]]
+*[[Vieta's formulas]]
+*[[Newton's method]]
+==References==
+{{reflist}}
+==External links==
+* Berkakis, Antonis: RealRoots, a free App for Android devices to compare Sturm's method and VAS
+* https://play.google.com/store/apps/details?id=org.kde.necessitas.berkakis.realroots
+*Encyclopedia of Mathematics  http://www.encyclopediaofmath.org/index.php
+* Kehagias, Spyros: RealRoots, a free App for iPhone, iPod Touch and iPad to compare Sturm's method and VAS http://itunes.apple.com/gr/app/realroots/id483609988?mt=8
+[[Category:Mathematical theorems]]