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| In [[mathematics]], the '''Sturm's sequence''' of a [[polynomial]] ''p'' is a sequence of polynomials associated to ''p'' and its derivative by a variant of [[Euclid's algorithm for polynomials]].
| | The author is recognized by the name of Numbers Wunder. California is where her house is but she needs to transfer simply because of her family. Body developing is 1 of the issues I love most. He utilized to be unemployed but now he is a computer operator but his marketing never comes.<br><br>my blog post: [http://www.videoworld.com/user/JMcQuade home std test] |
| '''Sturm's theorem''' expresses the number of distinct [[real number|real]] [[Root of a function|root]]s of ''p'' located in an [[interval (mathematics)|interval]] in terms of the number of changes of signs of the values of the Sturm's sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of ''p''.
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| Whereas the [[fundamental theorem of algebra]] readily yields the overall number of [[complex number|complex]] roots, counted with [[multiplicity (mathematics)|multiplicity]], Sturm's theorem yields the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it allows eventually to isolate the roots in arbitrary small intervals each containing exactly one root. This yields a symbolic [[root finding algorithm]], that is available in most [[computer algebra system]]s, although some more efficient methods are now usually preferred (see below).
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| Sturm's sequence and Sturm's theorems are named after [[Jacques Charles François Sturm]].
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| ==Sturm chains==
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| A '''Sturm chain''' or '''Sturm sequence''' is a finite sequence of polynomials
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| :<math>p_0, p_1, \dots, p_m</math>
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| of decreasing [[degree of a polynomial|degree]] with these following properties:
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| * <math>p_0= p</math> is [[square-free polynomial|square free]] (no square factors, i.e., no repeated roots);
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| * if <math>p(\xi)=0</math>, then <math>\operatorname{sign}(p_1(\xi))= \operatorname{sign}(p'(\xi));</math>
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| * if <math>p_i(\xi)=0</math> for <math>0<i<m</math> then <math>\operatorname{sign}(p_{i-1}(\xi))= -\operatorname{sign}(p_{i+1}(\xi));</math>
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| * <math>p_m</math> does not change its sign.
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| It can be noted that Sturm's sequence is a modification of [[Budan's theorem#Fourier sequence|Fourier's sequence]].
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| To obtain a Sturm chain, Sturm himself proposed to choose the intermediary results when applying [[Euclid's algorithm]] to ''p'' and its [[derivative]]:
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| :<math>\begin{align}
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| p_0(x) & := p(x),\\
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| p_1(x) & := p'(x),\\
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| p_2(x) & := -{\rm rem}(p_0, p_1) = p_1(x) q_0(x)- p_0(x),\\
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| p_3(x) & := -{\rm rem}(p_1,p_2) = p_2(x) q_1(x) - p_1(x),\\
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| &{}\ \ \vdots\\
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| 0 & = -\text{rem}(p_{m-1}, p_m),
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| \end{align} </math>
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| where rem<math> (p_i,p_j)</math> and <math>q_i</math> are the remainder and the quotient of the [[polynomial long division]] of <math>p_i</math> by <math>p_j</math>, and where ''m'' is the minimal number of polynomial divisions (never greater than the degree of ''p(x)'') needed to obtain a zero remainder. That is, successively take the remainders with [[polynomial#Divisibility|polynomial division]] and change their signs. Since <math>\operatorname{deg}(p_{i+1}) < \operatorname{deg} (p_i)</math> for <math>0 \le i < m</math>, the algorithm terminates. The final polynomial, ''p''<sub>''m''</sub>, is the [[Polynomial greatest common divisor|greatest common divisor]] of ''p'' and its derivative. If ''p'' is square free, it shares no roots with its derivative, hence ''p''<sub>''m''</sub> will be a non-zero constant polynomial. The Sturm chain, called the canonical Sturm chain, then is | |
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| :<math>p_0,p_1,p_2,\ldots,p_m . \,</math>
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| If ''p'' is not square-free, this sequence does not formally satisfy the definition of a Sturm chain above; nevertheless it still satisfies the conclusion of Sturm's theorem (see below).
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| ==Statement==
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| Let <math>p_0=p, p_1, \dots, p_m</math> be a Sturm chain, where ''p'' is a square-free polynomial, and let σ(ξ) denote the number of sign changes (ignoring zeroes) in the sequence
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| :<math>p_0(\xi), p_1(\xi), p_2(\xi),\ldots, p_m(\xi). \,\! </math>
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| Sturm's theorem then states that for two real numbers ''a'' < ''b'', the number of distinct roots of ''p'' in the half-open interval (''a'',''b''] is σ(''a'') − σ(''b'').
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| ===The non-square-free case===
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| Let <math>p_0=p, p_1, \dots, p_m</math> be the canonical Sturm sequence of a polynomial ''p'', not necessarily square-free. Then σ(''a'') − σ(''b'') is the number of distinct roots of ''p'' in (''a'',''b''] whenever ''a'' < ''b'' are real numbers such that neither ''a'' nor ''b'' is a multiple root of ''p''.
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| ==Example==
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| Suppose we wish to find the number of roots in some range for the polynomial <math>p(x)=x^4+x^3-x-1</math>. So <math>p_0(x)=p(x)=x^4+x^3-x-1</math> and <math>p_1(x)=p'(x)=4x^3+3x^2-1</math>. Using [[polynomial long division]] to divide <math>p_0</math> by <math>p_1</math> gives the remainder <math>-\tfrac{3}{16}x^2-\tfrac{3}{4}x-\tfrac{15}{16}</math>, and upon multiplying this remainder by −1 we obtain <math>p_2(x)=\tfrac{3}{16}x^2+\tfrac{3}{4}x+\tfrac{15}{16}</math>. Next dividing <math>p_1</math> by <math>p_2</math> and multiplying the remainder by −1, we obtain <math>p_3(x)=-32x-64</math>. And dividing <math>p_2</math> by <math>p_3</math> and multiplying the remainder by −1, we obtain <math>p_4(x)=-\tfrac{3}{16}</math>.
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| Thus the complete chain of Sturm polynomials is:
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| * <math>p_0(x) = x^4+x^3-x-1</math>
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| * <math>p_1(x) = 4x^3+3x^2-1</math>
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| * <math>p_2(x) = \tfrac{3}{16}x^2+\tfrac{3}{4}x+\tfrac{15}{16}</math>
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| * <math>p_3(x) = -32x-64</math>
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| * <math>p_4(x) = -\tfrac{3}{16}</math>
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| To find the number of roots between −∞ and ∞, first evaluate <math>p_0, p_1, p_2, p_3,</math> and <math>p_4</math> at −∞ and note the sequence of signs of the results: + − + + −, which contains 3 sign changes (+ to −, then − to +, then + to −). The same procedure for +∞ gives the sign sequence + + + − −, which contains just 1 sign change. Hence the number of roots of the original polynomial between −∞ and ∞ is {{nowrap|3 − 1 {{=}} 2.}} That this is correct can be seen by noting that <math>p(x)=x^4+x^3-x-1</math> can be factored as <math> (x^2-1)(x^2+x+1)</math>, where it is readily verifiable that <math>(x^2-1)</math> has the two roots −1 and 1 while <math> (x^2+x+1)</math> has no real roots. In more complicated examples in which there is no advance knowledge of the roots because factoring is either impossible or impractical, one can experiment with various finite bounds for the range to be considered, thus narrowing down the locations of the roots.
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| ==Proof==
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| Polynomials are [[continuous function]]s, and any sign change must occur at a root, so consider the behavior of a Sturm chain around the roots of its constituent polynomials.
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| First, note that two adjacent polynomials can share a common root only when it is a multiple root of ''p'' (in which case it is a root of every ''p<sub>i</sub>''). Indeed, if ''p''<sub>''i''</sub> and ''p''<sub>''i''−1</sub> are both zero at <math>\xi</math>, then ''p''<sub>''i''+1</sub> also have to be zero at <math>\xi</math>, since <math>\operatorname{sign}(p_{i-1}(\xi))= -\operatorname{sign}(p_{i+1}(\xi))</math>. The zero then propagates recursively up and down the chain, so that <math>\xi</math> is a root of all the polynomials ''p''<sub>0</sub>, ..., ''p<sub>m</sub>''.
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| Next, consider roots of polynomials in the interior (i.e., <math>0<i<m</math>) of the Sturm chain that are not multiple roots of ''p''. If <math>p_i(\xi)=0</math>, then from the previous paragraph it is true that <math>p_{i-1}(\xi)\ne 0</math> and <math>p_{i+1}(\xi)\ne 0</math>. Furthermore, <math>\operatorname{sign}(p_{i-1}(\xi))= -\operatorname{sign}(p_{i+1}(\xi))</math>. Since <math>p_{i-1}</math> and <math>p_{i+1}</math> are continuous,
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| <math>P_{i+1}(x)=-P_{i-1}(x)</math> for every ''x'' in the vicinity of <math>\xi</math>.
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| Similarly, the sign of <math>P_{i}(x)</math> is constant before and after <math>\xi</math>, and changes as <math>x</math> is crossing <math>\xi</math>. Thus, whenever <math>x</math> is crossing <math>\xi</math>, moving from left to right, the part ''p''<sub>''i''−1</sub>, ''p''<sub>''i''</sub>, ''p''<sub>''i''+1</sub> of the Sturm chain looses a sign change at one side, and acquires a new sign change at the other side. Consequently, the total number of sign changes is never affected by the polynomial variations in the interior of the chain, and only roots of the original polynomial, at the top of the chain, can affect the total number of sign changes.
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| Consider a root <math>\xi</math>, so <math>p(\xi)=0</math>, and assume first that it is a simple root. Then ''p'''s derivative, which is ''p''<sub>1</sub>, must be non-zero at <math>\xi</math>, so ''p'' must be either increasing or decreasing at <math>\xi</math>. If it's increasing, then its sign is changing from negative to positive when moving from left to right while its derivative (again, ''p''<sub>1</sub>) is positive, so the total number of sign changes decreases by one. Conversely, if it's decreasing, then its sign changes from positive to negative while its derivative is negative, so again the total number of sign changes decreases by one.
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| Finally, let <math>\xi</math> be a multiple root of ''p'', and let ''p''<sub>0</sub>, ..., ''p<sub>m</sub>'' be the canonical Sturm chain. Let ''d'' = gcd(''p'',''p{{'}}''), ''q'' = ''p''/''d'', and let ''q''<sub>0</sub>, ..., ''q<sub>m{{'}}</sub>'' be the canonical Sturm chain of ''q''. Then ''m'' = ''m{{'}}'' and ''p<sub>i</sub>'' = ''q<sub>i</sub>d'' for every ''i''. In particular, σ(''x'') is the same for both chains whenever ''x'' is not a root of ''d''. Then the number of sign changes (in either chain) around <math>\xi</math> decreases by one, since <math>\xi</math> is a simple root of ''q''.
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| In summary, only sign changes at roots of the original polynomial affect the total number of sign changes across the chain, and the total number of sign changes always decreases by one as roots are passed. The theorem follows directly.
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| ==History section and other related methods==
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| For counting and isolating the real roots, other methods are usually preferred, because they are computationally more efficient; these methods all use [[Descartes' rule of signs]] (just like Fourier<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> did back in 1820) and [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]]. Interestingly, the very first one of those methods<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> was initially called "modified Uspensky's algorithm" by its inventors, but it was later shown that there is no Uspensky's method;<ref name="akritas">{{cite book|last=Akritas|first=Alkiviadis G.|title=There is 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> afterwards, people started calling it either "Collins-Akritas method"<ref name=Descartes>{{cite book|last=Akritas|first=Alkiviadis G.|title=There is no "Descartes' method"|url=http://inf-server.inf.uth.gr/~akritas/articles/71.pdf|year=2008|publisher=In: M.J.Wester and M. Beaudin (Eds), Computer Algebra in Education, AullonaPress, USA, pp. 19-35}}</ref> or "Descartes' method" only to be shown that there is no Descartes' method<ref name="Descartes"/> either. Finally, François Boulier, of the University of Lille,<ref>{{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. 24, gave it the name "[[Vincent's theorem#Vincent–Collins–Akritas (VCA, 1976)|Vincent-Collins-Akritas]]" (VCA for short) to also give credit to Vincent. VCA is a bisection method; there exists also a continued fractions method based on [[Budan's theorem#Vincent's theorem (1834 and 1836)|Vincent's theorem]] namely, 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>
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| VAS is based on [[Budan's theorem#Budan's theorem|Budan's theorem]] whereas Sturm's method has been inspired by [[Budan's theorem#Fourier's theorem|Fourier's theorem]]. In fact Sturm himself,<ref>{{cite journal|last=Hourya|first=Benis-Sinaceur|title=Deux moments dans l'histoire du Théorème d'algèbre de Ch. F. Sturm|journal= In: Revue d'histoire des sciences|year=1988|volume=41|number=2|pages=99–132|url=http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1988_num_41_2_4093}}</ref> p. 108, acknowledges the great influence [[Budan's theorem#Fourier's theorem|Fourier's theorem]] had on him: « C'est en m'appuyant sur les principes qu'il a posés, et en imitant ses démonstrations, que j'ai trouvé les nouveaux théorèmes que je vais énoncer. » which translates to "It is by relying upon the principles he has laid out and by imitating his proofs that I have found the new theorems which I am about to announce."
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| ==Applications==
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| ===Number of real roots===
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| Sturm's theorem can be used to compute the total number of real roots of a polynomial.
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| This may be done by choosing -''a''=''b''=''M'' where ''M'' is larger than the absolute value of every root. For example, a bound due to [[Augustin Louis Cauchy|Cauchy]] says that all real roots of a polynomial with coefficients ''a''<sub>''i''</sub> are in the interval [−''M'', ''M''], where
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| :<math>M = 1 + \frac{\max_{i=0}^{n-1} |a_i|}{|a_n|} . \,\! </math>
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| Although theoretically the above approach is the simplest, in practice bounds on the positive (only) roots of polynomials are used and the positive roots are isolated and evaluated first; the negative roots are treated by first substituting x by -x, then compute a new (positive root) bound to isolate and evaluate the negative roots. Sturm's method and VCA need to compute only one bound to isolate the positive roots. By contrast, to isolate the positive roots VAS needs to compute various positive bounds, for the various polynomials that appear in the process. Efficient bounds on the values of the positive roots are described in P. S. Vigklas' Ph.D. Thesis<ref>{{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 in.<ref>{{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>
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| Another method is computationally simpler. One can use the fact that for large ''x'', the sign of
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| :<math>p(x)=a_n x^n+\cdots \,\! </math>
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| is <math>\text{sign}(a_n)</math>, whereas <math>\text{sign}(p(-x))</math> is <math>(-1)^n a_n.</math>.
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| In this way, simply counting the sign changes in the leading coefficients in the Sturm chain readily gives the number of distinct real roots of a polynomial.
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| Sturm's theorem allows also to determine the multiplicity of a given root, say ξ. Indeed, suppose that ''a''< ξ <''b'', with σ(''a'') − σ(''b'') = 1. Then, ξ has multiplicity ''k'' precisely when ξ is a root with multiplicity ''k'' − 1 of ''p''<sub>''m''</sub> (since it is the GCD of ''p'' and its derivative). Thus the multiplicity of ξ may be computed by recursively applying Sturm's theorem to ''p''<sub>''m''</sub>. However, this method is rarely used, as [[square-free factorization]] is computationally more efficient for this purpose.
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| ===Quotient===
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| The remainder is needed to compute the chain using Euclid's algorithm. For two polynomials <math>p(x)=\sum_{k=0}^i p_k x^k</math> and <math>\tilde p(x)=\sum_{k=0}^{i-1}\tilde p_k x^k</math> this is accomplished (for non-vanishing <math>\tilde p_{i-1}</math>) by
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| :<math>{\rm rem}(p, \tilde p)= \tilde p_{i-1}\cdot p(x)- p_i\cdot\tilde p(x)\left(x+\frac{p_{i-1}}{p_i}-\frac{\tilde p_{i-2}}{\tilde p_{i-1}}\right),</math>
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| where the quotient is built solely of the first two leading coefficients.
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| ==Generalized Sturm chains==
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| Let ξ be in the compact interval [''a'',''b'']. A '''generalized Sturm chain''' over [''a'',''b''] is a finite sequence of real polynomials (''X''<sub>0</sub>,''X''<sub>1</sub>,…,''X''<sub>''r''</sub>) such that:
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| #''X''(''a'')''X''(''b'') ≠ 0
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| #sign(''X''<sub>''r''</sub>) is constant on [''a'',''b'']
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| #If ''X''<sub>''i''</sub>(ξ) = 0 for 1 ≤ ''i'' ≤ ''r''−1, then ''X''<sub>''i''−1</sub>(ξ)''X''<sub>''i''+1</sub>(ξ) < 0.
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| One can check that each Sturm chain is indeed a generalized Sturm chain.
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| ==See also==
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| * [[Routh–Hurwitz theorem]]
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| * [[Hurwitz's theorem (complex analysis)]]
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| * [[Descartes' rule of signs]]
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| * [[Rouché's theorem]]
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| * [[Properties of polynomial roots]]
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| * [[Gauss–Lucas theorem]]
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| * [[Turán's inequalities]]
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| * D.G. Hook and P.R. McAree, "Using Sturm Sequences To Bracket Real Roots of Polynomial Equations" in Graphic Gems I (A. Glassner ed.), Academic Press, p. 416–422, 1990.
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| ==References==
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| {{reflist}}
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| * {{cite journal
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| |first1=Jacques Charles François
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| |last1=Sturm
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| |year=1829
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| |title=Mémoire sur la résolution des équations numériques
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| |journal=Bulletin des Sciences de Férussac
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| |volume=11
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| |pages=419–425
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| }}
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| * {{cite journal
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| |first1=J. J.
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| |last1=Sylvester
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| |year=1853
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| |title=On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions, and that of the greatest algebraical common measure
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| |journal=Phil. Trans. Roy. Soc. London
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| |volume=143
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| |pages=407–548
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| |jstor=108572
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| }}
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| * {{cite journal
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| |first1=Joseph Miller
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| |last1=Thomas
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| |title=Sturm's theorem for multiple roots
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| |year=1941
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| |journal=National Mathematics Magazine
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| |volume=15
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| |number=8
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| |pages=391–394
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| |jstor=3028551
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| |mr=0005945
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| }}
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| * {{citation
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| |first1=Lee E.
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| |last1=Heindel
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| |title=Integer arithmetic algorithms for polynomial real zero determination
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| |journal=Proc. SYMSAC '71
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| |year=1971
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| |page=415
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| |mr=0300434
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| }}
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| * {{cite journal
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| |first1=Don B.
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| |last1=Panton
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| |first2=William A.
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| |last2=Verdini
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| |title=A fortran program for applying Sturm's theorem in counting internal rates of return
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| |journal=J. Financ. Quant. Anal.
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| |year=1981
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| |volume=16
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| |number=3
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| |pages=381–388
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| |jstor=2330245
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| }}
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| * {{cite journal
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| |title=Reflections on a pair of theorems by Budan and Fourier
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| |journal=Math. Mag.
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| |first1=Alkiviadis G.
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| |last1=Akritas
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| |mr=0678195
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| |year=1982
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| |volume=55
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| |number=5
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| |pages=292–298
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| |jstor=2690097
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| }}
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| * {{cite journal
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| |title=Multivariate Sturm theory
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| |journal=Lecture Notes in Comp. Science
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| |year=1991
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| |volume=539
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| |pages=318–332
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| |mr=1229329
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| |doi=10.1007/3-540-54522-0_120
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| |first1=Paul
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| |last1=Petersen
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| }}
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| * {{cite book|first1=Chee
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| |last1=Yap
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| |url=http://www.cs.nyu.edu/yap/book/berlin/
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| |title=Fundamental Problems in Algorithmic Algebra
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| |publisher=[[Oxford University Press]]
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| |year=2000
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| |isbn=0-19-512516-9
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| }}
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| * {{cite book | last1=Rahman | first1=Q. I. | last2=Schmeisser | first2=G. | title=Analytic theory of polynomials | series=London Mathematical Society Monographs. New Series | volume=26 | location=Oxford | publisher=[[Oxford University Press]] | year=2002 | isbn=0-19-853493-0 | zbl=1072.30006 }}
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| *Baumol, William. ''Economic Dynamics'', section "Sturm's Theorem"
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| ==External links==
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| * [http://tog.acm.org/resources/GraphicsGems/gems/Sturm/ C code] from Graphic Gems by D.G. Hook and P.R. McAree.
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| {{DEFAULTSORT:Sturm's Theorem}}
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| [[Category:Theorems in real analysis]]
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| [[Category:Articles containing proofs]]
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| [[Category:Polynomials]]
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| [[Category:Computer algebra]]
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