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Diagram of a golden section search

The golden section search is a technique for finding the extremum (minimum or maximum) of a strictly unimodal function by successively narrowing the range of values inside which the extremum is known to exist. The technique derives its name from the fact that the algorithm maintains the function values for triples of points whose distances form a golden ratio. The algorithm is the limit of Fibonacci search (also described below) for a large number of function evaluations. Fibonacci search and Golden section search were discovered by Kiefer (1953). (see also Avriel and Wilde (1966)).

Basic idea

The diagram above illustrates a single step in the technique for finding a minimum. The functional values of $f (x)$ are on the vertical axis, and the horizontal axis is the x parameter. The value of $f (x)$ has already been evaluated at the three points: $x_{1}$ , $x_{2}$ , and $x_{3}$ . Since $f_{2}$ is smaller than either $f_{1}$ or $f_{3}$ , it is clear that a minimum lies inside the interval from $x_{1}$ to $x_{3}$ (since f is unimodal).

The next step in the minimization process is to "probe" the function by evaluating it at a new value of x, namely $x_{4}$ . It is most efficient to choose $x_{4}$ somewhere inside the largest interval, i.e. between $x_{2}$ and $x_{3}$ . From the diagram, it is clear that if the function yields $f_{4 a}$ then a minimum lies between $x_{1}$ and $x_{4}$ and the new triplet of points will be $x_{1}$ , $x_{2}$ , and $x_{4}$ . However if the function yields the value $f_{4 b}$ then a minimum lies between $x_{2}$ and $x_{3}$ , and the new triplet of points will be $x_{2}$ , $x_{4}$ , and $x_{3}$ . Thus, in either case, we can construct a new narrower search interval that is guaranteed to contain the function's minimum.

Probe point selection

From the diagram above, it is seen that the new search interval will be either between $x_{1}$ and $x_{4}$ with a length of a+c , or between $x_{2}$ and $x_{3}$ with a length of b . The golden section search requires that these intervals be equal. If they are not, a run of "bad luck" could lead to the wider interval being used many times, thus slowing down the rate of convergence. To ensure that b = a+c, the algorithm should choose $x_{4} = x_{1} + (x_{3} - x_{2})$ .

However there still remains the question of where $x_{2}$ should be placed in relation to $x_{1}$ and $x_{3}$ . The golden section search chooses the spacing between these points in such a way that these points have the same proportion of spacing as the subsequent triple $x_{1}, x_{2}, x_{4}$ or $x_{2}, x_{4}, x_{3}$ . By maintaining the same proportion of spacing throughout the algorithm, we avoid a situation in which $x_{2}$ is very close to $x_{1}$ or $x_{3}$ , and guarantee that the interval width shrinks by the same constant proportion in each step.

Mathematically, to ensure that the spacing after evaluating $f (x_{4})$ is proportional to the spacing prior to that evaluation, if $f (x_{4})$ is $f_{4 a}$ and our new triplet of points is $x_{1}$ , $x_{2}$ , and $x_{4}$ then we want:

\frac{c}{a} = \frac{a}{b} .

However, if $f (x_{4})$ is $f_{4 b}$ and our new triplet of points is $x_{2}$ , $x_{4}$ , and $x_{3}$ then we want:

\frac{c}{(b - c)} = \frac{a}{b} .

Eliminating c from these two simultaneous equations yields:

{(\frac{b}{a})}^{2} = \frac{b}{a} + 1

or

\frac{b}{a} = φ

where φ is the golden ratio:

φ = \frac{1 + \sqrt{5}}{2} = 1.618033988 \dots

The appearance of the golden ratio in the proportional spacing of the evaluation points is how this search algorithm gets its name.

Termination condition

In addition to a routine for reducing the size of the bracketing of the solution, a complete algorithm must have a termination condition. The one provided in the book Numerical Recipes in C is based on testing the gaps among $x_{1}$ , $x_{2}$ , $x_{3}$ and $x_{4}$ , terminating when within the relative accuracy bounds:

| x_{3} - x_{1} | < τ (| x_{2} | + | x_{4} |)

where $τ$ is a tolerance parameter of the algorithm and $| x |$ is the absolute value of $x$ . The check is based on the bracket size relative to its central value, because that relative error in $x$ is approximately proportional to the squared absolute error in $f (x)$ in typical cases. For that same reason, the Numerical Recipes text recommends that $τ = \sqrt{ϵ}$ where $ϵ$ is the required absolute precision of $f (x)$ .

Recursive algorithm

double phi = (1 + Math.sqrt(5)) / 2;
double resphi = 2 - phi;

// a and c are the current bounds; the minimum is between them.
// b is a center point
// f(x) is some mathematical function elsewhere defined
// a corresponds to x1; b corresponds to x2; c corresponds to x3
// x corresponds to x4
// tau is a tolerance parameter; see above

public double goldenSectionSearch(double a, double b, double c, double tau) {
    double x;
    if (c - b > b - a)
      x = b + resphi * (c - b);
    else
      x = b - resphi * (b - a);
    if (Math.abs(c - a) < tau * (Math.abs(b) + Math.abs(x))) 
      return (c + a) / 2; 
    assert(f(x) != f(b));
    if (f(x) < f(b)) {
      if (c - b > b - a) return goldenSectionSearch(b, x, c, tau);
      else return goldenSectionSearch(a, x, b, tau);
    }
    else {
      if (c - b > b - a) return goldenSectionSearch(a, b, x, tau);
      else return goldenSectionSearch(x, b, c, tau);
    }
  }

To realise the advantage of golden section search, the function $f (x)$ would be implemented with caching, so that in all invocations of goldenSectionSearch(..) above, except the first, $f (x_{2})$ would have already been evaluated previously — the result of the calculation will be re-used, bypassing the (perhaps expensive) explicit evaluation of the function. Together with a slightly smaller number of recursions, this 50% saving in the number of calls to $f (x)$ is the main algorithmic advantage over Ternary search.

Fibonacci search

A very similar algorithm can also be used to find the extremum (minimum or maximum) of a sequence of values that has a single local minimum or local maximum. In order to approximate the probe positions of golden section search while probing only integer sequence indices, the variant of the algorithm for this case typically maintains a bracketing of the solution in which the length of the bracketed interval is a Fibonacci number. For this reason, the sequence variant of golden section search is often called Fibonacci search.

Fibonacci search was first devised by Kiefer (1953) as a minimax search for the maximum (minimum) of a unimodal function in an interval.

References

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Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Template:Optimization algorithms

@@ Line 1: / Line 1: @@
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+[[Image:GoldenSectionSearch.png|thumb|325px|right|Diagram of a golden section search]]
+The '''golden section search''' is a technique for finding the [[extremum]] (minimum or maximum) of a strictly [[unimodal function]] by successively narrowing the range of values inside which the extremum is known to exist. The technique derives its name from the fact that the algorithm maintains the function values for triples of points whose distances form a [[golden ratio]]. The algorithm is the limit of  [[Fibonacci search]] (also described below) for a large number of function evaluations. Fibonacci search and Golden section search were discovered by [[Jack Kiefer (mathematician)|Kiefer]] (1953). (see also Avriel and Wilde (1966)).
+==Basic idea==
+The diagram above illustrates a single step in the technique for finding a minimum. The functional values of <math>f(x)</math> are on the vertical axis, and the horizontal axis is the ''x'' parameter. The value of <math>f(x)</math> has already been evaluated at the three points: <math>x_1</math>, <math>x_2</math>, and <math>x_3</math>. Since <math>f_2</math> is smaller than either <math>f_1</math> or <math>f_3</math>, it is clear that a minimum lies inside the interval from <math>x_1</math> to <math>x_3</math> (since ''f'' is [[Unimodal function|unimodal]]).
+The next step in the minimization process is to "probe" the function by evaluating it at a new value of ''x'', namely <math>x_4</math>. It is most efficient to choose <math>x_4</math> somewhere inside the largest interval, i.e. between <math>x_2</math> and <math>x_3</math>.  From the diagram, it is clear that if the function yields <math>f_{4a}</math> then a minimum lies between <math>x_1</math> and <math>x_4</math> and the new triplet of points will be <math>x_1</math>, <math>x_2</math>, and <math>x_4</math>. However if the function yields the value <math>f_{4b}</math> then a minimum lies between <math>x_2</math> and <math>x_3</math>, and the new triplet of points will be <math>x_2</math>, <math>x_4</math>, and <math>x_3</math>. Thus, in either case, we can construct a new narrower search interval that is guaranteed to contain the function's minimum.
+==Probe point selection==
+From the diagram above, it is seen that the new search interval will be either between <math>x_1</math> and <math>x_4</math> with a length of ''a''+''c''&nbsp;, or between <math>x_2</math> and <math>x_3</math> with a length of ''b''&nbsp;. The golden section search requires that these intervals be equal. If they are not, a run of "bad luck" could lead to the wider interval being used many times, thus slowing down the rate of convergence. To ensure that ''b'' = ''a''+''c'', the algorithm should choose <math>x_4=x_1+(x_3-x_2)</math>.
+However there still remains the question of where <math>x_2</math> should be placed in relation to <math>x_1</math> and <math>x_3</math>. The golden section search chooses the spacing between these points in such a way that these points have the same proportion of spacing as the subsequent triple <math>x_1,x_2,x_4</math> or <math>x_2,x_4,x_3</math>. By maintaining the same proportion of spacing throughout the algorithm, we avoid a situation in which <math>x_2</math> is very close to <math>x_1</math> or <math>x_3</math>, and guarantee that the interval width shrinks by the same constant proportion in each step.
+Mathematically, to ensure that the spacing after evaluating <math>f(x_4)</math> is proportional to the spacing prior to that evaluation, if <math>f(x_4)</math> is <math>f_{4a}</math> and our new triplet of points is <math>x_1</math>, <math>x_2</math>, and <math>x_4</math> then we want:
+:<math>\frac{c}{a}=\frac{a}{b}.</math>
+However, if <math>f(x_4)</math> is <math>f_{4b}</math> and our new triplet of points is <math>x_2</math>, <math>x_4</math>, and <math>x_3</math> then we want:
+:<math>\frac{c}{(b-c)}=\frac{a}{b}.</math>
+Eliminating ''c'' from these two simultaneous equations yields:
+:<math>\left(\frac{b}{a}\right)^2=\frac{b}{a}+1</math>
+or
+:<math>\frac{b}{a}=\varphi</math>
+where φ&nbsp; is the [[golden ratio]]:
+:<math>\varphi= \frac{1+\sqrt{5}}{2}= 1.618033988\ldots</math>
+The appearance of the golden ratio in the proportional spacing of the evaluation points is how this search [[algorithm]] gets its name.
+==Termination condition==
+In addition to a routine for reducing the size of the bracketing of the solution, a complete algorithm must have a termination condition. The one provided in the book ''Numerical Recipes in C'' is based on testing the gaps among <math>x_1</math>, <math>x_2</math>, <math>x_3</math> and <math>x_4</math>, terminating when within the relative accuracy bounds:
+:<math>|x_3 - x_1| < \tau (|x_2| + |x_4|)\,</math>
+where <math>\tau</math> is a tolerance parameter of the algorithm and <math>|x|</math> is the [[absolute value]] of <math>x</math>.  The check is based on the bracket size relative to its central value, because that relative error in <math>x</math> is approximately proportional to the squared absolute error in <math>f(x)</math> in typical cases. For that same reason, the Numerical Recipes text recommends that <math>\tau = \sqrt{\epsilon}</math> where <math>\epsilon</math> is the required absolute precision of <math>f(x)</math>.
+==Recursive algorithm==
+<source lang="java">
+double phi = (1 + Math.sqrt(5)) / 2;
+double resphi = 2 - phi;
+// a and c are the current bounds; the minimum is between them.
+// b is a center point
+// f(x) is some mathematical function elsewhere defined
+// a corresponds to x1; b corresponds to x2; c corresponds to x3
+// x corresponds to x4
+// tau is a tolerance parameter; see above
+public double goldenSectionSearch(double a, double b, double c, double tau) {
+    double x;
+    if (c - b > b - a)
+      x = b + resphi * (c - b);
+    else
+      x = b - resphi * (b - a);
+    if (Math.abs(c - a) < tau * (Math.abs(b) + Math.abs(x)))
+      return (c + a) / 2;
+    assert(f(x) != f(b));
+    if (f(x) < f(b)) {
+      if (c - b > b - a) return goldenSectionSearch(b, x, c, tau);
+      else return goldenSectionSearch(a, x, b, tau);
+    }
+    else {
+      if (c - b > b - a) return goldenSectionSearch(a, b, x, tau);
+      else return goldenSectionSearch(x, b, c, tau);
+    }
+  }</source>
+To realise the advantage of golden section search, the function <math>f(x)</math> would be implemented with caching, so that in all invocations of <code>goldenSectionSearch(..)</code> above, except the first, <math>f(x_2)</math> would have already been evaluated previously — the result of the calculation will be re-used, bypassing the (perhaps expensive) explicit evaluation of the function. Together with a slightly smaller number of recursions, this 50% saving in the number of calls to <math>f(x)</math> is the main algorithmic advantage over [[Ternary search]].
+==Fibonacci search==
+A very similar algorithm can also be used to find the [[extremum]] (minimum or maximum) of a [[sequence]] of values that has a single local minimum or local maximum. In order to approximate the probe positions of golden section search while probing only integer sequence indices, the variant of the algorithm for this case typically maintains a bracketing of the solution in which the length of the bracketed interval is a [[Fibonacci number]]. For this reason, the sequence variant of golden section search is often called ''[[Fibonacci search technique|Fibonacci search]]''.
+Fibonacci search was first devised by [[Jack Kiefer (statistician)|Kiefer]] (1953) as a [[minimax]] search for the maximum (minimum) of a unimodal function in an interval.
+;;;
+==See also==
+* [[Fibonacci search technique]]
+* [[Brent's method]]
+* [[Binary search]]
+==References==
+{{Reflist}}
+*{{citation
+ | last = Kiefer
+ | first = J.
+ | authorlink = Jack Kiefer (mathematician)
+ | title = Sequential minimax search for a maximum
+ | jstor = 2032161
+ | journal = [[Proceedings of the American Mathematical Society]]
+ | volume = 4
+ | year = 1953
+ | issue = 3
+ | pages = 502–506
+ | mr = 0055639
+ | doi = 10.2307/2032161}}
+*{{citation
+ | last1 = Avriel
+ | first1 = Mordecai
+ | last2 = Wilde
+ | first2 = Douglass J.
+ | title = Optimality proof for the symmetric Fibonacci search technique
+ | journal = [[Fibonacci Quarterly]]
+ | volume = 4
+ | year = 1966
+ | pages = 265–269
+ | mr = 0208812 }}
+*{{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press |  publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 10.2. Golden Section Search in One Dimension | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=492}}
+{{Optimization algorithms}}
+{{DEFAULTSORT:Golden Section Search}}
+[[Category:Golden ratio]]
+[[Category:Fibonacci numbers]]
+[[Category:Optimization algorithms and methods]]
+[[Category:Articles with example Java code]]