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| [[Image:GoldenSectionSearch.png|thumb|325px|right|Diagram of a golden section search]]
| | William Lagakos confirms that Wolf is correct in his analysis. Rich dairy products provide exactly the right amount of saturated fat required by the eating plan. Just because smoothies are created from natural basic elements, the chemical preservatives are way reduced also. Tim Mc - Graw is in the best shape of his life at 47 after giving up alcohol and drugs and adopting the low-carb Paleo diet and Cross - Fit workouts. Worrying about how healthy your food is isn’t really an issue with this diet. <br><br>So specifically exactly what really does one particular distinct eat and also by no implies eat right after getting Paleo Recipe Book. We feel sluggish and poor simply because we are not giving our bodies fuel. Ready-made paleo foods items are starting to find their way into conventional grocery stores. In addition, food is something that we can't turn down yet healthy foods is another matter to discuss. A diet that is rich with calcium may help to prevent tooth decay, but when a diet is low in calcium, as a majority of American diets are, the body takes that mineral from teeth and bones. <br><br>James, who famously returned to the Cleveland Cavaliers, is following in the fork-steps of his buddy, Miami Heat point guard Dwyane Wade, who has been adhering to the Paleo plan on the advice of fellow Heat player Ray Allen, who has been eating similarly since 2013. The results showed the Paleo dieters lost significantly more weight (14 pounds) compared to the low-fat dieters (5. Those that follow this program can expect a lot of red meat as well as fatty food. One thing that you will want to do is keep track of your weight to see if you are on the right rack to a healthier life. Varied cooking used in dessert, shakes, and drinks and baking. <br><br>Enjoy Paleo Chicken Stir Fry Recipe and Paleo Beef Spinach Salad Recipes. Chives: Used in salads, soups, stews and appetizers. Lose weight naturally by focusing on eating healthy Be re-educated on what it means to eat right Learn how to prepare healthy foods Feel encouraged to explore foods you may have never eaten before Discover how to shop differently for healthier meals and snacks Never get tired of eating healthy with so many recipes to choose from. Advocates who have adhered to the paleo diet can attest to the range of paleo foods there is to choose from -- many are already on your list of routinely consumed foods. The secret to poaching is to have a very tight lid that will let the whole meat piece cook evenly at low heat with the steam the liquid produces. <br><br>They consist of strict plans created by nutritionists who don’t enjoy the same things in life we do. * No need to rack your mind, working out caloric intake. If the preferred grocery store would not give natural merchandise, then it is actually in all probability the perfect time to store someplace else. You want to find recipes that will fit into your easy paleo diet plan. Summary Our bodies are designed to react to what we put in them. <br><br>Allen, 39, came back from the summer break in significantly better shape than he was the previous year after adopting a low-carb, sugar-free Paleo diet. "I'll be the first to admit I take it to extremes," he said. However, there are some exceptions, along with some specific foods that this diet is centered around; furthermore, it is crucial to stick to these foods. After all, if the animals have been consuming processed foods, then you won't get the levels of protein that you are looking for. One of these may be the foods by it tend to be scrumptious.<br><br>Here is more information on [http://gritsandgroceries.info/ paleo plan recipes] look at the web-site. |
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| 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)).
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| ==Basic idea==
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| 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]]).
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| 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.
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| ==Probe point selection==
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| 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'' , or between <math>x_2</math> and <math>x_3</math> 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 <math>x_4=x_1+(x_3-x_2)</math>.
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| 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.
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| 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:
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| :<math>\frac{c}{a}=\frac{a}{b}.</math>
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| 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:
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| :<math>\frac{c}{(b-c)}=\frac{a}{b}.</math>
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| Eliminating ''c'' from these two simultaneous equations yields:
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| :<math>\left(\frac{b}{a}\right)^2=\frac{b}{a}+1</math>
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| or
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| :<math>\frac{b}{a}=\varphi</math>
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| where φ is the [[golden ratio]]:
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| :<math>\varphi= \frac{1+\sqrt{5}}{2}= 1.618033988\ldots</math>
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| The appearance of the golden ratio in the proportional spacing of the evaluation points is how this search [[algorithm]] gets its name.
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| ==Termination condition==
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| 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:
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| :<math>|x_3 - x_1| < \tau (|x_2| + |x_4|)\,</math>
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| 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>.
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| ==Recursive algorithm==
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| <source lang="java">
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| double phi = (1 + Math.sqrt(5)) / 2;
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| double resphi = 2 - phi;
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| // a and c are the current bounds; the minimum is between them.
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| // b is a center point
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| // f(x) is some mathematical function elsewhere defined
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| // a corresponds to x1; b corresponds to x2; c corresponds to x3
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| // x corresponds to x4
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| // tau is a tolerance parameter; see above
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| public double goldenSectionSearch(double a, double b, double c, double tau) {
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| double x;
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| if (c - b > b - a)
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| x = b + resphi * (c - b);
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| else
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| x = b - resphi * (b - a);
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| if (Math.abs(c - a) < tau * (Math.abs(b) + Math.abs(x)))
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| return (c + a) / 2;
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| assert(f(x) != f(b));
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| if (f(x) < f(b)) {
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| if (c - b > b - a) return goldenSectionSearch(b, x, c, tau);
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| else return goldenSectionSearch(a, x, b, tau);
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| }
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| else {
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| if (c - b > b - a) return goldenSectionSearch(a, b, x, tau);
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| else return goldenSectionSearch(x, b, c, tau);
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| }
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| }</source>
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| 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]].
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| ==Fibonacci search==
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| 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]]''.
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| 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.
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| ;;;
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| ==See also==
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| * [[Fibonacci search technique]]
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| * [[Brent's method]]
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| * [[Binary search]]
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| ==References==
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| {{Reflist}}
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| *{{citation
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| | last = Kiefer
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| | first = J.
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| | authorlink = Jack Kiefer (mathematician)
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| | title = Sequential minimax search for a maximum
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| | jstor = 2032161
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| | journal = [[Proceedings of the American Mathematical Society]]
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| | volume = 4
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| | year = 1953
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| | issue = 3
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| | pages = 502–506
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| | mr = 0055639
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| | doi = 10.2307/2032161}}
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| *{{citation
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| | last1 = Avriel
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| | first1 = Mordecai
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| | last2 = Wilde
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| | first2 = Douglass J.
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| | title = Optimality proof for the symmetric Fibonacci search technique
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| | journal = [[Fibonacci Quarterly]]
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| | volume = 4
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| | year = 1966
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| | pages = 265–269
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| | mr = 0208812 }}
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| *{{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}}
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| {{Optimization algorithms}}
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| {{DEFAULTSORT:Golden Section Search}}
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| [[Category:Golden ratio]]
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| [[Category:Fibonacci numbers]]
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| [[Category:Optimization algorithms and methods]]
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| [[Category:Articles with example Java code]]
| |
William Lagakos confirms that Wolf is correct in his analysis. Rich dairy products provide exactly the right amount of saturated fat required by the eating plan. Just because smoothies are created from natural basic elements, the chemical preservatives are way reduced also. Tim Mc - Graw is in the best shape of his life at 47 after giving up alcohol and drugs and adopting the low-carb Paleo diet and Cross - Fit workouts. Worrying about how healthy your food is isn’t really an issue with this diet.
So specifically exactly what really does one particular distinct eat and also by no implies eat right after getting Paleo Recipe Book. We feel sluggish and poor simply because we are not giving our bodies fuel. Ready-made paleo foods items are starting to find their way into conventional grocery stores. In addition, food is something that we can't turn down yet healthy foods is another matter to discuss. A diet that is rich with calcium may help to prevent tooth decay, but when a diet is low in calcium, as a majority of American diets are, the body takes that mineral from teeth and bones.
James, who famously returned to the Cleveland Cavaliers, is following in the fork-steps of his buddy, Miami Heat point guard Dwyane Wade, who has been adhering to the Paleo plan on the advice of fellow Heat player Ray Allen, who has been eating similarly since 2013. The results showed the Paleo dieters lost significantly more weight (14 pounds) compared to the low-fat dieters (5. Those that follow this program can expect a lot of red meat as well as fatty food. One thing that you will want to do is keep track of your weight to see if you are on the right rack to a healthier life. Varied cooking used in dessert, shakes, and drinks and baking.
Enjoy Paleo Chicken Stir Fry Recipe and Paleo Beef Spinach Salad Recipes. Chives: Used in salads, soups, stews and appetizers. Lose weight naturally by focusing on eating healthy Be re-educated on what it means to eat right Learn how to prepare healthy foods Feel encouraged to explore foods you may have never eaten before Discover how to shop differently for healthier meals and snacks Never get tired of eating healthy with so many recipes to choose from. Advocates who have adhered to the paleo diet can attest to the range of paleo foods there is to choose from -- many are already on your list of routinely consumed foods. The secret to poaching is to have a very tight lid that will let the whole meat piece cook evenly at low heat with the steam the liquid produces.
They consist of strict plans created by nutritionists who don’t enjoy the same things in life we do. * No need to rack your mind, working out caloric intake. If the preferred grocery store would not give natural merchandise, then it is actually in all probability the perfect time to store someplace else. You want to find recipes that will fit into your easy paleo diet plan. Summary Our bodies are designed to react to what we put in them.
Allen, 39, came back from the summer break in significantly better shape than he was the previous year after adopting a low-carb, sugar-free Paleo diet. "I'll be the first to admit I take it to extremes," he said. However, there are some exceptions, along with some specific foods that this diet is centered around; furthermore, it is crucial to stick to these foods. After all, if the animals have been consuming processed foods, then you won't get the levels of protein that you are looking for. One of these may be the foods by it tend to be scrumptious.
Here is more information on paleo plan recipes look at the web-site.