golden_search_extrema.cpp File Reference

Find extrema of a univariate real function in a given interval using golden section search algorithm. More...

#include <cassert>
#include <cmath>
#include <functional>
#include <iostream>
#include <limits>

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Macros
#define	_USE_MATH_DEFINES
#define	EPSILON   1e-7
	solution accuracy limit

Functions
double	get_minima (const std::function< double(double)> &f, double lim_a, double lim_b)
	Get the minima of a function in the given interval. To get the maxima, simply negate the function. The golden ratio used here is:
void	test1 ()
	Test function to find minima for the function \(f(x)= (x-2)^2\) in the interval \([1,5]\) Expected result = 2.
void	test2 ()
	Test function to find maxima for the function \(f(x)= x^{\frac{1}{x}}\) in the interval \([-2,10]\) Expected result: \(e\approx 2.71828182845904509\).
void	test3 ()
	Test function to find maxima for the function \(f(x)= \cos x\) in the interval \([0,12]\) Expected result: \(\pi\approx 3.14159265358979312\).
int	main ()

Detailed Description

Find extrema of a univariate real function in a given interval using golden section search algorithm.

See also

brent_method_extrema.cpp

Author

Krishna Vedala

Function Documentation

get_minima()

double get_minima	(	const std::function< double(double)> &	f,
		double	lim_a,
		double	lim_b )

Get the minima of a function in the given interval. To get the maxima, simply negate the function. The golden ratio used here is:

\[ k=\frac{3-\sqrt{5}}{2} \approx 0.381966\ldots\]

Parameters

f	function to get minima for
lim_a	lower limit of search window
lim_b	upper limit of search window

Returns

local minima found in the interval

                                {
    uint32_t iters = 0;
    double c, d;
    double prev_mean, mean = std::numeric_limits<double>::infinity();
 
    // golden ratio value
    const double M_GOLDEN_RATIO = (1.f + std::sqrt(5.f)) / 2.f;
 
    // ensure that lim_a < lim_b
    if (lim_a > lim_b) {
        std::swap(lim_a, lim_b);
    } else if (std::abs(lim_a - lim_b) <= EPSILON) {
        std::cerr << "Search range must be greater than " << EPSILON << "\n";
        return lim_a;
    }
 
    do {
        prev_mean = mean;
 
        // compute the section ratio width
        double ratio = (lim_b - lim_a) / M_GOLDEN_RATIO;
        c = lim_b - ratio;  // right-side section start
        d = lim_a + ratio;  // left-side section end
 
        if (f(c) < f(d)) {
            // select left section
            lim_b = d;
        } else {
            // selct right section
            lim_a = c;
        }
 
        mean = (lim_a + lim_b) / 2.f;
        iters++;
 
        // continue till the interval width is greater than sqrt(system epsilon)
    } while (std::abs(lim_a - lim_b) > EPSILON);
 
    std::cout << " (iters: " << iters << ") ";
    return prev_mean;
}

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main()

int main

(

void

)

Main function

           {
    std::cout.precision(9);
 
    std::cout << "Computations performed with machine epsilon: " << EPSILON
              << "\n";
 
    test1();
    test2();
    test3();
 
    return 0;
}

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test1()

void test1

(

)

Test function to find minima for the function \(f(x)= (x-2)^2\) in the interval \([1,5]\)
Expected result = 2.

             {
    // define the function to minimize as a lambda function
    std::function<double(double)> f1 = [](double x) {
        return (x - 2) * (x - 2);
    };
 
    std::cout << "Test 1.... ";
 
    double minima = get_minima(f1, 1, 5);
 
    std::cout << minima << "...";
 
    assert(std::abs(minima - 2) < EPSILON);
    std::cout << "passed\n";
}

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test2()

void test2

(

)

Test function to find maxima for the function \(f(x)= x^{\frac{1}{x}}\) in the interval \([-2,10]\)
Expected result: \(e\approx 2.71828182845904509\).

             {
    // define the function to maximize as a lambda function
    // since we are maximixing, we negated the function return value
    std::function<double(double)> func = [](double x) {
        return -std::pow(x, 1.f / x);
    };
 
    std::cout << "Test 2.... ";
 
    double minima = get_minima(func, -2, 10);
 
    std::cout << minima << " (" << M_E << ")...";
 
    assert(std::abs(minima - M_E) < EPSILON);
    std::cout << "passed\n";
}

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test3()

void test3

(

)

Test function to find maxima for the function \(f(x)= \cos x\) in the interval \([0,12]\)
Expected result: \(\pi\approx 3.14159265358979312\).

             {
    // define the function to maximize as a lambda function
    // since we are maximixing, we negated the function return value
    std::function<double(double)> func = [](double x) { return std::cos(x); };
 
    std::cout << "Test 3.... ";
 
    double minima = get_minima(func, -4, 12);
 
    std::cout << minima << " (" << M_PI << ")...";
 
    assert(std::abs(minima - M_PI) < EPSILON);
    std::cout << "passed\n";
}

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