Brent Method Extrema

K
/**
* \file
* \brief Find real extrema of a univariate real function in a given interval
* using [Brent's method](https://en.wikipedia.org/wiki/Brent%27s_method).
*
* Refer the algorithm discoverer's publication
* [online](https://maths-people.anu.edu.au/~brent/pd/rpb011i.pdf) and also
* associated book:
* > R. P. Brent, Algorithms for Minimization without
* > Derivatives, Prentice-Hall, Englewood Cliffs, New Jersey, 1973
*
* \see golden_search_extrema.cpp
*
* \author [Krishna Vedala](https://github.com/kvedala)
*/
#define _USE_MATH_DEFINES  ///< required for MS Visual C++
#include <cassert>
#include <cmath>
#include <functional>
#include <iostream>
#include <limits>

#define EPSILON \
std::sqrt(  \
std::numeric_limits<double>::epsilon())  ///< system accuracy limit

/**
* @brief Get the real root of a function in the given interval.
*
* @param f function to get root for
* @param lim_a lower limit of search window
* @param lim_b upper limit of search window
* @return root found in the interval
*/
double get_minima(const std::function<double(double)> &f, double lim_a,
double lim_b) {
uint32_t iters = 0;

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;
}

// golden ratio value
const double M_GOLDEN_RATIO = (3.f - std::sqrt(5.f)) / 2.f;

double v = lim_a + M_GOLDEN_RATIO * (lim_b - lim_a);
double u, w = v, x = v;
double fu, fv = f(v);
double fw = fv, fx = fv;

double mid_point = (lim_a + lim_b) / 2.f;
double p = 0, q = 0, r = 0;

double d, e = 0;
double tolerance, tolerance2;

do {
mid_point = (lim_a + lim_b) / 2.f;
tolerance = EPSILON * std::abs(x);
tolerance2 = 2 * tolerance;

if (std::abs(e) > tolerance2) {
// fit parabola
r = (x - w) * (fx - fv);
q = (x - v) * (fx - fw);
p = (x - v) * q - (x - w) * r;
q = 2.f * (q - r);
if (q > 0)
p = -p;
else
q = -q;
r = e;
e = d;
}

if (std::abs(p) < std::abs(0.5 * q * r) && p < q * (lim_b - x)) {
// parabolic interpolation step
d = p / q;
u = x + d;
if (u - lim_a < tolerance2 || lim_b - u < tolerance2)
d = x < mid_point ? tolerance : -tolerance;
} else {
// golden section interpolation step
e = (x < mid_point ? lim_b : lim_a) - x;
d = M_GOLDEN_RATIO * e;
}

// evaluate not too close to x
if (std::abs(d) >= tolerance)
u = d;
else if (d > 0)
u = tolerance;
else
u = -tolerance;
u += x;
fu = f(u);

// update variables
if (fu <= fx) {
if (u < x)
lim_b = x;
else
lim_a = x;
v = w;
fv = fw;
w = x;
fw = fx;
x = u;
fx = fu;
} else {
if (u < x)
lim_a = u;
else
lim_b = u;
if (fu <= fw || x == w) {
v = w;
fv = fw;
w = u;
fw = fu;
} else if (fu <= fv || v == x || v == w) {
v = u;
fv = fu;
}
}

iters++;
} while (std::abs(x - mid_point) > (tolerance - (lim_b - lim_a) / 2.f));

std::cout << " (iters: " << iters << ") ";

return x;
}

/**
* @brief Test function to find root for the function
* \f$f(x)= (x-2)^2\f$
* in the interval \f$[1,5]\f$
* \n Expected result = 2
*/
void test1() {
// 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";
}

/**
* @brief Test function to find root for the function
* \f$f(x)= x^{\frac{1}{x}}\f$
* in the interval \f$[-2,10]\f$
* \n Expected result: \f$e\approx 2.71828182845904509\f$
*/
void test2() {
// 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, 5);

std::cout << minima << " (" << M_E << ")...";

assert(std::abs(minima - M_E) < EPSILON);
std::cout << "passed\n";
}

/**
* @brief Test function to find *maxima* for the function
* \f$f(x)= \cos x\f$
* in the interval \f$[0,12]\f$
* \n Expected result: \f$\pi\approx 3.14159265358979312\f$
*/
void test3() {
// 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";
}

/** Main function */
int main() {
std::cout.precision(18);

std::cout << "Computations performed with machine epsilon: " << EPSILON
<< "\n";

test1();
test2();
test3();

return 0;
}