C++, , durand_kerner_roots.cpp

/**
* @file
* \brief Compute all possible approximate roots of any given polynomial using
* [Durand Kerner
* algorithm](https://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method)
* \author [Krishna Vedala](https://github.com/kvedala)
*
* Test the algorithm online:
* https://gist.github.com/kvedala/27f1b0b6502af935f6917673ec43bcd7
*
* Try the highly unstable Wilkinson's polynomial:
* 
* ./numerical_methods/durand_kerner_roots 1 -210 20615 -1256850 53327946
* -1672280820 40171771630 -756111184500 11310276995381 -135585182899530
* 1307535010540395 -10142299865511450 63030812099294896 -311333643161390640
* 1206647803780373360 -3599979517947607200 8037811822645051776
* -12870931245150988800 13803759753640704000 -8752948036761600000
* 2432902008176640000
* 
* Sample implementation results to compute approximate roots of the equation
* \f$x^4-1=0\f$:\n
* <img
* src="https://raw.githubusercontent.com/TheAlgorithms/C-Plus-Plus/docs/images/numerical_methods/durand_kerner_error.svg"
* width="400" alt="Error evolution during root approximations computed every
* iteration."/> <img
* src="https://raw.githubusercontent.com/TheAlgorithms/C-Plus-Plus/docs/images/numerical_methods/durand_kerner_roots.svg"
* width="400" alt="Roots evolution - shows the initial approximation of the
* roots and their convergence to a final approximation along with the iterative
* approximations" />
*/

#include <algorithm>
#include <cassert>
#include <cmath>
#include <complex>
#include <cstdlib>
#include <ctime>
#include <fstream>
#include <iostream>
#include <valarray>
#ifdef _OPENMP
#include <omp.h>
#endif

#define ACCURACY 1e-10 /**< maximum accuracy limit */

/**
* Evaluate the value of a polynomial with given coefficients
* \param[in] coeffs coefficients of the polynomial
* \param[in] x point at which to evaluate the polynomial
* \returns \f$f(x)\f$
**/
std::complex<double> poly_function(const std::valarray<double> &coeffs,
std::complex<double> x) {
double real = 0.f, imag = 0.f;
int n;

// #ifdef _OPENMP
// #pragma omp target teams distribute reduction(+ : real, imag)
// #endif
for (n = 0; n < coeffs.size(); n++) {
std::complex<double> tmp =
coeffs[n] * std::pow(x, coeffs.size() - n - 1);
real += tmp.real();
imag += tmp.imag();
}

return std::complex<double>(real, imag);
}

/**
* create a textual form of complex number
* \param[in] x point at which to evaluate the polynomial
* \returns pointer to converted string
*/
const char *complex_str(const std::complex<double> &x) {
#define MAX_BUFF_SIZE 50
static char msg[MAX_BUFF_SIZE];

std::snprintf(msg, MAX_BUFF_SIZE, "% 7.04g%+7.04gj", x.real(), x.imag());

return msg;
}

/**
* check for termination condition
* \param[in] delta point at which to evaluate the polynomial
* \returns false if termination not reached
* \returns true if termination reached
*/
bool check_termination(long double delta) {
static long double past_delta = INFINITY;
if (std::abs(past_delta - delta) <= ACCURACY || delta < ACCURACY)
return true;
past_delta = delta;
return false;
}

/**
* Implements Durand Kerner iterative algorithm to compute all roots of a
* polynomial.
*
* \param[in] coeffs coefficients of the polynomial
* \param[out] roots the computed roots of the polynomial
* \param[in] write_log flag whether to save the log file (default = false)
* \returns pair of values - number of iterations taken and final accuracy
* achieved
*/
std::pair<uint32_t, double> durand_kerner_algo(
const std::valarray<double> &coeffs,
std::valarray<std::complex<double>> *roots, bool write_log = false) {
long double tol_condition = 1;
uint32_t iter = 0;
int n;
std::ofstream log_file;

if (write_log) {
/*
* store intermediate values to a CSV file
*/
log_file.open("durand_kerner.log.csv");
if (!log_file.is_open()) {
perror("Unable to create a storage log file!");
std::exit(EXIT_FAILURE);
}
log_file << "iter#,";

for (n = 0; n < roots->size(); n++) log_file << "root_" << n << ",";

log_file << "avg. correction";
log_file << "\n0,";
for (n = 0; n < roots->size(); n++)
log_file << complex_str((*roots)[n]) << ",";
}

bool break_loop = false;
while (!check_termination(tol_condition) && iter < INT16_MAX &&
!break_loop) {
tol_condition = 0;
iter++;
break_loop = false;

if (log_file.is_open())
log_file << "\n" << iter << ",";

#ifdef _OPENMP
#pragma omp parallel for shared(break_loop, tol_condition)
#endif
for (n = 0; n < roots->size(); n++) {
if (break_loop)
continue;

std::complex<double> numerator, denominator;
numerator = poly_function(coeffs, (*roots)[n]);
denominator = 1.0;
for (int i = 0; i < roots->size(); i++)
if (i != n)
denominator *= (*roots)[n] - (*roots)[i];

std::complex<long double> delta = numerator / denominator;

if (std::isnan(std::abs(delta)) || std::isinf(std::abs(delta))) {
std::cerr << "\n\nOverflow/underrun error - got value = "
<< std::abs(delta) << "\n";
// return std::pair<uint32_t, double>(iter, tol_condition);
break_loop = true;
}

(*roots)[n] -= delta;

#ifdef _OPENMP
#pragma omp critical
#endif
tol_condition = std::max(tol_condition, std::abs(std::abs(delta)));
}
// tol_condition /= (degree - 1);

if (break_loop)
break;

if (log_file.is_open()) {
for (n = 0; n < roots->size(); n++)
log_file << complex_str((*roots)[n]) << ",";
}

#if defined(DEBUG) || !defined(NDEBUG)
if (iter % 500 == 0) {
std::cout << "Iter: " << iter << "\t";
for (n = 0; n < roots->size(); n++)
std::cout << "\t" << complex_str((*roots)[n]);
std::cout << "\t\tabsolute average change: " << tol_condition
<< "\n";
}
#endif

if (log_file.is_open())
log_file << tol_condition;
}

return std::pair<uint32_t, long double>(iter, tol_condition);
}

/**
* Self test the algorithm by checking the roots for \f$x^2+4=0\f$ to which the
* roots are \f$0 \pm 2i\f$
*/
void test1() {
const std::valarray<double> coeffs = {1, 0, 4};  // x^2 - 2 = 0
std::valarray<std::complex<double>> roots(2);
std::valarray<std::complex<double>> expected = {
std::complex<double>(0., 2.),
std::complex<double>(0., -2.)  // known expected roots
};

/* initialize root approximations with random values */
for (int n = 0; n < roots.size(); n++) {
roots[n] = std::complex<double>(std::rand() % 100, std::rand() % 100);
roots[n] -= 50.f;
roots[n] /= 25.f;
}

auto result = durand_kerner_algo(coeffs, &roots, false);

for (int i = 0; i < roots.size(); i++) {
// check if approximations are have < 0.1% error with one of the
// expected roots
bool err1 = false;
for (int j = 0; j < roots.size(); j++)
err1 |= std::abs(std::abs(roots[i] - expected[j])) < 1e-3;
assert(err1);
}

std::cout << "Test 1 passed! - " << result.first << " iterations, "
<< result.second << " accuracy"
<< "\n";
}

/**
* Self test the algorithm by checking the roots for \f$0.015625x^3-1=0\f$ to
* which the roots are \f$(4+0i),\,(-2\pm3.464i)\f$
*/
void test2() {
const std::valarray<double> coeffs = {// 0.015625 x^3 - 1 = 0
1. / 64., 0., 0., -1.};
std::valarray<std::complex<double>> roots(3);
const std::valarray<std::complex<double>> expected = {
std::complex<double>(4., 0.), std::complex<double>(-2., 3.46410162),
std::complex<double>(-2., -3.46410162)  // known expected roots
};

/* initialize root approximations with random values */
for (int n = 0; n < roots.size(); n++) {
roots[n] = std::complex<double>(std::rand() % 100, std::rand() % 100);
roots[n] -= 50.f;
roots[n] /= 25.f;
}

auto result = durand_kerner_algo(coeffs, &roots, false);

for (int i = 0; i < roots.size(); i++) {
// check if approximations are have < 0.1% error with one of the
// expected roots
bool err1 = false;
for (int j = 0; j < roots.size(); j++)
err1 |= std::abs(std::abs(roots[i] - expected[j])) < 1e-3;
assert(err1);
}

std::cout << "Test 2 passed! - " << result.first << " iterations, "
<< result.second << " accuracy"
<< "\n";
}

/***
* Main function.
* The comandline input arguments are taken as coeffiecients of a
*polynomial. For example, this command
* sh
* ./durand_kerner_roots 1 0 -4
* 
* will find roots of the polynomial \f$1\cdot x^2 + 0\cdot x^1 + (-4)=0\f$
**/
int main(int argc, char **argv) {
/* initialize random seed: */
std::srand(std::time(nullptr));

if (argc < 2) {
test1();  // run tests when no input is provided
test2();  // and skip tests when input polynomial is provided
std::cout << "Please pass the coefficients of the polynomial as "
"commandline "
"arguments.\n";
return 0;
}

int n, degree = argc - 1;              // detected polynomial degree
std::valarray<double> coeffs(degree);  // create coefficiencts array

// number of roots = degree - 1
std::valarray<std::complex<double>> s0(degree - 1);

std::cout << "Computing the roots for:\n\t";
for (n = 0; n < degree; n++) {
coeffs[n] = strtod(argv[n + 1], nullptr);
if (n < degree - 1 && coeffs[n] != 0)
std::cout << "(" << coeffs[n] << ") x^" << degree - n - 1 << " + ";
else if (coeffs[n] != 0)
std::cout << "(" << coeffs[n] << ") x^" << degree - n - 1
<< " = 0\n";

/* initialize root approximations with random values */
if (n < degree - 1) {
s0[n] = std::complex<double>(std::rand() % 100, std::rand() % 100);
s0[n] -= 50.f;
s0[n] /= 50.f;
}
}

// numerical errors less when the first coefficient is "1"
// hence, we normalize the first coefficient
{
double tmp = coeffs[0];
coeffs /= tmp;
}

clock_t end_time, start_time = clock();
auto result = durand_kerner_algo(coeffs, &s0, true);
end_time = clock();

std::cout << "\nIterations: " << result.first << "\n";
for (n = 0; n < degree - 1; n++)
std::cout << "\t" << complex_str(s0[n]) << "\n";
std::cout << "absolute average change: " << result.second << "\n";
std::cout << "Time taken: "
<< static_cast<double>(end_time - start_time) / CLOCKS_PER_SEC
<< " sec\n";

return 0;
}