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296 lines
10 KiB
C++
296 lines
10 KiB
C++
//
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// NumericalAnalysis.cpp
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//
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// Created by Marc Melikyan on 11/13/20.
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//
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#include "numerical_analysis.h"
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#include "../lin_alg/lin_alg.h"
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#include <climits>
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#include <cmath>
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#include <iostream>
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#include <string>
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namespace MLPP {
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double NumericalAnalysis::numDiff(double (*function)(double), double x) {
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double eps = 1e-10;
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return (function(x + eps) - function(x)) / eps; // This is just the formal def. of the derivative.
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}
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double NumericalAnalysis::numDiff_2(double (*function)(double), double x) {
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double eps = 1e-5;
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return (function(x + 2 * eps) - 2 * function(x + eps) + function(x)) / (eps * eps);
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}
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double NumericalAnalysis::numDiff_3(double (*function)(double), double x) {
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double eps = 1e-5;
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double t1 = function(x + 3 * eps) - 2 * function(x + 2 * eps) + function(x + eps);
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double t2 = function(x + 2 * eps) - 2 * function(x + eps) + function(x);
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return (t1 - t2) / (eps * eps * eps);
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}
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double NumericalAnalysis::constantApproximation(double (*function)(double), double c) {
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return function(c);
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}
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double NumericalAnalysis::linearApproximation(double (*function)(double), double c, double x) {
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return constantApproximation(function, c) + numDiff(function, c) * (x - c);
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}
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double NumericalAnalysis::quadraticApproximation(double (*function)(double), double c, double x) {
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return linearApproximation(function, c, x) + 0.5 * numDiff_2(function, c) * (x - c) * (x - c);
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}
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double NumericalAnalysis::cubicApproximation(double (*function)(double), double c, double x) {
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return quadraticApproximation(function, c, x) + (1 / 6) * numDiff_3(function, c) * (x - c) * (x - c) * (x - c);
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}
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double NumericalAnalysis::numDiff(double (*function)(std::vector<double>), std::vector<double> x, int axis) {
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// For multivariable function analysis.
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// This will be used for calculating Jacobian vectors.
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// Diffrentiate with respect to indicated axis. (0, 1, 2 ...)
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double eps = 1e-10;
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std::vector<double> x_eps = x;
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x_eps[axis] += eps;
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return (function(x_eps) - function(x)) / eps;
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}
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double NumericalAnalysis::numDiff_2(double (*function)(std::vector<double>), std::vector<double> x, int axis1, int axis2) {
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//For Hessians.
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double eps = 1e-5;
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std::vector<double> x_pp = x;
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x_pp[axis1] += eps;
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x_pp[axis2] += eps;
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std::vector<double> x_np = x;
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x_np[axis2] += eps;
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std::vector<double> x_pn = x;
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x_pn[axis1] += eps;
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return (function(x_pp) - function(x_np) - function(x_pn) + function(x)) / (eps * eps);
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}
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double NumericalAnalysis::numDiff_3(double (*function)(std::vector<double>), std::vector<double> x, int axis1, int axis2, int axis3) {
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// For third order derivative tensors.
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// NOTE: Approximations do not appear to be accurate for sinusodial functions...
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// Should revisit this later.
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double eps = INT_MAX;
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std::vector<double> x_ppp = x;
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x_ppp[axis1] += eps;
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x_ppp[axis2] += eps;
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x_ppp[axis3] += eps;
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std::vector<double> x_npp = x;
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x_npp[axis2] += eps;
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x_npp[axis3] += eps;
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std::vector<double> x_pnp = x;
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x_pnp[axis1] += eps;
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x_pnp[axis3] += eps;
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std::vector<double> x_nnp = x;
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x_nnp[axis3] += eps;
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std::vector<double> x_ppn = x;
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x_ppn[axis1] += eps;
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x_ppn[axis2] += eps;
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std::vector<double> x_npn = x;
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x_npn[axis2] += eps;
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std::vector<double> x_pnn = x;
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x_pnn[axis1] += eps;
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double thirdAxis = function(x_ppp) - function(x_npp) - function(x_pnp) + function(x_nnp);
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double noThirdAxis = function(x_ppn) - function(x_npn) - function(x_pnn) + function(x);
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return (thirdAxis - noThirdAxis) / (eps * eps * eps);
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}
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double NumericalAnalysis::newtonRaphsonMethod(double (*function)(double), double x_0, double epoch_num) {
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double x = x_0;
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for (int i = 0; i < epoch_num; i++) {
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x -= function(x) / numDiff(function, x);
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}
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return x;
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}
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double NumericalAnalysis::halleyMethod(double (*function)(double), double x_0, double epoch_num) {
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double x = x_0;
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for (int i = 0; i < epoch_num; i++) {
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x -= ((2 * function(x) * numDiff(function, x)) / (2 * numDiff(function, x) * numDiff(function, x) - function(x) * numDiff_2(function, x)));
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}
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return x;
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}
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double NumericalAnalysis::invQuadraticInterpolation(double (*function)(double), std::vector<double> x_0, double epoch_num) {
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double x = 0;
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std::vector<double> currentThree = x_0;
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for (int i = 0; i < epoch_num; i++) {
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double t1 = ((function(currentThree[1]) * function(currentThree[2])) / ((function(currentThree[0]) - function(currentThree[1])) * (function(currentThree[0]) - function(currentThree[2])))) * currentThree[0];
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double t2 = ((function(currentThree[0]) * function(currentThree[2])) / ((function(currentThree[1]) - function(currentThree[0])) * (function(currentThree[1]) - function(currentThree[2])))) * currentThree[1];
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double t3 = ((function(currentThree[0]) * function(currentThree[1])) / ((function(currentThree[2]) - function(currentThree[0])) * (function(currentThree[2]) - function(currentThree[1])))) * currentThree[2];
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x = t1 + t2 + t3;
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currentThree.erase(currentThree.begin());
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currentThree.push_back(x);
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}
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return x;
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}
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double NumericalAnalysis::eulerianMethod(double (*derivative)(double), std::vector<double> q_0, double p, double h) {
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double max_epoch = (p - q_0[0]) / h;
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double x = q_0[0];
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double y = q_0[1];
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for (int i = 0; i < max_epoch; i++) {
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y = y + h * derivative(x);
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x += h;
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}
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return y;
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}
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double NumericalAnalysis::eulerianMethod(double (*derivative)(std::vector<double>), std::vector<double> q_0, double p, double h) {
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double max_epoch = (p - q_0[0]) / h;
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double x = q_0[0];
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double y = q_0[1];
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for (int i = 0; i < max_epoch; i++) {
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y = y + h * derivative({ x, y });
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x += h;
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}
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return y;
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}
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double NumericalAnalysis::growthMethod(double C, double k, double t) {
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/*
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dP/dt = kP
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dP/P = kdt
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integral(1/P)dP = integral(k) dt
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ln|P| = kt + C_initial
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|P| = e^(kt + C_initial)
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|P| = e^(C_initial) * e^(kt)
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P = +/- e^(C_initial) * e^(kt)
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P = C * e^(kt)
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*/
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// auto growthFunction = [&C, &k](double t) { return C * exp(k * t); };
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return C * std::exp(k * t);
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}
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std::vector<double> NumericalAnalysis::jacobian(double (*function)(std::vector<double>), std::vector<double> x) {
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std::vector<double> jacobian;
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jacobian.resize(x.size());
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for (int i = 0; i < jacobian.size(); i++) {
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jacobian[i] = numDiff(function, x, i); // Derivative w.r.t axis i evaluated at x. For all x_i.
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}
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return jacobian;
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}
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std::vector<std::vector<double>> NumericalAnalysis::hessian(double (*function)(std::vector<double>), std::vector<double> x) {
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std::vector<std::vector<double>> hessian;
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hessian.resize(x.size());
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for (int i = 0; i < hessian.size(); i++) {
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hessian[i].resize(x.size());
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}
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for (int i = 0; i < hessian.size(); i++) {
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for (int j = 0; j < hessian[i].size(); j++) {
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hessian[i][j] = numDiff_2(function, x, i, j);
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}
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}
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return hessian;
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}
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std::vector<std::vector<std::vector<double>>> NumericalAnalysis::thirdOrderTensor(double (*function)(std::vector<double>), std::vector<double> x) {
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std::vector<std::vector<std::vector<double>>> tensor;
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tensor.resize(x.size());
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for (int i = 0; i < tensor.size(); i++) {
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tensor[i].resize(x.size());
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for (int j = 0; j < tensor[i].size(); j++) {
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tensor[i][j].resize(x.size());
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}
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}
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for (int i = 0; i < tensor.size(); i++) { // O(n^3) time complexity :(
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for (int j = 0; j < tensor[i].size(); j++) {
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for (int k = 0; k < tensor[i][j].size(); k++)
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tensor[i][j][k] = numDiff_3(function, x, i, j, k);
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}
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}
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return tensor;
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}
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double NumericalAnalysis::constantApproximation(double (*function)(std::vector<double>), std::vector<double> c) {
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return function(c);
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}
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double NumericalAnalysis::linearApproximation(double (*function)(std::vector<double>), std::vector<double> c, std::vector<double> x) {
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LinAlg alg;
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return constantApproximation(function, c) + alg.matmult(alg.transpose({ jacobian(function, c) }), { alg.subtraction(x, c) })[0][0];
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}
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double NumericalAnalysis::quadraticApproximation(double (*function)(std::vector<double>), std::vector<double> c, std::vector<double> x) {
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LinAlg alg;
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return linearApproximation(function, c, x) + 0.5 * alg.matmult({ (alg.subtraction(x, c)) }, alg.matmult(hessian(function, c), alg.transpose({ alg.subtraction(x, c) })))[0][0];
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}
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double NumericalAnalysis::cubicApproximation(double (*function)(std::vector<double>), std::vector<double> c, std::vector<double> x) {
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/*
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Not completely sure as the literature seldom discusses the third order taylor approximation,
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in particular for multivariate cases, but ostensibly, the matrix/tensor/vector multiplies
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should look something like this:
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(N x N x N) (N x 1) [tensor vector mult] => (N x N x 1) => (N x N)
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Perform remaining multiplies as done for the 2nd order approximation.
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Result is a scalar.
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*/
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LinAlg alg;
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std::vector<std::vector<double>> resultMat = alg.tensor_vec_mult(thirdOrderTensor(function, c), alg.subtraction(x, c));
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double resultScalar = alg.matmult({ (alg.subtraction(x, c)) }, alg.matmult(resultMat, alg.transpose({ alg.subtraction(x, c) })))[0][0];
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return quadraticApproximation(function, c, x) + (1 / 6) * resultScalar;
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}
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double NumericalAnalysis::laplacian(double (*function)(std::vector<double>), std::vector<double> x) {
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LinAlg alg;
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std::vector<std::vector<double>> hessian_matrix = hessian(function, x);
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double laplacian = 0;
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for (int i = 0; i < hessian_matrix.size(); i++) {
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laplacian += hessian_matrix[i][i]; // homogenous 2nd derivs w.r.t i, then i
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}
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return laplacian;
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}
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std::string NumericalAnalysis::secondPartialDerivativeTest(double (*function)(std::vector<double>), std::vector<double> x) {
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LinAlg alg;
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std::vector<std::vector<double>> hessianMatrix = hessian(function, x);
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/*
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The reason we do this is because the 2nd partial derivative test is less conclusive for functions of variables greater than
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2, and the calculations specific to the bivariate case are less computationally intensive.
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*/
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if (x.size() == 2) {
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double det = alg.det(hessianMatrix, hessianMatrix.size());
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double secondDerivative = numDiff_2(function, x, 0, 0);
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if (secondDerivative > 0 && det > 0) {
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return "min";
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} else if (secondDerivative < 0 && det > 0) {
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return "max";
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} else if (det < 0) {
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return "saddle";
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} else {
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return "test was inconclusive";
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}
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} else {
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if (alg.positiveDefiniteChecker(hessianMatrix)) {
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return "min";
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} else if (alg.negativeDefiniteChecker(hessianMatrix)) {
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return "max";
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} else if (!alg.zeroEigenvalue(hessianMatrix)) {
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return "saddle";
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} else {
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return "test was inconclusive";
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}
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}
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}
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} //namespace MLPP
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