pmlpp/mlpp/numerical_analysis/numerical_analysis.cpp

//
//  NumericalAnalysis.cpp
//
//  Created by Marc Melikyan on 11/13/20.
//

#include "numerical_analysis.h"
#include "../lin_alg/lin_alg.h"

#include <climits>
#include <cmath>
#include <iostream>
#include <string>

/*
real_t MLPPNumericalAnalysis::numDiff(real_t (*function)(real_t), real_t x) {
	real_t eps = 1e-10;
	return (function(x + eps) - function(x)) / eps; // This is just the formal def. of the derivative.
}

real_t MLPPNumericalAnalysis::numDiff_2(real_t (*function)(real_t), real_t x) {
	real_t eps = 1e-5;
	return (function(x + 2 * eps) - 2 * function(x + eps) + function(x)) / (eps * eps);
}

real_t MLPPNumericalAnalysis::numDiff_3(real_t (*function)(real_t), real_t x) {
	real_t eps = 1e-5;
	real_t t1 = function(x + 3 * eps) - 2 * function(x + 2 * eps) + function(x + eps);
	real_t t2 = function(x + 2 * eps) - 2 * function(x + eps) + function(x);
	return (t1 - t2) / (eps * eps * eps);
}

real_t MLPPNumericalAnalysis::constantApproximation(real_t (*function)(real_t), real_t c) {
	return function(c);
}

real_t MLPPNumericalAnalysis::linearApproximation(real_t (*function)(real_t), real_t c, real_t x) {
	return constantApproximation(function, c) + numDiff(function, c) * (x - c);
}

real_t MLPPNumericalAnalysis::quadraticApproximation(real_t (*function)(real_t), real_t c, real_t x) {
	return linearApproximation(function, c, x) + 0.5 * numDiff_2(function, c) * (x - c) * (x - c);
}

real_t MLPPNumericalAnalysis::cubicApproximation(real_t (*function)(real_t), real_t c, real_t x) {
	return quadraticApproximation(function, c, x) + (1 / 6) * numDiff_3(function, c) * (x - c) * (x - c) * (x - c);
}

real_t MLPPNumericalAnalysis::numDiff(real_t (*function)(std::vector<real_t>), std::vector<real_t> x, int axis) {
	// For multivariable function analysis.
	// This will be used for calculating Jacobian vectors.
	// Diffrentiate with respect to indicated axis. (0, 1, 2 ...)
	real_t eps = 1e-10;
	std::vector<real_t> x_eps = x;
	x_eps[axis] += eps;

	return (function(x_eps) - function(x)) / eps;
}

real_t MLPPNumericalAnalysis::numDiff_2(real_t (*function)(std::vector<real_t>), std::vector<real_t> x, int axis1, int axis2) {
	//For Hessians.
	real_t eps = 1e-5;

	std::vector<real_t> x_pp = x;
	x_pp[axis1] += eps;
	x_pp[axis2] += eps;

	std::vector<real_t> x_np = x;
	x_np[axis2] += eps;

	std::vector<real_t> x_pn = x;
	x_pn[axis1] += eps;

	return (function(x_pp) - function(x_np) - function(x_pn) + function(x)) / (eps * eps);
}

real_t MLPPNumericalAnalysis::numDiff_3(real_t (*function)(std::vector<real_t>), std::vector<real_t> x, int axis1, int axis2, int axis3) {
	// For third order derivative tensors.
	// NOTE: Approximations do not appear to be accurate for sinusodial functions...
	// Should revisit this later.
	real_t eps = 1e-5;

	std::vector<real_t> x_ppp = x;
	x_ppp[axis1] += eps;
	x_ppp[axis2] += eps;
	x_ppp[axis3] += eps;

	std::vector<real_t> x_npp = x;
	x_npp[axis2] += eps;
	x_npp[axis3] += eps;

	std::vector<real_t> x_pnp = x;
	x_pnp[axis1] += eps;
	x_pnp[axis3] += eps;

	std::vector<real_t> x_nnp = x;
	x_nnp[axis3] += eps;

	std::vector<real_t> x_ppn = x;
	x_ppn[axis1] += eps;
	x_ppn[axis2] += eps;

	std::vector<real_t> x_npn = x;
	x_npn[axis2] += eps;

	std::vector<real_t> x_pnn = x;
	x_pnn[axis1] += eps;

	real_t thirdAxis = function(x_ppp) - function(x_npp) - function(x_pnp) + function(x_nnp);
	real_t noThirdAxis = function(x_ppn) - function(x_npn) - function(x_pnn) + function(x);
	return (thirdAxis - noThirdAxis) / (eps * eps * eps);
}

real_t MLPPNumericalAnalysis::newtonRaphsonMethod(real_t (*function)(real_t), real_t x_0, real_t epoch_num) {
	real_t x = x_0;
	for (int i = 0; i < epoch_num; i++) {
		x -= function(x) / numDiff(function, x);
	}
	return x;
}

real_t MLPPNumericalAnalysis::halleyMethod(real_t (*function)(real_t), real_t x_0, real_t epoch_num) {
	real_t x = x_0;
	for (int i = 0; i < epoch_num; i++) {
		x -= ((2 * function(x) * numDiff(function, x)) / (2 * numDiff(function, x) * numDiff(function, x) - function(x) * numDiff_2(function, x)));
	}
	return x;
}

real_t MLPPNumericalAnalysis::invQuadraticInterpolation(real_t (*function)(real_t), std::vector<real_t> x_0, int epoch_num) {
	real_t x = 0;
	std::vector<real_t> currentThree = x_0;
	for (int i = 0; i < epoch_num; i++) {
		real_t t1 = ((function(currentThree[1]) * function(currentThree[2])) / ((function(currentThree[0]) - function(currentThree[1])) * (function(currentThree[0]) - function(currentThree[2])))) * currentThree[0];
		real_t t2 = ((function(currentThree[0]) * function(currentThree[2])) / ((function(currentThree[1]) - function(currentThree[0])) * (function(currentThree[1]) - function(currentThree[2])))) * currentThree[1];
		real_t t3 = ((function(currentThree[0]) * function(currentThree[1])) / ((function(currentThree[2]) - function(currentThree[0])) * (function(currentThree[2]) - function(currentThree[1])))) * currentThree[2];
		x = t1 + t2 + t3;

		currentThree.erase(currentThree.begin());
		currentThree.push_back(x);
	}
	return x;
}

real_t MLPPNumericalAnalysis::eulerianMethod(real_t (*derivative)(real_t), std::vector<real_t> q_0, real_t p, real_t h) {
	int max_epoch = static_cast<int>((p - q_0[0]) / h);
	real_t x = q_0[0];
	real_t y = q_0[1];
	for (int i = 0; i < max_epoch; i++) {
		y = y + h * derivative(x);
		x += h;
	}
	return y;
}

real_t MLPPNumericalAnalysis::eulerianMethod(real_t (*derivative)(std::vector<real_t>), std::vector<real_t> q_0, real_t p, real_t h) {
	int max_epoch = static_cast<int>((p - q_0[0]) / h);
	real_t x = q_0[0];
	real_t y = q_0[1];
	for (int i = 0; i < max_epoch; i++) {
		y = y + h * derivative({ x, y });
		x += h;
	}
	return y;
}

real_t MLPPNumericalAnalysis::growthMethod(real_t C, real_t k, real_t t) {
	//dP/dt = kP
	//dP/P = kdt
	//integral(1/P)dP = integral(k) dt
	//ln|P| = kt + C_initial
	//|P| = e^(kt + C_initial)
	//|P| = e^(C_initial) * e^(kt)
	//P = +/- e^(C_initial) * e^(kt)
	//P = C * e^(kt)
	

	// auto growthFunction = [&C, &k](real_t t) { return C * exp(k * t); };
	return C * std::exp(k * t);
}

std::vector<real_t> MLPPNumericalAnalysis::jacobian(real_t (*function)(std::vector<real_t>), std::vector<real_t> x) {
	std::vector<real_t> jacobian;
	jacobian.resize(x.size());
	for (uint32_t i = 0; i < jacobian.size(); i++) {
		jacobian[i] = numDiff(function, x, i); // Derivative w.r.t axis i evaluated at x. For all x_i.
	}
	return jacobian;
}
std::vector<std::vector<real_t>> MLPPNumericalAnalysis::hessian(real_t (*function)(std::vector<real_t>), std::vector<real_t> x) {
	std::vector<std::vector<real_t>> hessian;
	hessian.resize(x.size());

	for (uint32_t i = 0; i < hessian.size(); i++) {
		hessian[i].resize(x.size());
	}

	for (uint32_t i = 0; i < hessian.size(); i++) {
		for (uint32_t j = 0; j < hessian[i].size(); j++) {
			hessian[i][j] = numDiff_2(function, x, i, j);
		}
	}

	return hessian;
}

std::vector<std::vector<std::vector<real_t>>> MLPPNumericalAnalysis::thirdOrderTensor(real_t (*function)(std::vector<real_t>), std::vector<real_t> x) {
	std::vector<std::vector<std::vector<real_t>>> tensor;
	tensor.resize(x.size());

	for (uint32_t i = 0; i < tensor.size(); i++) {
		tensor[i].resize(x.size());
		for (uint32_t j = 0; j < tensor[i].size(); j++) {
			tensor[i][j].resize(x.size());
		}
	}

	for (uint32_t i = 0; i < tensor.size(); i++) { // O(n^3) time complexity :(
		for (uint32_t j = 0; j < tensor[i].size(); j++) {
			for (uint32_t k = 0; k < tensor[i][j].size(); k++)
				tensor[i][j][k] = numDiff_3(function, x, i, j, k);
		}
	}

	return tensor;
}

real_t MLPPNumericalAnalysis::constantApproximation(real_t (*function)(std::vector<real_t>), std::vector<real_t> c) {
	return function(c);
}

real_t MLPPNumericalAnalysis::linearApproximation(real_t (*function)(std::vector<real_t>), std::vector<real_t> c, std::vector<real_t> x) {
	MLPPLinAlg alg;
	return constantApproximation(function, c) + alg.matmult(alg.transpose({ jacobian(function, c) }), { alg.subtraction(x, c) })[0][0];
}

real_t MLPPNumericalAnalysis::quadraticApproximation(real_t (*function)(std::vector<real_t>), std::vector<real_t> c, std::vector<real_t> x) {
	MLPPLinAlg alg;
	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];
}

real_t MLPPNumericalAnalysis::cubicApproximation(real_t (*function)(std::vector<real_t>), std::vector<real_t> c, std::vector<real_t> x) {
	//Not completely sure as the literature seldom discusses the third order taylor approximation,
	//in particular for multivariate cases, but ostensibly, the matrix/tensor/vector multiplies
	//should look something like this:

	//(N x N x N) (N x 1) [tensor vector mult] => (N x N x 1) => (N x N)
	//Perform remaining multiplies as done for the 2nd order approximation.
	//Result is a scalar.
	
	MLPPLinAlg alg;
	std::vector<std::vector<real_t>> resultMat = alg.tensor_vec_mult(thirdOrderTensor(function, c), alg.subtraction(x, c));
	real_t resultScalar = alg.matmult({ (alg.subtraction(x, c)) }, alg.matmult(resultMat, alg.transpose({ alg.subtraction(x, c) })))[0][0];

	return quadraticApproximation(function, c, x) + (1 / 6) * resultScalar;
}

real_t MLPPNumericalAnalysis::laplacian(real_t (*function)(std::vector<real_t>), std::vector<real_t> x) {
	std::vector<std::vector<real_t>> hessian_matrix = hessian(function, x);
	real_t laplacian = 0;

	for (uint32_t i = 0; i < hessian_matrix.size(); i++) {
		laplacian += hessian_matrix[i][i]; // homogenous 2nd derivs w.r.t i, then i
	}

	return laplacian;
}

std::string MLPPNumericalAnalysis::secondPartialDerivativeTest(real_t (*function)(std::vector<real_t>), std::vector<real_t> x) {
	MLPPLinAlg alg;
	std::vector<std::vector<real_t>> hessianMatrix = hessian(function, x);
	
	// The reason we do this is because the 2nd partial derivative test is less conclusive for functions of variables greater than
	// 2, and the calculations specific to the bivariate case are less computationally intensive.
	
	if (x.size() == 2) {
		real_t det = alg.det(hessianMatrix, hessianMatrix.size());
		real_t secondDerivative = numDiff_2(function, x, 0, 0);
		if (secondDerivative > 0 && det > 0) {
			return "min";
		} else if (secondDerivative < 0 && det > 0) {
			return "max";
		} else if (det < 0) {
			return "saddle";
		} else {
			return "test was inconclusive";
		}
	} else {
		if (alg.positiveDefiniteChecker(hessianMatrix)) {
			return "min";
		} else if (alg.negativeDefiniteChecker(hessianMatrix)) {
			return "max";
		} else if (!alg.zeroEigenvalue(hessianMatrix)) {
			return "saddle";
		} else {
			return "test was inconclusive";
		}
	}
}
*/

void MLPPNumericalAnalysis::_bind_methods() {
}