Solving Non-Linear Equations

Solving Non-Linear Equations

 

Intro

This posting will give you some methods to help you find a nonlinear equation's approximated value. Before organizing our purpose, let's see this example problem.

 

Example MATLAB function

%================================================
%Handong Global University
%------------------------------------------------
%Name:      Taesan Kim
%ID:        22300203
%Create:    2024.09.05
%Modifire:  2024.09.05
%------------------------------------------------
%Plot of Trajectory Formula
%================================================
clc, clear all, close all;
t = 0:1e-3:45;
t0 = 3.0; %Find the most closest value with initial value 10.
sol = fzero(@fn,t0); %fzero find x which make @function = 0

axis equal;
plot(t, fn(t)); grid on
xlabel('x[-]'); ylabel('f(t)');

%Find theta which is solution of fnTrajectory = 0
disp(['Solution for t where fn = 0: ', num2str(sol), ' degrees']);

%Trajectory formula
function z=fn(t)
    x=20; y=2; v=17; g=9.8;
    z=x.*tand(t)-g.*x.*x./(2.*v.*v.*cosd(t).*cosd(t))-y;
end

 

 

 

Purpose

We want to derive a non-linear equation's solution not using fzero function. We will learn how to make fzero() to solve a non-linear equation with one variable, f(x) = 0. Then, we will expand the algorithm to solve a system of non-linear equations. This posting will organize 3 methods including the Bisection method, Newton-Raphson, and Secant's method.

 

 

 

 

Bisection method

 

 

 

 

 

 

 

Newton-Raphson method

 

 

Secant method

 

 

 

Header code

/*
================================================
Handong Global University
------------------------------------------------
Name:		Taesan Kim
ID:			22300203
Create:		2024.07.19
Modifier:	2024.07.19
------------------------------------------------
비선형 근삿값 계산을 지원한다.
================================================
*/

#ifndef MYNP22300203_H
#define MYNP22300203_H
#include <stdio.h>
#include <math.h>


/**
* Name:		Non-linear approximation method newtonRaphson
* Date:		2024/09/09
* Input:	
* - func:	non-linear function
* - dfunc:	differential non-linear function
* - x0:		initial value
* - tol:	allowence tolerance
* Return:	None(Print error)
**/
void newtonRaphson(double func(double x), double dfunc(double x), double x0, double tol);

/**
* Name:		Non-linear approximation method secant
* Date:		2024/09/09
* Input:
* - func:	non-linear function
* - x0, x1:		initial value
* - tol:	allowence tolerance
* Return:	xk
**/
float secantfzero(double func(double x), double dfunc(double x), double x0, double x1, double tol);

/**
* Name:		Non-linear approximation method biSection
* Date:		2024/09/09
* Input:
* - func:	non-linear function
* - tol:	allowence tolerance
* - a, b:	2D area [a, b]
* Return:	None(Print error)
**/
void biSection(double func(double x), double tol, double a, double b);

#endif

 

 

Source code

#include "myNP.h"



void biSection(double func(double x), double tol, double a, double b)
{
	printf("===================================================================\n");
	printf("	Bi-Section Method Results\n");
	printf("===================================================================\n");
	printf("Bi-Section Method Result:\n");
	int Nmax = 1000;
	double ep = 10.0;
	double xk = 0.0;
	int i = 0;
	if (func(a) * func(b) > 0)
		printf("Solution dont exists in this area!");
	else
		while (Nmax > i && ep > tol)
		{
			xk = 0.5 * (a + b);
			if (func(a) * func(xk) < 0)
			{
				b = xk;
			}
			else
			{
				a = xk;
			}
			i++;
			ep = fabs(func(xk));
			printf("Iteration: %2d		 Xn(k): %f	Tolerance: %f\n", i, xk, ep);
		}
	if (i == Nmax) printf("Solution did not converge !!!");
	printf("Final Solution: %f\n", xk);
}

void newtonRaphson(double func(double x), double dfunc(double x), double x0, double tol)
{
	printf("===================================================================\n");
	printf("	Newton-Raphson Method Results\n");
	printf("===================================================================\n");
	printf("Newton-Raphson Method Results:\n");
	int Nmax = 10000;
	double ep = 10.0;
	double xk = x0;
	double h = 0.0; int i = 0;

	while (i<Nmax && ep> tol)
	{
		if (dfunc(xk) == 0) printf("Error!! dF=0 !!");
		h = -func(xk) / dfunc(xk);
		xk = xk + h;
		i++;
		ep = fabs(func(xk));
		printf("Iteration: %2d		 Xn(k): %f	Tolerance: %f\n", i, xk, ep);
	}
	if (i == Nmax) printf("Solution did not converge !!!");
	printf("Final Solution: %f\n", xk);
}

float secantfzero(double func(double x), double dfunc(double x), double x0, double x1, double tol)
{
	printf("===================================================================\n");
	printf("	Secant Method Results\n");
	printf("===================================================================\n");
	printf("Secant Method Results:\n");
	//initialize

	

	double xk = x0; //최종 xk를 구해준다.
	double temp = 0.0;
	double ep = 10.0;
	int Nmax = 1000; int i = 0;
	while (ep > tol && i < Nmax)
	{
		if ((func(x1) - func(x0)) * func(x1) == 0)
		{
			printf("Error!! dF=0 !!");
			break;
		}
		temp = xk;
		xk = x1 - (x1 - x0) / (func(x1) - func(x0)) * func(x1);
		x0 = x1;
		x1 = temp;
		i++;
		ep = fabs(func(xk));
		printf("Iteration: %2d		 Xn(k): %f	Tolerance: %f\n", i, xk, ep);
	}
	return xk;
	printf("Final Solution: %f\n", xk);
}

 

 

Main code

#include "../../include/myNP.h"

#define pi 3.141592653589793238462643383279

/**
* Name:		example non-linear function
* Date:		2024/09/09
* Input:	x
* Return:	function output
**/
double func(double x)
{
	//return 8 - 4.5 * (x - sin(x));
	x = x * pi / 180; // deg2rad
	return 20 * tanf(x) - 9.8 * 200 / (17*17 * cosf(x)*cosf(x))-2;
}

/**
* Name:		example non-linear differential function
* Date:		2024/09/09
* Input:	x
* Return:	differential function output
**/
double dfunc(double x)
{
	//return -4.5 * (1 - cos(x));
	return 20 * (1/cosf(x)) * (1/cos(x)) - 9.8 * 400 * tanf(x) * (1/cosf(x)) * (1/cosf(x)) /289;
}



int main()
{
	biSection(func, 1e-5, 20,30);
	newtonRaphson(func, dfunc, 28, 1e-5);
	printf("%f", secantfzero(func, dfunc, 29, 30, 1e-5));
}

*you should set your myNP.cpp, myNP.h file in the include folder.