## Introduction

This article tries to familiarize the beginner with numerical methods. I am working a lot with numerical analysis and methods, and I want to share with you some of my experiences and the results that I encountered. This is intended to be the first article in a series of **Numerical Analysis Methods and Their Implementation in C++**.

After finishing reading this article, you may wonder *"Why reinvent the wheel?"*. Of course, there are a great number of references on the internet that handle the issues presented here, but my intent is not to reinvent the wheel, but to try and explain in detail some classical methods in Numerical Analysis, what they do, why they are useful, how they are implemented, and above all, how to use them.

In this article, we are going to focus on two famous equations in Mathematics, which are of huge importance in almost every domain of science: F(x) = 0 and F(x) = x. Many problems in Mathematics, Chemistry, Physics are reduced to solving one of these two equations, and that's why scientists over time have tried hard to offer good solutions to resolve these equations. We will go through some numerical methods that try to solve them, and we will try to make a short analysis of those methods, showing the advantages/disadvantages of every one of them.

## The methods and their implementation

The first issue of our article is the problem of solving the equation F(x) = 0, where F(x) can be any kind of function. For instance, we can have F(x) = x^2 + 6x + 6, or F(x) = cos(x). This is a well known problem in Mathematics, and it is known also that there is no way of finding out exactly the solution or solutions of every equation of this form. That's why mathematicians have been trying to offer **approximate** solutions for this equation. Please also note that the solutions of the equation F(x) = 0 are usually called the **roots** of the function F(x).

### Newton's method

Newton's approach is the following: we start with an initial value for the solution (also called initial approximation), then we replace the function by its tangent, and we compute the root of this tangent which will be a better approximation for the function's root. We repeat this process until we find a suitable solution (one that is close enough to the actual solution and fits very well the equation F(x) = 0). It is obvious that this process is, in fact, an iterative process. Note also that the function F must be a real valued, differentiable function in order to apply Newton's algorithm.

In detail, if we have a current approximation xCrt, the next approximation nNxt will be computed using the following formula:

xNxt = xCrt - (F(xCrt) / F`(xCrt))

where F` denotes the derivative of the function F. The iteration process stops when we have gone through a maximum permitted number of iterations and we still can't find the solution, or when we have found an approximation which is close enough to the actual solution of the equation. Here is the code that implements Newton's method:

int NewtonMethodForEquation(double& x)
{
int n = 1;
while( ( fabs(F(x)) > error ) && ( n <= MAXITER ) )
{
x = x - ( F(x) / Fd(x) );
n++;
}
return n;
}

In the above code snippet, `Fd`

denotes the derivative of the function F.

### Secant method

The secant method is another approach for solving the equation F(x) = 0. The method is almost identical with Newton's method, except the fact that we choose two initial approximations instead of one before we start the iteration process. Suppose we have the current approximations xCrt0 and xCrt1. The next approximation xNxt will be computed this time using the following formula:

xNxt = xCrt1 - (F(xCrt1)(xCrt1 - xCrt0)) / (F(xCrt1) - F(xCrt0))

Note that this method doesn't require the derivative of the function F, like Newton's method did. Here is the code that implements the Secant method:

int SecantMethodForEquation(double& x, double x0, double x1)
{
int n = 2;
while( ( fabs(F(x1)) > error ) && ( n <= MAXITER ) )
{
x = x1 - (F(x1) * (x1 - x0)) / (F(x1) - F(x0));
x0 = x1;
x1 = x;
n++;
}
return n;
}

Another problem that comes into attention some times is solving the equation F(x) = x. If we write the equation like this: F(x) - x = 0 and we note G(x) = F(x) - x, then the equation becomes G(x) = 0. But the equation in the form F(x) = x presents a particular interest for mathematicians. It is said that if x0 is a solution of the equation F(x) = x, then x0 is called a **fixed point** of the function F(x). Of course, we can apply the methods learned before for the equation G(x) = 0, but our interest is to present methods for solving the equation F(x) = x.

### Successive approximations method

This method, as simple as it may be, is of huge importance in Mathematics, being widely used in many fixed point theories. Let's see how the method works. First, like before, we choose an initial approximation x0, and we start the iterative process. If xCrt denotes the current approximation, then we compute xNxt like this:

xNxt = F(xCrt)

This is a pretty simple formula, and against all odds, it has been proved that the method usually converges after a number of iterations, leading to a good approximation of the equation solution. The source code is pretty straightforward:

int SuccessiveApproxForEquation(double& x)
{
int n = 1;
while( ( fabs(x - F1(x)) > error ) && ( n <= MAXITER ))
{
x = F1(x);
n++;
}
return n;
}

This method, as simple and useful as it may be, has one big disadvantage: its convergence rate is very slow, meaning that the number of iterations passed until we get a solution will be pretty high. If the initial approximation is chosen close to the actual solution of the equation, the iteration process will be fast enough and the algorithm will find a suitable solution. But, if the initial approximation is chosen at random, the process may not find a solution at all (depending on the number of maximum iterations permitted, and how close to the actual solution the initial approximation is).

To overcome this problem, some mathematicians tried to speed up the process of finding the right solution for the iterative methods. So some algorithms were developed to do just that (those algorithms are usually called convergence acceleration algorithms). The most important algorithms of convergence acceleration are Aitken's algorithm and Overholt's algorithm.

The idea behind a convergence acceleration algorithm is the following: if we look closer at the iteration process, we see that if we choose every approximation value at each step, we can form a real valued number sequence: x0, x1, x2, ... ,xn which, after a number of iterations, converges to the solution (x) of the equation F(x) = x. A convergence acceleration algorithm transforms the number sequence x0, x1, x2, ... ,xn into another real valued number sequence y0, y1, y2, ... yn, which has the very important property that it converges **faster** to the solution x.

We present in the following section, three such algorithms for convergence acceleration, and we stick to our problem of solving the equation F(x) = x.

### Aitken's method

Aitken's method is an iterative process similar to the ones presented. I will not go into the mathematical details again. Instead, I prefer to present the code for Aitken's method:

int AitkenMethodForEquation(double& y, double x0)
{
int n = 1;
double x;
do
{
x = F1(x0);
y = x + 1 / ((1 / (F1(x) - x)) - (1 / (x - x0)));
n++;
x0 = x;
}
while((fabs(y - F1(y)) > error) && (n <= MAXITER));
return n;
}

### Steffenson's method

This method is a simplified version of Aitken's method, observing that if we apply Aitken's formula for the values xCrt, F(xCrt), F(F(xCrt)), we obtain:

xNxt = (xCrt F(F(xCrt)) - (F(xCrt)) ^ 2) / (F(F(xCrt)) - 2F(xCrt) + xCrt)

In a simplified form, this is written as:

xCrt = F(xCrt) + 1 / ((1 / (F(F(xCrt)) - F(xCrt))) - (1 / (F(xCrt) - xCrt)))

The code for the algorithm is:

int SteffensenMethodForEquation(double& x)
{
int n = 0;
do
{
x = F1(x) + 1 / ( (1 / (F1(F1(x)) - F1(x)) ) - (1 / (F1(x) - x) ) );
n++;
}
while((fabs(x - F1(x)) > error) && (n <= MAXITER));
return n;
}

### Overholt's method

Overholt's method is, by far, the fastest method for solving the equation F(x) = x. I am going to present the code for the algorithm, the method itself being pretty straightforward from the source code:

int OverholtMethodForEquation(double &x, double x0, int s)
{
int m = 0;
x = x0;
do
{
V[0][0] = x;
for(int n = 1; n <= s; n++)
V[0][n] = F1(V[0][n - 1]);
for(int k = 0; k <= s - 2; k++)
for(int n = 0; n <= s - k - 2; n++)
V[k + 1][n] = (pow(V[0][n + k + 2] - V[0][n + k + 1], k+1) * V[k][n] -
pow(V[0][n + k + 1] - V[0][n + k], k + 1) * V[k][n + 1]) /
(pow(V[0][n + k + 2] - V[0][n + k + 1], k + 1) -
pow(V[0][n + k + 1] - V[0][n + k], k + 1));
m++;
x = V[s - 1][0];
}
while((fabs(x - F1(x)) > error) && (m <= MAXITER));
return m;
}

In the source code above, `V`

is a global variable declared as:

double V[100][100];

## Using the code

Now, let's see how to effectively use the methods described above, and on the way, we will make a short analysis of these methods.

In order to use the methods, we must first define some constants:

const int MAXITER = 100;
const double error = 0.0001;

`MAXITER`

represents the maximum number of iterations an algorithm is permitted to pass, meaning that if one of our algorithms has passed `MAXITER`

iterations and it still didn't find a suitable solution, the algorithm will stop nevertheless. `error`

represents the minimum accepted error for the solution, meaning that the approximate solution must be close enough to the real solution of the equation with respect to this error. The smaller the value of this error, the closer to the real solution the approximate solution will be.

Let's take an example equation of the first kind, for instance, x*e^x - 1 = 0. So, we have F(x) = x*e^x - 1. We define the function in our code, and also its derivative, because we will need it in order to apply Newton's method:

const double e = 2.718281828459;
#define F(x) ( x * pow(e, x) - 1 )
#define Fd(x) ( (x + 1) * pow(e, x) )

You can change the function `F`

and its derivative `Fd`

, and you will solve any kind of equation you like:

double x;
int n;
cout << "Newton's method: " << endl << endl;
cout << "Give the initial approximation: ";
cin >> x;
n = NewtonMethodForEquation(x);
if(n > MAXITER)
cout << "In " << MAXITER << " iterations no solution was found!" << endl;
else
cout << "The solution is: " << x << " and it was found in "
<< n << " iterations" << endl;
double x0, x1;
cout << "Secant method: " << endl << endl;
cout << "Give the first initial approximation: ";
cin >> x0;
cout << "Give the second initial approximation: ";
cin >> x1;
n = SecantMethodForEquation(x, x0, x1);
if(n > MAXITER)
cout << "In " << MAXITER << " iterations no solution was found!" << endl;
else
cout << "The solution is: " << x << " and it was found in "
<< n << " iterations" << endl;

Now you can play with the algorithms, giving various initial approximations and decreasing or increasing the `error`

value to see how they are behaving. We can see that, overall, Newton's method is faster than Secant method.

For the second type of equation, let's take, for example, F(x) = e^(-x), and the equation becomes e^(-x) = x.

#define F1(x) ( pow(e, -x) )

And, here is how we apply our algorithms:

cout << "Successive approximations method: " << endl << endl;
cout << "Give the initial approximation: ";
cin >> x;
n = SuccessiveApproxForEquation(x);
if(n > MAXITER)
cout << "In " << MAXITER << " iterations no solution was found!" << endl;
else
cout << "The solution is: " << x << " and it was found in "
<< n << " iterations" << endl;
cout << "Aitken's method: " << endl << endl;
cout << "Give the initial approximation: ";
cin >> x;
double y;
n = AitkenMethodForEquation(y, x);
if(n > MAXITER)
cout << "In " << MAXITER
<< " iterations no solution was found!" << endl;
else
cout << "The solution is: " << y << " and it was found in "
<< n << " iterations" << endl;
cout << "Steffensen's method: " << endl << endl;
cout << "Give the initial approximation: ";
cin >> x;
n = SteffensenMethodForEquation(x);
if(n > MAXITER)
cout << "In " << MAXITER
<< " iterations no solution was found!" << endl;
else
cout << "The solution is: " << x << " and it was found in "
<< n << " iterations" << endl;
cout << "Overholt's method: " << endl << endl;
cout << "Give the initial approximation: ";
cin >> x;
double rez;
n = OverholtMethodForEquation(rez, x, 2);
if(n > MAXITER)
cout << "In " << MAXITER
<< " iterations no solution was found!" << endl;
else
cout << "The solution is: " << rez << " and it was found in "
<< n << " iterations" << endl;

Try and play with these algorithms too, and you will see that, indeed, Overholt's method is a stable and fast method for solving equations of type F(x) = x.

## Last words

In this article, I tried to present to you two of the most famous and useful equations in Mathematics, very useful in IT too, and I tried to present some numerical algorithms to solve them, very easy to understand, and also very easy to use in your applications.

## History

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