Numerical analysis

Introduction

In the world of computer science and programming, numerical analysis plays a crucial role in solving complex mathematical problems. This field focuses on the development and analysis of algorithms that use numerical approximation techniques to tackle mathematical analysis problems. Unlike symbolic manipulations, numerical analysis deals with numerical values and their computations.
In this article, we will dive into the realm of numerical analysis, exploring its applications, algorithms, and providing code examples in popular programming languages such as C#, JavaScript, Python, and PHP.

Understanding Numerical Analysis

Numerical analysis encompasses a wide range of techniques used to solve mathematical problems that are difficult or impossible to solve analytically. By using numerical approximation, we can obtain valuable insights and solutions to problems in various scientific and engineering fields.

Applications of Numerical Analysis

Numerical analysis finds applications in numerous domains, including:

Engineering: Numerical analysis is used to solve complex structural engineering problems, such as determining the stress distribution in a bridge or the behavior of a fluid flow in a pipe.

Physics: Numerical methods are employed in physics to simulate physical phenomena, such as the motion of celestial bodies, fluid dynamics, or quantum mechanics.

Finance: Financial institutions rely on numerical analysis to evaluate investment risks, determine asset pricing, and model complex financial systems.

Computer Graphics: Techniques from numerical analysis are used to render realistic images, simulate physical effects, and create visually stunning graphics in video games and movies.

Numerical Algorithms and Code Examples

Let's explore some common numerical algorithms and provide code examples in different programming languages.

Root-Finding Algorithms:

Root-finding algorithms are used to find the roots or solutions of equations. One popular method is the Bisection Method. Here's an example implementation in C#:

Interpolation Methods:

Interpolation methods approximate a function based on a set of known data points. A popular technique is the Newton's Divided Difference Interpolation. Here's an example implementation in JavaScript:

Numerical Integration:

Numerical integration techniques approximate the definite integral of a function. The Trapezoidal Rule is a commonly used method. Here's an example implementation in Python:

Linear Systems Solvers:

Numerical analysis also deals with solving linear systems of equations. One widely used method is Gaussian Elimination with Backward Substitution. Here's an example implementation in PHP:

Links

Code Examples

C#
public static double BisectionMethod(Func<double, double> function, double a, double b, double epsilon)
{
    double c;
    do
    {
        c = (a + b) / 2;
        if (function(c) == 0 || Math.Abs(b - a) < epsilon)
            break;
        if (function(a) * function(c) < 0)
            b = c;
        else
            a = c;
    } while (true);
    return c;
}

JavaScript
function dividedDifferenceInterpolation(x, y, target)
{
    let n = x.length;
    let coefficients = [...y];
    for (let j = 1; j < n; j++)
    {
        for (let i = n - 1; i >= j; i--)
        {
            coefficients[i] = (coefficients[i] - coefficients[i - 1]) / (x[i] - x[i - j]);
        }
    }
    let result = coefficients[0];
    let product = 1;
    for (let i = 1; i < n; i++)
    {
        product *= (target - x[i - 1]);
        result += coefficients[i] * product;
    }
    return result;
}

Python
def trapezoidal_rule(f, a, b, n):
    h = (b - a) / n
    result = (f(a) + f(b)) / 2
    for i in range(1, n):
        result += f(a + i * h)
    return result * h

PHP
function gaussianElimination($matrix, $vector)
{
    $n = count($matrix);
    for ($i = 0; $i < $n; $i++)
    {
        $pivot = $matrix[$i][$i];
        for ($j = $i + 1; $j < $n; $j++)
        {
            $ratio = $matrix[$j][$i] / $pivot;
            for ($k = $i; $k < $n; $k++)
            {
                $matrix[$j][$k] -= $matrix[$i][$k] * $ratio;
            }
            $vector[$j] -= $vector[$i] * $ratio;
        }
    }
    $solution = array_fill(0, $n, 0);
    for ($i = $n - 1; $i >= 0; $i--)
    {
        $sum = 0;
        for ($j = $i + 1; $j < $n; $j++)
        {
            $sum += $matrix[$i][$j] * $solution[$j];
        }
        $solution[$i] = ($vector[$i] - $sum) / $matrix[$i][$i];
    }
    return $solution;
}

Conclusion