Numerical methods

Numerical methods are used to find approximate solutions to mathematical problems where no simple analytical formula exists. They play an important role in applied mathematics, physics, and engineering, where exact calculations are often impossible or very difficult.

 

A numerical method consists of replacing a theoretical infinite process with a computable approximation that can be performed on a computer. This makes it possible to calculate values of functions, integrals, derivatives, and differential equations based on a set of known data points or a given formula.

 

 

Root-finding methods

Root-finding methods are used to locate points where a function satisfies \( \large f(x) = 0 \). In practice, iterative procedures are applied that approach the root step by step. The most common ones are the bisection method, Newton-Raphson, and the secant method.

 

 

Numerical integration

When an integral cannot be solved symbolically, it can be calculated by approximating the area under the graph. This is done by dividing the interval into smaller parts and summing the areas of simple figures that each approximate the curve locally.

 

$$ \large \int_a^b f(x)\,dx \approx \sum_{i=1}^{n} A_i $$

 

The most commonly used methods are the trapezoidal rule and Simpson’s rule, which differ in how they approximate the function between the given points.

 

 

Numerical differentiation

A derivative function can also be estimated from discrete points.

Here, difference quotients are used, where two or more nearby points are used to calculate an approximate value of \( \large f'(x) \).

 

$$ \large f'(x) \approx \frac{f(x+h) - f(x)}{h} $$

 

 

Numerical differential equations

Differential equations can rarely be solved exactly, but they can be approximated by calculating function values step by step. The simplest methods, such as Euler’s method and Runge–Kutta, compute the function at a point based on its known value and slope at a previous point.

 

$$ \large y_{n+1} = y_n + h \cdot f(x_n, y_n) $$

 

These methods form the basis of many numerical simulations and computations where an exact formula does not exist.