Table of Contents
Numerical methods are algorithms used to solve mathematical problems numerically rather than analytically. MATLAB provides a versatile environment for implementing these methods efficiently. This article presents practical examples of how to implement common numerical techniques in MATLAB.
Solving Nonlinear Equations
The bisection method is a simple technique for finding roots of nonlinear equations. MATLAB code for the bisection method involves iteratively narrowing down an interval where the function changes sign.
Example MATLAB code:
Function: f(x) = x^3 – x – 2
Code:
“`matlab a = 1; b = 2; tol = 1e-5; while (b – a)/2 > tol c = (a + b)/2; if f(c) == 0 break; elseif f(a)*f(c) < 0 b = c; else a = c; end end root = c; “`
Numerical Integration
The trapezoidal rule approximates the integral of a function by dividing the area into trapezoids. MATLAB can implement this method with simple loops or built-in functions.
Example MATLAB code:
Function: f(x) = sin(x)
Code:
“`matlab a = 0; b = pi; n = 100; h = (b – a)/n; integral = 0; for i = 1:n x1 = a + (i – 1)*h; x2 = a + i*h; integral = integral + (f(x1) + f(x2))/2 * h; end “`
Solving Differential Equations
Euler’s method is a straightforward technique for solving initial value problems for ordinary differential equations. MATLAB’s implementation involves updating the solution iteratively.
Example MATLAB code:
Equation: dy/dx = y, with y(0) = 1
Code:
“`matlab x0 = 0; y0 = 1; h = 0.1; x_end = 1; x = x0:h:x_end; y = zeros(size(x)); y(1) = y0; for i = 1:length(x)-1 y(i+1) = y(i) + h * y(i); end “`