Numerical differentiation and integration

Part A: Numerical differentiation

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Part B: Numerical integration

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Source code used in slides

Composite trapezoidal integration

MATLAB

Python

function In = trapezcomp(f, a, b, n) % Composite trapezoidal function integration % % INPUTS: % f: the function to integrate % a: lower bound of integration % b: upper bound % n: number of panels to create between ``a`` and ``b`` % Initialization h = (b-a)/n; x = a; % Composite rule In =f(a); for k=2:n x = x + h; In = In + 2. * f(x); end In = (In + f(b)) * h * 0.5; % end of function Then use this function as follows: >> f = @(x)sin(x); >> I1 = trapezcomp( f, 0, pi/2, 1 ) % 0.7854 >> I2 = trapezcomp( f, 0, pi/2, 2 ) % 0.9481

import numpy as np def trapezcomp(f, a, b, n): """ Composite trapezoidal function integration INPUTS: f: the function to integrate a: lower bound of integration b: upper bound n: number of panels to create between ``a`` and ``b`` """ # Initialization h = (b - a) / n x = a # Composite rule In = f(a) for k in range(1, n): x = x + h In += 2*f(x) return (In + f(b))*h*0.5 if __name__ == '__main__': def func(x): return np.sin(x) print(trapezcomp(func, 0, np.pi/2, 1)) # 0.785398163397 print(trapezcomp(func, 0, np.pi/2, 2)) # 0.948059448969

Romberg integration

MATLAB	Python
Then use this function as follows: >> f = @(x)sin(x); >> format long >>