Numerical differentiation and integration
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Part A: Numerical differentiation
<pdfreflow> class_date = 01 to 04 November button_label = Create my course notes! show_page_layout = 1 show_frame_option = 1 pdf_file = E-Numerical-differentiation-01-Nov-2010.pdf </pdfreflow>
Part B: Numerical integration
<pdfreflow> class_date = 08 to 15 November button_label = Create my course notes! show_page_layout = 1 show_frame_option = 1 pdf_file = E-Numerical-integration-10-Nov-2010.pdf </pdfreflow>
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 |
|---|---|
function I = romberg(f, a, b, p)
I = zeros(p, p);
for k=1:p
% Composite trapezoidal rule for 2^k panels
I(k,1) = trapezcomp(f, a, b, 2^(k-1));
% Romberg recursive formula
for j=1:k-1
I(k,j+1) = (4^j * I(k,j) - I(k-1,j)) / (4^j - 1);
end
disp(I(k,1:k)); % display intermediate results
end
% end of function
Then use this function as follows: >> f = @(x)sin(x);
>> format long
>>
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