# Finite Differences with Toeplitz Matrix

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Join For Free*f(x)*as matrix-vector products between a Toeplitz matrix and a vector of equally spaced values of

*f*. Let's see how to generate the matrices we need using the function toeplitz(...) provided by numpy:

from numpy import * from scipy.linalg import toeplitz import pylab def forward(size): """ returns a toeplitz matrix for forward differences """ r = zeros(size) c = zeros(size) r[0] = -1 r[size-1] = 1 c[1] = 1 return toeplitz(r,c) def backward(size): """ returns a toeplitz matrix for backward differences """ r = zeros(size) c = zeros(size) r[0] = 1 r[size-1] = -1 c[1] = -1 return toeplitz(r,c).T def central(size): """ returns a toeplitz matrix for central differences """ r = zeros(size) c = zeros(size) r[1] = .5 r[size-1] = -.5 c[1] = -.5 c[size-1] = .5 return toeplitz(r,c).T # testing the functions printing some 4-by-4 matrices print 'Forward matrix' print forward(4) print 'Backward matrix' print backward(4) print 'Central matrix' print central(4)

The result of the test above is as follows:

Forward matrix [[-1. 1. 0. 0.] [ 0. -1. 1. 0.] [ 0. 0. -1. 1.] [ 1. 0. 0. -1.]] Backward matrix [[ 1. 0. 0. -1.] [-1. 1. 0. 0.] [ 0. -1. 1. 0.] [ 0. 0. -1. 1.]] Central matrix [[ 0. 0.5 0. -0.5] [-0.5 0. 0.5 0. ] [ 0. -0.5 0. 0.5] [ 0.5 0. -0.5 0. ]]

We can observe that the matrix-vector product between those matrices and the vector of equally spaced values of

*f(x)*implements, respectively, the following equations:

Forward difference,

Backward difference,

And central difference,

Â

where h is the step size between the samples. Those equations are called Finite Differences and they give us an approximate derivative of *f*. So, let's approximate some derivatives!

x = linspace(0,10,15) y = cos(x) # recall, the derivative of cos(x) is sin(x) # we need the step h to compute f'(x) # because the product gives h*f'(x) h = x[1]-x[2] # generating the matrices Tf = forward(15)/h Tb = backward(15)/h Tc = central(15)/h pylab.subplot(211) # approximation and plotting pylab.plot(x,dot(Tf,y),'g',x,dot(Tb,y),'r',x,dot(Tc,y),'m') pylab.plot(x,sin(x),'b--',linewidth=3) pylab.axis([0,10,-1,1]) # the same experiment with more samples (h is smaller) x = linspace(0,10,50) y = cos(x) h = x[1]-x[2] Tf = forward(50)/h Tb = backward(50)/h Tc = central(50)/h pylab.subplot(212) pylab.plot(x,dot(Tf,y),'g',x,dot(Tb,y),'r',x,dot(Tc,y),'m') pylab.plot(x,sin(x),'b--',linewidth=3) pylab.axis([0,10,-1,1]) pylab.legend(['Forward', 'Backward', 'Central', 'True f prime'],loc=4) pylab.show()The resulting plot would appear as follows:

As the theory suggests, the approximation is better when h is smaller and the central differences are more accurate (note that, they have an higher order of accuracy respect to the backward and forward ones).

Published at DZone with permission of Giuseppe Vettigli, DZone MVB. See the original article here.

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