# Finite Differences with Toeplitz Matrix

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Join For FreeA Toeplitz matrix is a band matrix in which each descending diagonal
from left to right is constant. In this post we will see how to
approximate the derivative of a function

The result of the test above is as follows:

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

Forward difference,

Backward difference,

And central difference,

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).

*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,

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).

Matrix (protocol)

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