Differentiation-Matrix.md

Differentiation Matrix using Chebyshev Nodes

This document explains the derivation of the differentiation matrix based on Chebyshev nodes, which is useful for pseudospectral methods, interpolation, and solving boundary value problems.

1. Lagrange Polynomials

Given ( n+1 ) Chebyshev nodes ( x_0, x_1, \dots, x_n ) (often taken as ( x_j = \cos\left(\frac{j\pi}{n}\right) ), ( j = 0, 1, \dots, n )), the Lagrange polynomial ( L_k(x) ) associated with each node is defined as:

[ L_k(x) = \prod_{\substack{0 \leq j \leq n \ j \neq k}} \frac{x - x_j}{x_k - x_j}. ]

Each ( L_k(x) ) is a polynomial of degree ( n ) that satisfies:

[ L_k(x_j) = \delta_{kj} = \begin{cases} 1 & \text{if} \ j = k, \ 0 & \text{if} \ j \neq k. \end{cases} ]

2. Derivative of Lagrange Polynomials

To obtain the differentiation matrix, we need to differentiate ( L_k(x) ), which can be written as:

[ L_k'(x) = \sum_{\substack{0 \leq j \leq n \ j \neq k}} \left( \prod_{\substack{0 \leq m \leq n \ m \neq k, m \neq j}} \frac{x - x_m}{x_k - x_m} \right) \cdot \frac{1}{x_k - x_j}. ]

This gives the derivative of the Lagrange polynomial ( L_k(x) ), but we are primarily interested in evaluating it at the Chebyshev nodes ( x_i ).

2.1. Derivative at the Chebyshev Nodes

For Chebyshev nodes, we can simplify the derivative at node ( x_i ) (for ( i \neq k )) as:

[ L_k'(x_i) = \frac{(-1)^{i+k}}{x_i - x_k} \cdot \frac{c_i}{c_k}, \quad i \neq k, ]

where:

• ( c_i = 2 ) for ( i = 0 ) and ( i = n ),
• ( c_i = 1 ) for all other nodes.

For the diagonal entries where ( i = k ), the derivative is:

[ L_k'(x_k) = -\sum_{j \neq k} L_k'(x_j). ]

3. The Differentiation Matrix

We can now assemble the differentiation matrix ( D ) using the formulas above:

• For ( i \neq j ):

[ D_{ij} = \frac{(-1)^{i+j}}{x_i - x_j}, \quad i \neq j, ]

• For diagonal entries ( i = j ):

[ D_{ii} = -\sum_{j \neq i} D_{ij}. ]

This differentiation matrix can then be used for numerical differentiation of functions sampled at Chebyshev nodes.

Example Code

Here’s an example code in MATLAB to compute the differentiation matrix for Chebyshev nodes:

function D = chebyshev_diff_matrix(n)
    % Compute the Chebyshev nodes
    x = cos(pi*(0:n)/n)';
    
    % Compute the differentiation matrix
    c = [2; ones(n-1,1); 2].*(-1).^(0:n)';
    X = repmat(x,1,n+1);
    dX = X - X' + eye(n+1);
    D  = (c*(1./c)')./(dX + eye(n+1));
    
    % Set the diagonal elements
    D = D - diag(sum(D,2));
end