Gauss-Laguerre Formula: MATLAB Implementation and Applications

This text provides comprehensive information about the Gauss-Laguerre formula, a mathematical formula used for calculating orthogonal polynomials. Its significance lies in solving numerous physics and engineering problems, particularly in quantum mechanics and numerical integration. For MATLAB implementation of the Gauss-Laguerre formula, here's an enhanced code example: The code demonstrates the recursive generation of Laguerre polynomials using the three-term recurrence relation. Key implementation details include: x = linspace(-1,1,1000); % Define evaluation points across interval [-1,1] n = 5; % Set polynomial degree to 5 alpha = 0; % Parameter alpha for generalized Laguerre polynomials beta = 0; % Parameter beta for generalized Laguerre polynomials L = zeros(length(x), n+1); % Preallocate matrix for polynomial values % Initialize first two polynomials with normalization factors L(:,1) = 1/sqrt(2); % L0(x) normalized L(:,2) = x/sqrt(3); % L1(x) normalized % Recursive computation using three-term recurrence relation for k = 2:n % Calculate recurrence coefficients a = k*(alpha + beta + k - 1)/(k + alpha + beta)/(k + alpha); b = (2*k + alpha + beta - 1)*(alpha + beta + k)/(k + alpha + beta)/(k + alpha)/(k + beta); % Compute next polynomial using recurrence: L_{k+1} = (x*L_k - a*L_{k-1})/sqrt(b) L(:,k+1) = (x.*L(:,k) - a*L(:,k-1))/sqrt(b); end This implementation efficiently computes orthogonal polynomials using vectorized operations, where each column of matrix L contains values of Laguerre polynomials L0 through L5 evaluated at points x. The normalization factors ensure orthonormality with respect to the appropriate weight function. This example helps understand both the practical application and implementation methodology of Gauss-Laguerre formulas in numerical computations, demonstrating how to handle polynomial recurrence relations efficiently in MATLAB.