MATLAB Program for Computing Eigenvalues Using Lanczos Algorithm

The following is a MATLAB program example that uses the Lanczos algorithm to compute matrix eigenvalues. The Lanczos algorithm is a widely used method for solving large-scale eigenvalue problems, which gradually approximates matrix eigenvalues through an iterative process. The core principle involves constructing a tridiagonal matrix through orthogonalization with the matrix's Krylov subspace, then solving for eigenvalues. Key implementation details include vector initialization, iterative orthogonalization steps, and tridiagonal matrix construction using MATLAB's diagonal matrix functions. MATLAB code implementation: % Define matrix A A = [1 2 3; 4 5 6; 7 8 9]; % Set number of Lanczos iterations k k = 3; % Initialize starting vector v0 v0 = ones(3,1); % Compute initial vectors and coefficients v1 = A*v0; alpha = dot(v1,v0); v1 = v1 - alpha*v0; beta = norm(v1); v1 = v1/beta; v2 = A*v1 - alpha*v0 - beta*v1; % Lanczos iteration loop for i = 2:k alpha = dot(v2,v1); v2 = v2 - alpha*v1 - beta*v0; beta = norm(v2); v2 = v2/beta; v0 = v1; v1 = v2; end % Construct tridiagonal matrix T using diagonal functions T = diag(alpha*ones(k,1),0) + diag(beta*ones(k-1,1),1) + diag(beta*ones(k-1,1),-1); % Solve eigenvalues and eigenvectors of T [V,D] = eig(T); % Extract eigenvalue approximations for matrix A eig_val = diag(D); disp('Approximated eigenvalues of matrix A:'); disp(eig_val); This program serves as a reference implementation. For practical applications, adjustments should be made according to specific requirements. Carefully review the algorithmic flow and ensure understanding of the Lanczos method's mathematical foundation and computational steps.