MATLAB Implementation of Lanczos Algorithm for Symmetric Matrices

This article explores how to implement the Lanczos algorithm for symmetric matrices using MATLAB. We begin by defining symmetric matrices and explaining the fundamental principles of the Lanczos algorithm, which efficiently tridiagonalizes symmetric matrices through iterative orthogonal projection. The implementation leverages MATLAB's built-in matrix operations and handles the three-term recurrence relation using vectorized computations.

We then detail the programming methodology, including key steps such as initialization of starting vectors, iterative Arnoldi-like reduction process, and maintaining orthogonality through reorthogonalization techniques. The code structure demonstrates how to store tridiagonal elements efficiently using sparse matrix representation and computes Ritz values/vectors for eigenvalue approximations.

A practical example illustrates the application of our MATLAB program to solve real-world problems, comparing computational efficiency with full eigenvalue decomposition. The implementation includes error handling for numerical stability and discusses convergence criteria based on residual norms. This guide aims to enhance understanding of both Lanczos algorithm mechanics and effective MATLAB programming practices for large-scale symmetric eigenvalue problems.