Fast Arnoldi Algorithm for Generating Krylov Subspaces

In mathematics and computer science, the Arnoldi algorithm is an efficient iterative method for generating Krylov subspaces. A Krylov subspace is a linear subspace generated by a matrix A and a vector v, with broad applications across multiple disciplines. The Arnoldi algorithm employs an iterative orthogonalization process where each iteration adds a new vector to the Krylov subspace until reaching the desired dimension. The algorithm typically uses modified Gram-Schmidt orthogonalization to maintain numerical stability while constructing an orthonormal basis. Key computational steps include matrix-vector products and orthogonal projections, making it particularly suitable for large sparse matrices. This algorithm finds extensive applications in engineering, physics, and finance - for instance, in calculating transmission line performance in circuit analysis or solving large-scale linear systems Ax=b through methods like GMRES. The implementation often involves Hessenberg reduction and eigenvalue approximation for improved computational efficiency.