Investigating the Effects of MINRES with Local Reorthogonalization

The minimum residual method, MINRES, is an iterative method for solving a n x n linear system, Ax = b, where is A is a symmetric matrix. It searches for a vector, xk, in the k th Krylov subspace that minimizes the residual, rk = b − Axk. Due to the symmetric nature, the vectors are computed to be the Lanczos vectors, which simplifies to a three-term recurrence, where vk+1 is computed as a linear combination of the previous two Lanczos vectors vk and vk−1. To update the new vector in the (k + 1) th Krylov subspace, a QR decomposition is maintained and each iteration, we just must compute a new plane or Givens rotation [3]. In exact arithmetic, all the Lanczos vectors are orthogonal to each other. However, the problem that arises is numerically, this orthogonality is not maintained. This is contrasted by the generalized minimum residual method, GMRES, which is an iterative method for solving the same linear system that minimizes the residual, except now A may be an unsymmetric matrix. Here, the new vector vk+1 in the orthogonal basis for the Krylov subspace is calculated in the Arnoldi recurrence process, which requires storing all the previous vectors. Thus, GMRES requires more storage and work than MINRES does. However, some users prefer to use GMRES with a good preconditioner, M , even on symmetric systems, even though MINRES is designed to take advantage of the symmetric properties. The reason for this being that since GMRES uses a modified Gram-Schmidt orthogonalization in the Arnoldi process, the vectors in the basis, Vk, for the k th Krylov subspace remain orthogonal and so do not lose the numerical orthogonality property as happens in MINRES. Thus, fewer total iterations may be required of GMRES on the symmetric saddle-point systems than MINRES. Therefore, we investigate a process to reorthogonalize the vectors in MINRES, by storing an input parameter, localSize of them and explicitly making the new vk+1 orthogonal to the previous localSize Lanczos vectors. The theory that we will test is that this will decrease the number of iterations and make MINRES the more preferable method on symmetric systems, as it was originally designed.