The Block Lanczos Method for Computing Eigenvalues

TL;DR: A block biconjugate gradient algorithm for general matrices is developed, and block conjugate gradient, minimum residual, and minimum error algorithms for symmetric semidefinite matrices are developed.

TL;DR: Four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture are presented and may help advance the development of future out-of-core sparse SVD methods, which can be used to handle extremely large sparsematrices associated with extremely large databases in query-based information-retrieval applications.

TL;DR: An "industrial strength" algorithm for solving sparse symmetric generalized eigenproblems that can live as a black box eigensolver inside a large applications code and is a novel combination of new techniques and extensions of old techniques.

TL;DR: This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems, that have a proven track record for real-world problems and treats both the theoretical foundations of the subject and practical computational issues.

TL;DR: A particular computation algorithm for the method without reorthogonalization is shown to have remarkably good error properties, and this suggests that this variant of the Lanczos process is likely to become an extremely useful algorithm for finding several extreme eigenvalues, and their eigenvectors if needed, of very large sparse symmetric matrices.

TL;DR: In this paper, the convergence rate of the conjugate gradient method is analyzed in terms of the eigenvalues of the matrix A and the matrix B. The convergence rate for the positive eigenvalue is not perturbed so much by the existence of negative eigen values having large modulus.