Fast Computation of a Restricted Subset of Eigenpairs of a Varying Hermitian Matrix
Standard algorithms require an order of dm 2 flops to compute d eigenpairs of a m × m semi-definite positive hermitian matrix, R. A larger amount of computations (of same order) is required to find d left singular pairs of a square matrix, D, such that R = DD H ; this is known to be numerically more stable. Note that D is not unique, the best choice in the sequel turns out not to be the Cholesky factor of R. The algorithm proposed in this contribution will compute d left singular pairs of D in a faster way, such that it requires an order of αmd 2 flops. This algorithm takes advantage of the fact that the matrix considered is nothing else but a matrix whose d eigenpairs are known, perturbed by a rank-one modification. It is pointed out in the sequel why the technique of Bunch et al  is not relevant for O(md 2) algorithms, neither for square-root implementations.
KeywordsCholesky Factor Ritz Vector Subspace Iteration Positive Definite Hermitian Matrix Singular Triplet
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