Many algorithms exist for computing the symmetric eigendecomposition, the singular value decomposition and the generalized singular value decomposition. In this thesis, we present several new algorithms and improvements on old algorithms, analyzing them with respect to their speed, accuracy, and storage requirements. We rst discuss the variations on the bisection algorithm for nding eigenvalues of symmetric tridiagonal matrices. We show the challenges in implementing a correct al-gorithm with oating point arithmetic. We show how reasonable looking but incorrect implementations can fail. We carefully de ne correctness, and present several implementa-tions that we rigorously prove correct. We then discuss a fast implementation of bisection using parallel pre x. We show many numerical examples of the instability of this algorithm, and then discuss its forward error and backward error analysis. We also discuss possible ways to stabilize it by using iterative re nement. Finally, we discuss how to use a divide-and-conquer algorithm to compute the sin-gular value decomposition and solve the linear least squares problem, and how to implement

Abstract. We discuss a new method for the iterative computation of some of the generalized singular values and vectors of a large sparse matrix. Our starting point is the augmented matrix formulation of the GSVD. The subspace expansion is performed by (approximately) solving a Jacobi–Davidson type correction equation, while we give several alternatives for the subspace extraction. Numerical experiments indicate the efficiency of the method. Key words. Generalized singular value decomposition (GSVD), partial GSVD, Jacobi–Davidson, subspace method, augmented matrix, correction equation, (inexact) accelerated Newton, refined extraction, harmonic extraction. AMS subject classifications. 65F15, 65F50, (65F30). 1. Introduction. The generalized singular value decomposition (GSVD) was introduced by Van Loan [15] and further developed by Paige and Saunders [9]. Let A ∈ R m×n and B ∈ R p×n be given. The generalized singular values of the pair (A, B) are [15, Def. 1] Σ(A, B) = {σ ≥ 0 | A T A − σ 2 B T B singular}. The (diagonal form of the) GSVD of A and B is given by [15, Th. 2], [9, p. 399]

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