Computing the generalized singular values/vectors of large sparse or structured matrix pairs

Abstract

We present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair {A, B }. The algorithm is based on the CS decomposition and the Lanczos bidiagonalization process. At each iteration step of the Lanczos process, the solution to a linear least squares problem with (A T , B T ) T as the coefficient matrix is approximately computed, and this consists the only interface of the algorithm with the matrix pair {A, B }. Numerical results are also given to demonstrate the feasibility and efficiency of the algorithm.

Necessary Conditions and Tight Two-level Convergence Bounds for Parareal and Multigrid Reduction in Time

“…If we consider real-valued operators, then (from Remark 4) we seek the maximum generalized singular value of {Ψ − Φ k , I − Ψ} and {Ψ − Φ k , I + Ψ}. For sparse matrices, such as those that arise with explicit time stepping of differential discretizations, iterative methods have been developed to compute these values and vectors for relatively cheap and without forming (I − Ψ) −1 (for example, see [45]). Even if Φ and Ψ are implicit and thus contain inverses, iterative methods to compute the largest generalized singular value are typically applicable if the action of Φ and Ψ are available.…”

A QR-type reduction for computing the SVD of a general matrix product/quotient

“…The proof is similar to that of Lemma 2 and hence is omitted here. 21 = I , as shown in Lemma 2/ Lemma 3. Now we explain how the matrixˆsA or Aˆs, in which ∈ R n×n , s ∈ {±1}, and is nonsingular ifŝ = −1, can be reduced to the form…”

“…The purpose of this section is to show how the matrix A can be reduced to the form Q (1) 21 = I , by performing at most m QR-factorizations. Section 2.1 explains how the problem can be solved for a sequence of 2 or 3 matrices.…”

“…(b) As A 1 and A 3 are nonsingular, Q (2) 11 , Q (2) 22 , R (2) and Q (1) 12 , Q (1) 21 , R (1) are nonsingular as well. Since…”

“…The fact that the Cosine-Sine Decomposition (CSD) of a partitioned columnorthogonal matrix A C corresponds to the QSVD of the couple (A, C), forms the basis for two backward stable algorithms, proposed in [16] and [18]. Recently, the computation of the QSVD via the CSD and the Lanczos bidiagonalization process has been studied in [21]. The Kogbetliantz algorithm has been generalized in [15] for the computation of the QSVD.…”