Approximating the leading singular triplets of a large matrix function

作者：Gaaf Sarah W; Simoncini Valeria^*

来源：Applied Numerical Mathematics, 2017, 113: 26-43.

DOI：10.1016/j.apnum.2016.10.015

• Matrix functions
• Singular values
• 2-norm
• Lanczos bidiagonalization
• Inexact iteration

Given a large square matrix A and a sufficiently regular function f so that f (A) is well defined, we are interested in the approximation of the leading singular values and corresponding left and right singular vectors of f (A), and in particular in the approximation of vertical bar vertical bar f (A) vertical bar, where vertical bar vertical bar . vertical bar vertical bar is the matrix norm induced by the Euclidean vector norm. Since neither f (A) nor f (A)v can be computed exactly, we introduce a new inexact Golub-Kahan-Lanczos bidiagonalization procedure, where the inexactness is related to the inaccuracy of the operations f (A)v, f (A)*v. Particular outer and inner stopping criteria are devised so as to cope with the lack of a true residual. Numerical experiments with the new algorithm on typical application problems are reported.

• 出版日期2017-3