Computing tall skinny solutions of AX - XB = C

Authors:
Jan Brandts
Affiliations:
Mathematical Institute, Utrecht University, P.O. Box 80,010, 3508 TA, Utrecht, The Netherlands
Venue:
Mathematics and Computers in Simulation - MODELLING 2001 - Second IMACS conference on mathematical modelling and computational methods in mechanics, physics, biomechanics and geodynamics
Year:
2003

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Abstract

We will concentrate on the numerical computation of so-called tall and skinny solutions X of the Sylvester equations AX - XB = C. By this we mean that A is an n × n matrix with cheaply applicable action (A is for example sparse), and B a k × k matrix, with k ≤ n. This type of Sylvester equation plays an important role in the computation of invariant subspaces when block Rayleigh quotient or block Jacobi-Davidson methods are used.There exist essentially two types of subspace projection methods for such equations. One in which the tall skinny matrices are treated as rigid objects, and one in which they are treated as a basis of a subspace. In this paper we review the two different types, design new methods, and aim to identify which application should make use of which solution method.