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Math. Nachr. 208 (1999), 5-29 Verification Algorithms for Generalized Singular Values

Summary: Math. Nachr. 208 (1999), 5-29
Verification Algorithms for Generalized Singular Values
By GÖTZ ALEFELD of Karlsruhe, ROLF HOFFMANNof Karlsdorf
and GÜNTER MAYER of Rostock
Dedicated to Prof. Dr. G. WILDENHAIN, Rostock, on the occasion 01 his 6(jh birthday
(Received January 7, 1997)
(Revised Version December 18, 1998)
Abstract. By means of interval arithmetic tools we present new algorithms for verifying and
encIosing generalized singular values and corresponding vectors for a matrix pair (A, B) E 1R.pxn x
JRqxn. To this end, we state and prove a fundamental theorem in interval analysis which shows a way
how enclosures can be constructed if approximations are known. Furthermore, we perform a careful
comparison of the new method with those introduced in [11].
1. Introduction
We first address the main purpose of our paper. It consists in constructing tight
intervals which contain the components c, s of a generalized singular value and the
components of a column of U, V and X, respectively. This means, in particular, that
we provide a method which verifies a generalized singular value and the corresponding
column vectors of U, V, X. Gathering these quantities into a vector z* we will inter-
prete z* as a fixed point of some nmction t. We will expand t into a Taylor series at an
approximation Z of z*. The interval arithmetic evaluation of the resulting expression


Source: Alefeld, Götz - Institut für Angewandte und Numerische Mathematik & Fakultät für Mathematik, Universität Karlsruhe


Collections: Mathematics