ON USING H OLDER NORMS IN THE QUASI-MINIMAL RESIDUAL APPROACH

1 Introduction.
(1.1)
There is a vast number of Krylov subspace methods for the iterative solution of large sparse linear systems
Ax = b
with general non-Hermitian coe cient matrices A. Among those involving short recurrences, i.e., a constant amount of work and storage per iteration, the ones based on the quasi-minimal residual (QMR-)approach 3, 4, 6, 7] are getting a lot of attention. The QMR-approach is designed to produce a smooth and nearly monotone convergence behavior in the residual norm by minimizing a factor of
BIT 2000, Vol. xxx, No. xxx, pp. xx{xx
0006-3835/00/4004-0001 $15.00 c Swets & Zeitlinger
ON USING HOLDER NORMS IN THE QUASI-MINIMAL RESIDUAL APPROACH
H. MARTIN BUCKER1 and MANFRED SAUREN2
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2
H. M. BUCKER AND M. SAUREN
the residual vector in each iteration. The key behind the approach is the e cient solution of a sequence of least-squares problems of the form (1.2) yn = argmin k 1e(1n+1) ? Hnyk2 n upper Hessenberg matrix of full column rank, and n denotes the iteration number. The standard technique to solve (1.2) is to compute a QR decomposition of Hn by means of Givens rotations leading to short recurrences in 4, 6, 7, 11]. This technique is restricted to the minimization in the 2-norm because it relies upon unitary transformations. In contrast to the 2-norm, general (Holder) p-norms are not preserved under unitary transformations. Therefore, it seemed reasonable to suppose that the derivation of any iterative method based on a generalization of the minimization problem (1.2) from the Euclidean norm to a general p-norm, though possible, would lead to long recurrences rather than to short recurrences. Due to the Faber{Manteu el theorem 2], such a method would be senseless because long recurrences can be used to minimize not only a factor of the residual but the residual itself; see 5, 8, 9, 12] for an overview of Krylov subspace methods. We will refer to the above mentioned generalization of the standard QMR-approach as the p-norm QMR-approach. In this note, it is shown that methods based on the p-norm QMR-approach that still involve short recurrences can be derived if the upper Hessenberg matrix reduces to a lower bidiagonal matrix. The key ingredient to the approach is the e cient solution of a sequence of p-norm minimization problems. More precisely, the solution is shown to satisfy an explicit update formula involving only a xed number of preceding solutions. The structure of this note is as follows. We begin this note in x2 showing how a sequence of p-norm minimization problems similar to (1.2) can be solved e ciently if the Hessenberg matrix is bidiagonal. Given a suitable process to span the underlying Krylov spaces, the p-norm QMR-approach is introduced in x3 in order to de ne iterative methods for the solution of linear systems. The following section, x4, is devoted to derive an actual implementation of the resulting method. Numerical experiments are reported in x5. Throughout this note, the system of linear equations (1.1) is assumed to have a nonsingular, in general non-Hermitian, N N coe cient matrix A with real or complex entries and a right-hand side b 2 C N . Furthermore, the notation Kn (y ; A) := spanfy; Ay; : : : ; An?1 yg is used to denote the nth Krylov subspace of C N generated by y 2 C N and A. The symbol e(n) = (0; : : : ; 0; 1; 0; : : : ; 0)T 2 R n with 1 at position j denotes the j j th canonical unit vector. As usual, the p-norm is de ned by where 1 6= 0, e(n+1) = (1; 0; : : : ; 0)T 1
1
y
Institute for Scienti c Computing, Aachen University of Technology 52056 Aachen, Germany. email: buecker@sc.rwth-aachen.de 2 MicroStrategy Deutschland GmbH, Kolner Stra e 263 51149 Koln, Germany. email: msauren@microstrategy.com
Abstract.
Given a suitable process to span a basis for the underlying Krylov subspaces, the quasi-minimal residual (QMR-)approach has often been applied to derive iterative methods for the solution of linear systems. The QMR-approach is only reasonable if the resulting methods are based on short recurrences. The key ingredient of the QMR-approach is the e cient solution of a sequence of least-squares problems by computing the QR decomposition of an upper Hessenberg matrix by means of Givens rotations. Since (Holder) p-norms are not preserved under unitary transformations, a generalization of the minimization problem from the Euclidean norm to general p-norms while still leading to methods based on short recurrences appeared infeasible. Here, it is shown that this kind of generalization is possible if the upper Hessenberg matrix reduces to a lower bidiagonal matrix. AMS subject classi cation: 65F10. Key words: Krylov subspace iteration, non-Hermitian linear systems, p-norm quasiminimal residual approach.
Received September 2000. Revised September 200x. Communicated by xxx work was partially supported by the Graduiertenkolleg \Informatik und Technik", Aachen University of Technology, 52056 Aachen, Germany, and Zentralinstitut fur Angewandte Mathematik, Forschungszentrum Julich, 52425 Julich, Germany.