Obtaining bounds on the two norm of a matrix from the splitting lemma

Doron Chen, John R. Gilbert, Sivan Toledo

Research output: Contribution to journalArticlepeer-review

Abstract

The splitting lemma is one of the two main tools of support theory, a framework for bounding the condition number of definite and semidefinite preconditioned linear systems. The splitting lemma allows the analysis of a complicated system to be partitioned into analyses of simpler systems. The other tool is the symmetric-product-support lemma, which provides an explicit spectral bound on a preconditioned matrix. The symmetric-product-support lemma shows that under suitable conditions on the null spaces of A and B, the finite eigenvalues of the pencil (A,B) are bounded by ∥W∥2 2, where U = VW, A = UUT, and B = VVT. To apply the lemma, one has to construct a W satisfying these conditions, and to bound its 2-norm. In this paper we show that in all its existing applications, the splitting lemma can be viewed as a mechanism to bound ∥W∥ 22, for a given W. We also show that this bound is sometimes tighter than other easily-computed bounds on ∥W∥ 22, such as ∥W∥F2, and ∥W∥1, ∥W∥. The paper shows that certain regular splittings have useful algebraic and combinatorial interpretations. In particular, we derive six separate algebraic bounds on the 2-norm of a real matrix; to the best of our knowledge, these bounds are new.

Original languageEnglish
Pages (from-to)28-46
Number of pages19
JournalElectronic Transactions on Numerical Analysis
Volume21
StatePublished - 2005

Keywords

  • Matrix norm bounds
  • Norm bounds for sparse matrices
  • Splitting lemma
  • Support preconditioning
  • Support theory
  • Two-norm

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