Homotopic residual correction processes


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Pan V., Kunin M., Rosholt R., Kodal H.

MATHEMATICS OF COMPUTATION, vol.75, no.253, pp.345-368, 2006 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 75 Issue: 253
  • Publication Date: 2006
  • Doi Number: 10.1090/s0025-5718-05-01771-0
  • Journal Name: MATHEMATICS OF COMPUTATION
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.345-368
  • Keywords: esidual correction, Newton's iteration, homotopic (continuation) algorithms, (generalized) inverse matrix, STRUCTURED MATRICES, NEWTONS ITERATION, GENERALIZED INVERSE, COMPUTATIONS, EQUATIONS
  • Kocaeli University Affiliated: Yes

Abstract

We present and analyze homotopic (continuation) residual correction algorithms for the computation of matrix inverses. For complex indefinite Hermitian input matrices, our homotopic methods substantially accelerate the known nonhomotopic algorithms. Unlike the nonhomotopic case our algorithms require no pre-estimation of the smallest singular value of an input matrix. Furthermore, we guarantee rapid convergence to the inverses of well-conditioned structured matrices even where no good initial approximation is available. In particular we yield the inverse of a well-conditioned n X n matrix with a structure of Toeplitz/Hankel type in O(n log(3) n) flops. For a large class of input matrices, our methods can be extended to computing numerically the generalized inverses. Our numerical experiments confirm the validity of our analysis and the efficiency of the presented algorithms for well-conditioned input matrices and furnished us with the proper values of the parameters that de. ne our algorithms.