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Adaptive Lanczos methods for recursive condition estimation

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Abstract

Estimates for the condition number of a matrix are useful in many areas of scientific computing, including: recursive least squares computations, optimization, eigenanalysis, and general nonlinear problems solved by linearization techniques where matrix modification techniques are used. The purpose of this paper is to propose anadaptiveLanczosestimator scheme, which we callale, for tracking the condition number of the modified matrix over time. Applications to recursive least squares (RLS) computations using the covariance method with sliding data windows are considered.ale is fast for relatively smalln-parameter problems arising in RLS methods in control and signal processing, and is adaptive over time, i.e., estimates at timet are used to produce estimates at timet+1. Comparisons are made with other adaptive and non-adaptive condition estimators for recursive least squares problems. Numerical experiments are reported indicating thatale yields a very accurate recursive condition estimator.

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Authors and Affiliations

  1. Department of Mathematics, NC State University, 27695-8205, Raleigh, NC, USA

    William R. Ferng

  2. Department of Computer Science, Stanford University, 94305, Stanford, CA, USA

    Gene H. Golub

  3. Department of Mathematics and Computer Science, Wake Forest University, P.O. Box 7388, 27109, Winston-Salem, N.C., USA

    Robert J. Plemmons

Authors
  1. William R. Ferng
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  2. Gene H. Golub
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  3. Robert J. Plemmons
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Additional information

Research supported by the US Air Force under grant no. AFOSR-88-0285.

Research supported by the US Army under grant no. DAAL03-90-G-105.

Research supported by the US Air Force under grant no. AFOSR-88-0285.

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Ferng, W.R., Golub, G.H. & Plemmons, R.J. Adaptive Lanczos methods for recursive condition estimation. Numer Algor 1, 1–19 (1991). https://doi.org/10.1007/BF02145580

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