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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Error analysis of $QR$ updating with exponential windowing
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by G. W. Stewart PDF
Math. Comp. 59 (1992), 135-140 Request permission


Exponential windowing is a widely used technique for suppressing the effects of old data as new data is added to a matrix. Specifically, given an $n \times p$ matrix ${X_n}$ and a "forgetting factor" $\beta \in (0,1)$, one works with the matrix ${\operatorname {diag}}({\beta ^{n - 1}},{\beta ^{n - 2}}, \ldots ,1){X_n}$. In this paper we examine an updating algorithm for computing the QR factorization of ${\operatorname {diag}}({\beta ^{n - 1}},{\beta ^{n - 2}}, \ldots ,1){X_n}$ and show that it is unconditionally stable in the presence of rounding errors.
  • Gene H. Golub and Charles F. Van Loan, Matrix computations, 2nd ed., Johns Hopkins Series in the Mathematical Sciences, vol. 3, Johns Hopkins University Press, Baltimore, MD, 1989. MR 1002570
  • G. W. Stewart, An updating algorithm for subspace tracking, Technical Report CS-TR 2494, Department of Computer Science, Univ. of Maryland, 1990 (to appear in IEEE Trans. Acoust. Speech Signal Process.).
  • G. W. Stewart, Updating a rank-revealing $ULV$ decomposition, SIAM J. Matrix Anal. Appl. 14 (1993), no.Β 2, 494–499. MR 1211802, DOI 10.1137/0614034
  • J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Math. Comp. 59 (1992), 135-140
  • MSC: Primary 65F25; Secondary 65G05
  • DOI:
  • MathSciNet review: 1134738