Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials
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- by J. Rissanen PDF
- Math. Comp. 27 (1973), 147-154 Request permission
Abstract:
Algorithms are given for calculating the block triangular factors $A,\hat A,B = {A^{ - 1}}$ and $\hat B = {\hat A^{ - 1}}$ and the block diagonal factor D in the factorizations $R = AD\hat A$ and $BR\hat B = D$ of block Hankel and Toeplitz matrices R. The algorithms require $O({p^3}{n^2})$ operations when R is an $n \times n$-matrix of $p \times p$-blocks. As an application, an iterative method is described for factoring $p \times p$-matrix valued positive polynomials $R = \sum \nolimits _{i = - m}^m {R_i}{x^i},{R_{ - i}} = {R’_i}$, as $\bar A(x)\bar A’({x^{ - 1}})$, where $\bar A(x)$ is outer.References
-
N. Levinson, "The Wiener RMS error criterion in filter design and prediction," Appendix B of N. Wiener’s book Extrapolation, Interpolation, and Smoothing Stationary Time Series with Engineering Applications, Wiley, New York, 1949, pp. 129-148.
- Ralph A. Wiggins and Enders A. Robinson, Recursive solution to the multichannel filtering problem, J. Geophys. Res. 70 (1965), 1885–1891. MR 183107
- James L. Phillips, The triangular decomposition of Hankel matrices, Math. Comp. 25 (1971), 559–602. MR 295553, DOI 10.1090/S0025-5718-1971-0295553-0
- Allen Devinatz, The factorization of operator valued functions, Ann. of Math. (2) 73 (1961), 458–495. MR 126702, DOI 10.2307/1970313
- Friedrich L. Bauer, Ein direktes Iterationsverfahren zur Hurwitz-Zerlegung eines Polynoms, Arch. Elek. Übertr. 9 (1955), 285–290 (German). MR 76447
- Handbook for automatic computation. Vol. II, Die Grundlehren der mathematischen Wissenschaften, Band 186, Springer-Verlag, New York-Heidelberg, 1971. Linear algebra; Compiled by J. H. Wilkinson and C. Reinsch. MR 0461856
- J. Rissanen and L. Barbosa, Properties of infinite covariance matrices and stability of optimum predictors, Information Sci. 1 (1968/1969), 221–236. MR 0243711, DOI 10.1016/s0020-0255(69)80009-5
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Math. Comp. 27 (1973), 147-154
- MSC: Primary 65F30
- DOI: https://doi.org/10.1090/S0025-5718-1973-0329235-5
- MathSciNet review: 0329235