Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials
HTML articles powered by AMS MathViewer
 by J. Rissanen PDF
 Math. Comp. 27 (1973), 147154 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. 129148.
 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/S00255718197102955530
 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 HurwitzZerlegung 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, SpringerVerlag, New YorkHeidelberg, 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/s00200255(69)800095
Additional Information
 © Copyright 1973 American Mathematical Society
 Journal: Math. Comp. 27 (1973), 147154
 MSC: Primary 65F30
 DOI: https://doi.org/10.1090/S00255718197303292355
 MathSciNet review: 0329235