Recursive algorithms for the matrix Padé problem

Author:
Adhemar Bultheel

Journal:
Math. Comp. **35** (1980), 875-892

MSC:
Primary 41A21; Secondary 65D15

DOI:
https://doi.org/10.1090/S0025-5718-1980-0572862-4

MathSciNet review:
572862

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A matrix triangularization interpretation is given for the recursive algorithms computing the Padé approximants along a certain path in the Padé table, which makes it possible to unify all known algorithms in this field [5], [6]. For the normal Padé table, all these results carry over to the matrix Padé problem in a straightforward way. Additional features, resulting from the noncommutativity are investigated. A generalization of the Trench-Zohar algorithm and related recursions are studied in greater detail.

**[1]**Hirotugu Akaike,*Block Toeplitz matrix inversion*, SIAM J. Appl. Math.**24**(1973), 234–241. MR**0362864**, https://doi.org/10.1137/0124024**[2]**Glen Baxter,*Polynomials defined by a difference system*, J. Math. Anal. Appl.**2**(1961), 223–263. MR**0126125**, https://doi.org/10.1016/0022-247X(61)90033-6**[3]**D. BESSIS, "Topics in the theory of Padé approximants," in*Padé Approximants*(P. R. Graves-Morris, Ed.), The institute of Physics, London-Bristol, 1973, pp. 19-44.**[4]**A. Bultheel,*Division algorithms for continued fractions and the Padé table*, J. Comput. Appl. Math.**6**(1980), no. 4, 259–266. MR**601538**, https://doi.org/10.1016/0771-050X(80)90034-0**[5]**A. BULTHEEL, "Fast factorization algorithms and Padé approximation: a survey." (Submitted.)**[6]**A. Bultheel,*Recursive algorithms for the Padé table: two approaches*, Padé approximation and its applications (Proc. Conf., Univ. Antwerp, Antwerp, 1979) Lecture Notes in Math., vol. 765, Springer, Berlin, 1979, pp. 211–230. MR**561452****[7]**Philippe Delsarte, Yves V. Genin, and Yves G. Kamp,*Orthogonal polynomial matrices on the unit circle*, IEEE Trans. Circuits and Systems**CAS-25**(1978), no. 3, 149–160. MR**0481886**, https://doi.org/10.1109/TCS.1978.1084452**[8]**Bradley W. Dickinson, Martin Morf, and Thomas Kailath,*A minimal realization algorithm for matrix sequences*, IEEE Trans. Automatic Control**AC-19**(1974), 31–38. MR**0363556****[9]**L. Ya. Geronimus,*Orthogonal polynomials: Estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval*, Authorized translation from the Russian, Consultants Bureau, New York, 1961. MR**0133643****[10]**William B. Gragg,*Laurent, Fourier, and Chebyshev-Padé tables*, Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New York, 1977, pp. 61–72. MR**0477569****[11]**Norman Levinson,*The Wiener RMS (root mean square) error criterion in filter design and prediction*, J. Math. Phys. Mass. Inst. Tech.**25**(1947), 261–278. MR**0019257**, https://doi.org/10.1002/sapm1946251261**[12]**A. MORF, A. VIEIRA & D. T. LEE,*Ladder Forms for Identification and Speech Processing*, Proc. Conf. on Decision and Control, New Orleans, Louisiana, Dec. 7-9, 1977.**[13]**J. Rissanen,*Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials*, Math. Comp.**27**(1973), 147–154. MR**0329235**, https://doi.org/10.1090/S0025-5718-1973-0329235-5**[14]**G. SZEGÖ,*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939.**[15]**William F. Trench,*An algorithm for the inversion of finite Toeplitz matrices*, J. Soc. Indust. Appl. Math.**12**(1964), 515–522. MR**0173681****[16]**William F. Trench,*An algorithm for the inversion of finite Hankel matrices*, J. Soc. Indust. Appl. Math.**13**(1965), 1102–1107. MR**0189232****[17]**Ralph A. Wiggins and Enders A. Robinson,*Recursive solution to the multichannel filtering problem*, J. Geophys. Res.**70**(1965), 1885–1891. MR**0183107****[18]**Shalhav Zohar,*Toeplitz matrix inversion: The algoritm of W. F. Trench*, J. Assoc. Comput. Mach.**16**(1969), 592–601. MR**0247762**, https://doi.org/10.1145/321541.321549

Retrieve articles in *Mathematics of Computation*
with MSC:
41A21,
65D15

Retrieve articles in all journals with MSC: 41A21, 65D15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1980-0572862-4

Keywords:
Hankel and Toeplitz matrices,
triangular decomposition of matrices,
fast algorithms,
rational approximation

Article copyright:
© Copyright 1980
American Mathematical Society