Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Solution of linear equations with rational Toeplitz matrices

Author: Bradley W. Dickinson
Journal: Math. Comp. 34 (1980), 227-233
MSC: Primary 65F05
MathSciNet review: 551300
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We associate a sequence of Toeplitz matrices with the rational formal power series $ T(z)$. An algorithm for solving linear equations with a Toeplitz matrix from this sequence is given. The algorithm requires $ O(n)$ operations to solve a set of n equations, for n sufficiently large.

References [Enhancements On Off] (What's this?)

  • [1] E. H. BAREISS, "Numerical solution of linear equations with Toeplitz and vector Toeplitz matrices," Numer. Math., v. 13, 1969, pp. 404-424. MR 0255027 (40:8234)
  • [2] K. M. DAY, "Toeplitz matrices generated by the Laurent series expansion of an arbitrary rational function," Trans. Amer. Math. Soc., v. 206, 1975, pp. 224-245. MR 0379803 (52:708)
  • [3] B. W. DICKINSON, "Efficient solution of banded Toeplitz systems," IEEE Trans. Acoust. Speech Signal Process., v. ASSP-27, 1979, pp. 421-423.
  • [4] B. FRIEDLANDER, M. MORF, T. KAILATH & L. LJUNG, "New inversion formulas for matrices classified in term of their distance from Toeplitz matrices," J. Linear Algebra Appl. (To appear.) MR 545721 (80k:15006)
  • [5] N. LEVINSON, "The Wiener rms (root mean square) error criterion in filter design and prediction," J. Mathematical Phys., v. 25, 1947, pp. 261-278. MR 0019257 (8:391e)
  • [6] M. MORF, Fast Algorithms for Multivariable Systems, Ph. D. thesis, Stanford University, 1974.
  • [7] J. RISSANEN, "Algorithms for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials," Math. Comp., v. 27, 1973, pp. 147-154. MR 0329235 (48:7577)
  • [8] J. F. TRAUB, "Associated polynomials and uniform methods for the solution of linear problems," SIAM Rev., v. 8, 1966, pp. 277-301. MR 0207238 (34:7054)
  • [9] W. F. TRENCH, "An algorithm for the inversion of finite Toeplitz matrices," SIAM J. Appl. Math., v. 12, 1964, pp. 515-522. MR 0173681 (30:3891)
  • [10] W. F. TRENCH, "Weighting coefficients for the prediction of stationary time series from the finite past," SIAM J. Appl. Math., v. 15, 1967, pp. 1502-1510. MR 0225458 (37:1051)
  • [11] W. F. TRENCH, "Inversion of Toeplitz band matrices," Math. Comp., v. 28, 1974, pp. 1089-1095. MR 0347066 (49:11786)
  • [12] S. ZOHAR, "Toeplitz matrix inversion: the algorithm of W. F. Trench," J. Assoc. Comput. Mach., v. 16, 1967, pp. 592-601. MR 0247762 (40:1023)
  • [13] S. ZOHAR, "The solution of a Toeplitz set of linear equations," J. Assoc. Comput. Mach., v. 21, 1974, pp. 272-276. MR 0343567 (49:8308)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65F05

Retrieve articles in all journals with MSC: 65F05

Additional Information

Keywords: Toeplitz matrix, linear equations
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society