Inversion of Toeplitz band matrices

Author:
William F. Trench

Journal:
Math. Comp. **28** (1974), 1089-1095

MSC:
Primary 65F05

MathSciNet review:
0347066

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Abstract: An algorithm for inverting Toeplitz matrices is simplified for Toeplitz band matrices. In some cases, the simplification yields formulas for the elements in the first row and column of the inverse, from which the remaining elements can be easily calculated. Two examples are given. In any case, the simplification yields a recursive method for computing the first row and column of the inverse of an *n*th order Toeplitz band matrix with operations, where are required with the more general algorithm.

**[1]**Hirotugu Akaike,*Block Toeplitz matrix inversion*, SIAM J. Appl. Math.**24**(1973), 234–241. MR**0362864****[2]**Erwin H. Bareiss,*Numerical solution of linear equations with Toeplitz and vector Toeplitz matrices*, Numer. Math.**13**(1969), 404–424. MR**0255027****[3]**W. D. Hoskins and P. J. Ponzo,*Some properties of a class of band matrices*, Math. Comp.**26**(1972), 393–400. MR**0303703**, 10.1090/S0025-5718-1972-0303703-3**[4]**Lars Rehnqvist,*Inversion of certain symmetric band matrices*, Nordisk Tidskr. Informationsbehandling (BIT)**12**(1972), 90–98. MR**0311087****[5]**William F. Trench,*An algorithm for the inversion of finite Toeplitz matrices*, J. Soc. Indust. Appl. Math.**12**(1964), 515–522. MR**0173681****[6]**William F. Trench,*On the extrapolation of a special class of stationary time series*, Ann. Math. Statist.**36**(1965), 1426–1432. MR**0185783****[7]**William F. Trench,*Weighting coefficients for the prediction of stationary time series from the finite past*, SIAM J. Appl. Math.**15**(1967), 1502–1510. MR**0225458****[8]**Shalhav Zohar,*Toeplitz matrix inversion: The algoritm of W. F. Trench*, J. Assoc. Comput. Mach.**16**(1969), 592–601. MR**0247762**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1974-0347066-8

Keywords:
Toeplitz matrix,
band matrix,
inversion

Article copyright:
© Copyright 1974
American Mathematical Society