LEVDURB Procedure

Cholesky-factors symmetric, positive, definite Toeplitz matrices using the Levinson-Durbin algorithm.

Usage

LEVDURB, r, alpha, t, c

Input Parameters

r—An array consisting of the first row of the Toeplitz matrix.

Output Parameters

alpha—An array consisting of the elements of the diagonal matrix factor.

t—The upper triangular matrix factor, also known as the Cholesky decomposition matrix.

c—An array of reflection coefficients.

Keywords

None.

Discussion

Given the first row of the Toeplitz matrix  R, the procedure finds an upper triangular matrix  T and a diagonal matrix D = diag{a(n), a(n – 1), ..., a(0)} that satisfy the following equation:

D = TRT ^T

The so-called reflection coefficients of the Toeplitz matrix are also returned. LEVDURB is one of a suite of functions (including JURYRC,  LEVCORR,  LEVDURB, and TOEPSOL) used to solve Toeplitz linear equations and factorization problems.

LEVDURB performs the inverse operation of LEVCORR by accepting the elements of the matrices D and  T  and then computing the first row of the Toeplitz matrix R = T ^–1DT ^–T.

Example

This example illustrates the inverse relationship between LEVDURB and LEVCORR.

; First order autoregressive autocorrelation sequence used

; to form the first row of Toeplitz matrix.

r = [1.0d0, 0.9d0, 0.9d0^2, 0.9d0^3, 0.9d0^4, 0.9d0^5]

PM, TRANSPOSE(r), Title = 'First Row of Original Toeplitz Matrix'

; Compute Cholesky decomposition T –1DT –T of the Toeplitz matrix.

LEVDURB, r, alpha, t, c

; Compute the first row of the Toeplitz matrix R = T –1DT –T from

; the Cholesky decomposition.

LEVCORR, r_out, alpha, t

PM, TRANSPOSE(r_out), $
Title = 'First Row of Returned Toeplitz Matrix'

See Also

JURYRC,  LEVCORR,  TOEPSOL

For Additional Information

Proakis and Manolakis, 1992. Roberts and Mullis, 1987 p. 522.