F04FFF solves the equations
where
T is the
n by
n symmetric positive definite Toeplitz matrix
and
b is an
n-element vector.
The routine uses the method of Levinson (see
Levinson (1947) and
Golub and Van Loan (1996)). Optionally, the reflection coefficients for each step may also be returned.
Bunch J R (1985) Stability of methods for solving Toeplitz systems of equations
SIAM J. Sci. Statist. Comput. 6 349–364
Bunch J R (1987) The weak and strong stability of algorithms in numerical linear algebra
Linear Algebra Appl. 88/89 49–66
Cybenko G (1980) The numerical stability of the Levinson–Durbin algorithm for Toeplitz systems of equations
SIAM J. Sci. Statist. Comput. 1 303–319
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Levinson N (1947) The Weiner RMS error criterion in filter design and prediction
J. Math. Phys. 25 261–278
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
The computed solution of the equations certainly satisfies
where
r is approximately bounded by
c being a modest function of
n,
ε being the
machine precision and
CT being the condition number of
T with respect to inversion. This bound is almost certainly pessimistic, but it seems unlikely that the method of Levinson is backward stable, so caution should be exercised when
T is ill-conditioned. The following bound on
T-1 holds:
(See
Golub and Van Loan (1996).) The norm of
T-1 may also be estimated using routine
F04YDF. For further information on stability issues see
Bunch (1985),
Bunch (1987),
Cybenko (1980) and
Golub and Van Loan (1996).
If
yi is the solution of the equations
then the partial correlation coefficient
pi is defined as the
ith element of
yi.
This example finds the solution of the equations
Tx=b, where