F04MFF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F04MFF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F04MFF updates the solution of the equations Tx=b, where T is a real symmetric positive definite Toeplitz matrix.

2  Specification

SUBROUTINE F04MFF ( N, T, B, X, P, WORK, IFAIL)
INTEGER  N, IFAIL
REAL (KIND=nag_wp)  T(0:*), B(*), X(*), P, WORK(*)

3  Description

F04MFF solves the equations
Tnxn=bn,
where Tn is the n by n symmetric positive definite Toeplitz matrix
Tn= τ0 τ1 τ2 τn-1 τ1 τ0 τ1 τn-2 τ2 τ1 τ0 τn-3 . . . . τn-1 τn-2 τn-3 τ0
and bn is the n-element vector bn=β1β2βnT, given the solution of the equations
Tn-1xn-1=bn-1.
This routine will normally be used to successively solve the equations
Tkxk=bk,   k= 1,2,,n.
If it is desired to solve the equations for a single value of n, then routine F04FFF may be called. This routine uses the method of Levinson (see Levinson (1947) and Golub and Van Loan (1996)).

4  References

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

5  Parameters

1:     N – INTEGERInput
On entry: the order of the Toeplitz matrix T.
Constraint: N0. When N=0, then an immediate return is effected.
2:     T(0:*) – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array T must be at least max1,N.
On entry: Ti must contain the values τi, i=0,1,,N-1.
Constraint: T0>0.0. Note that if this is not true, then the Toeplitz matrix cannot be positive definite.
3:     B(*) – REAL (KIND=nag_wp) arrayInput
Note: the dimension of the array B must be at least max1,N.
On entry: the right-hand side vector bn.
4:     X(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array X must be at least max1,N.
On entry: with N>1 the (n-1) elements of the solution vector xn-1 as returned by a previous call to F04MFF. The element XN need not be specified.
On exit: the solution vector xn.
5:     P – REAL (KIND=nag_wp)Output
On exit: the reflection coefficient pn-1. (See Section 8.)
6:     WORK(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array WORK must be at least max1,2×N-1.
On entry: with N>2 the elements of WORK should be as returned from a previous call to F04MFF with (N-1) as the parameter N.
On exit: the first (N-1) elements of WORK contain the solution to the Yule–Walker equations
Tn-1yn-1=-tn-1,
where tn-1=τ1τ2τn-1T.
7:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
IFAIL=-1
On entry,N<0,
orT00.0.
IFAIL=1
The Toeplitz matrix Tn is not positive definite to working accuracy. If, on exit, P is close to unity, then Tn was probably close to being singular.

7  Accuracy

The computed solution of the equations certainly satisfies
r=Tnxn-bn,
where r1 is approximately bounded by
r1cεCTn,
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 Tn is ill-conditioned. The following bound on Tn-1 holds:
max1i=1 n-11-pi2 , 1i=1 n-11-pi Tn-11i=1 n-1 1+pi 1-pi .
(See Golub and Van Loan (1996).) The norm of Tn-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).

8  Further Comments

The number of floating point operations used by this routine is approximately 8n.
If yi is the solution of the equations
Tiyi=-τ1τ2τiT,
then the reflection coefficient pi is defined as the ith element of yi.

9  Example

This example finds the solution of the equations Tkxk=bk, k=1,2,3,4, where
T4= 4 3 2 1 3 4 3 2 2 3 4 3 1 2 3 4   and  b4= 1 1 1 1 .

9.1  Program Text

Program Text (f04mffe.f90)

9.2  Program Data

Program Data (f04mffe.d)

9.3  Program Results

Program Results (f04mffe.r)


F04MFF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012