F04MEF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


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

F04MEF updates the solution to the Yule–Walker equations for a real symmetric positive definite Toeplitz system.

2  Specification

REAL (KIND=nag_wp)  T(0:N), X(*), V, WORK(N-1)

3  Description

F04MEF solves the equations
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 tn is the vector
tnT =τ1τ2τn,
given the solution of the equations
Tn- 1xn- 1=-tn- 1.
The routine will normally be used to successively solve the equations
Tkxk=-tk,  k=1,2,,n.
If it is desired to solve the equations for a single value of n, then routine F04FEF may be called. This routine uses the method of Durbin (see Durbin (1960) 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
Durbin J (1960) The fitting of time series models Rev. Inst. Internat. Stat. 28 233
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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:N) – REAL (KIND=nag_wp) arrayInput
On entry: T0 must contain the value τ0 of the diagonal elements of T, and the remaining N elements of T must contain the elements of the vector tn.
Constraint: T0>0.0. Note that if this is not true, then the Toeplitz matrix cannot be positive definite.
3:     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 F04MEF. The element XN need not be specified.
Constraint: XN-1<1.0. Note that this is the partial (auto)correlation coefficient, or reflection coefficient, for the n-1th step. If the constraint does not hold, then Tn cannot be positive definite.
On exit: the solution vector xn. The element XN returns the partial (auto)correlation coefficient, or reflection coefficient, for the nth step. If XN1.0, then the matrix Tn+1 will not be positive definite to working accuracy.
4:     V – REAL (KIND=nag_wp)Input/Output
On entry: with N>1 the mean square prediction error for the (n-1)th step, as returned by a previous call to F04MEF.
On exit: the mean square prediction error, or predictor error variance ratio, νn, for the nth step. (See Section 8 and the Introduction to Chapter G13.)
5:     WORK(N-1) – REAL (KIND=nag_wp) arrayWorkspace
6:     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:
On entry,N<0,
orN>1 and XN-11.0.
The Toeplitz matrix Tn+1 is not positive definite to working accuracy. If, on exit, XN is close to unity, then the principal minor was probably close to being singular, and the sequence τ0,τ1,,τN may be a valid sequence nevertheless. X returns the solution of the equations
and V returns vn, but it may not be positive.

7  Accuracy

The computed solution of the equations certainly satisfies
where r1 is approximately bounded by
r1cε i=1n1+pi-1 ,
c being a modest function of n, ε being the machine precision and pk being the kth element of xk. This bound is almost certainly pessimistic, but it has not yet been established whether or not the method of Durbin is backward stable. For further information on stability issues see Bunch (1985), Bunch (1987), Cybenko (1980) and Golub and Van Loan (1996). The following bounds on Tn-11 hold:
max1vn-1,1i=1 n-11-pi Tn-11i=1 n-1 1+pi 1-pi ,
where vn is the mean square prediction error for the nth step. (See Cybenko (1980).) Note that vn<vn-1. The norm of Tn-1 may also be estimated using routine F04YDF.

8  Further Comments

The number of floating point operations used by this routine is approximately 4n.
The mean square prediction errors, vi, is defined as
vi=τ0+ ti-1T xi-1/τ0.
Note that vi=1-pi2vi-1.

9  Example

This example finds the solution of the Yule–Walker equations Tkxk=-tk, k=1,2,3,4 where
T4= 4 3 2 1 3 4 3 2 2 3 4 3 1 2 3 4   and  t4= 3 2 1 0 .

9.1  Program Text

Program Text (f04mefe.f90)

9.2  Program Data

Program Data (f04mefe.d)

9.3  Program Results

Program Results (f04mefe.r)

F04MEF (PDF version)
F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

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