NAG Library Routine Document
F04MEF
1 Purpose
F04MEF updates the solution to the Yule–Walker equations for a real symmetric positive definite Toeplitz system.
2 Specification
INTEGER 
N, IFAIL 
REAL (KIND=nag_wp) 
T(0:N), X(*), V, WORK(N1) 

3 Description
F04MEF solves the equations
where
${T}_{n}$ is the
$n$ by
$n$ symmetric positive definite Toeplitz matrix
and
${t}_{n}$ is the vector
given the solution of the equations
The routine will normally be used to successively solve the equations
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: $\mathrm{N}$ – INTEGERInput

On entry: the order of the Toeplitz matrix $T$.
Constraint:
${\mathbf{N}}\ge 0$. When ${\mathbf{N}}=0$, then an immediate return is effected.
 2: $\mathrm{T}\left(0:{\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry:
${\mathbf{T}}\left(0\right)$ must contain the value
${\tau}_{0}$ of the diagonal elements of
$T$, and the remaining
N elements of
T must contain the elements of the vector
${t}_{n}$.
Constraint:
${\mathbf{T}}\left(0\right)>0.0$. Note that if this is not true, then the Toeplitz matrix cannot be positive definite.
 3: $\mathrm{X}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array
X
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: with ${\mathbf{N}}>1$ the ($n1$) elements of the solution vector ${x}_{n1}$ as returned by a previous call to F04MEF. The element ${\mathbf{X}}\left({\mathbf{N}}\right)$ need not be specified.
Constraint:
$\left{\mathbf{X}}\left({\mathbf{N}}1\right)\right<1.0$. Note that this is the partial (auto)correlation coefficient, or reflection coefficient, for the $\left(n1\right)$th step. If the constraint does not hold, then ${T}_{n}$ cannot be positive definite.
On exit: the solution vector ${x}_{n}$. The element ${\mathbf{X}}\left({\mathbf{N}}\right)$ returns the partial (auto)correlation coefficient, or reflection coefficient, for the $n$th step. If $\left{\mathbf{X}}\left({\mathbf{N}}\right)\right\ge 1.0$, then the matrix ${T}_{n+1}$ will not be positive definite to working accuracy.
 4: $\mathrm{V}$ – REAL (KIND=nag_wp)Input/Output

On entry: with ${\mathbf{N}}>1$ the mean square prediction error for the ($n1$)th step, as returned by a previous call to F04MEF.
On exit: the mean square prediction error, or predictor error variance ratio,
${\nu}_{n}$, for the
$n$th step. (See
Section 9 and the Introduction to
Chapter G13.)
 5: $\mathrm{WORK}\left({\mathbf{N}}1\right)$ – REAL (KIND=nag_wp) arrayWorkspace

 6: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ 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 $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry,  ${\mathbf{N}}<0$, 
or  ${\mathbf{T}}\left(0\right)\le 0.0$, 
or  ${\mathbf{N}}>1$ and $\left{\mathbf{X}}\left({\mathbf{N}}1\right)\right\ge 1.0$. 
 ${\mathbf{IFAIL}}=1$

The Toeplitz matrix
${T}_{n+1}$ is not positive definite to working accuracy. If, on exit,
${\mathbf{X}}\left({\mathbf{N}}\right)$ is close to unity, then the principal minor was probably close to being singular, and the sequence
${\tau}_{0},{\tau}_{1},\dots ,{\tau}_{{\mathbf{N}}}$ may be a valid sequence nevertheless.
X returns the solution of the equations
and
V returns
${v}_{n}$, but it may not be positive.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
The computed solution of the equations certainly satisfies
where
${\Vert r\Vert}_{1}$ is approximately bounded by
$c$ being a modest function of
$n$,
$\epsilon $ being the
machine precision and
${p}_{k}$ being the
$k$th element of
${x}_{k}$. 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
${\Vert {T}_{n}^{1}\Vert}_{1}$ hold:
where
${v}_{n}$ is the mean square prediction error for the
$n$th step. (See
Cybenko (1980).) Note that
${v}_{n}<{v}_{n1}$. The norm of
${T}_{n}^{1}$ may also be estimated using routine
F04YDF.
8 Parallelism and Performance
F04MEF is not threaded by NAG in any implementation.
F04MEF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The number of floatingpoint operations used by this routine is approximately $4n$.
The mean square prediction errors,
${v}_{i}$, is defined as
Note that
${v}_{i}=\left(1{p}_{i}^{2}\right){v}_{i1}$.
10 Example
This example finds the solution of the Yule–Walker equations
${T}_{k}{x}_{k}={t}_{k}$,
$k=1,2,3,4$ where
10.1 Program Text
Program Text (f04mefe.f90)
10.2 Program Data
Program Data (f04mefe.d)
10.3 Program Results
Program Results (f04mefe.r)