NAG Library Routine Document
F04FEF
1 Purpose
F04FEF solves 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(N), P(*), V(*), VLAST, WORK(N1) 
LOGICAL 
WANTP, WANTV 

3 Description
F04FEF solves the equations
where
$T$ is the
$n$ by
$n$ symmetric positive definite Toeplitz matrix
and
$t$ is the vector
The routine uses the method of Durbin (see
Durbin (1960) and
Golub and Van Loan (1996)). Optionally the mean square prediction errors and/or the partial correlation coefficients for each step can be returned.
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
$t$.
Constraint:
${\mathbf{T}}\left(0\right)>0.0$. Note that if this is not true, then the Toeplitz matrix cannot be positive definite.
 3: $\mathrm{X}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the solution vector $x$.
 4: $\mathrm{WANTP}$ – LOGICALInput

On entry: must be set to .TRUE. if the partial (auto)correlation coefficients are required, and must be set to .FALSE. otherwise.
 5: $\mathrm{P}\left(*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array
P
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ if
${\mathbf{WANTP}}=\mathrm{.TRUE.}$, and at least
$1$ otherwise.
On exit: with
WANTP as .TRUE., the
$i$th element of
P contains the partial (auto)correlation coefficient, or reflection coefficient,
${p}_{i}$ for the
$i$th step. (See
Section 9 and
Chapter G13.) If
WANTP is .FALSE., then
P is not referenced. Note that in any case,
${x}_{n}={p}_{n}$.
 6: $\mathrm{WANTV}$ – LOGICALInput

On entry: must be set to .TRUE. if the mean square prediction errors are required, and must be set to .FALSE. otherwise.
 7: $\mathrm{V}\left(*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array
V
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ if
${\mathbf{WANTV}}=\mathrm{.TRUE.}$, and at least
$1$ otherwise.
On exit: with
WANTV as .TRUE., the
$i$th element of
V contains the mean square prediction error, or predictor error variance ratio,
${v}_{i}$, for the
$i$th step. (See
Section 9 and
Chapter G13.) If
WANTV is .FALSE., then
V is not referenced.
 8: $\mathrm{VLAST}$ – REAL (KIND=nag_wp)Output

On exit: the value of ${v}_{n}$, the mean square prediction error for the final step.
 9: $\mathrm{WORK}\left({\mathbf{N}}1\right)$ – REAL (KIND=nag_wp) arrayWorkspace

 10: $\mathrm{IFAIL}$ – 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, because for this routine the values of the output parameters may be useful even if
${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is
$1$.
When the value $\mathbf{1}\text{ or}1$ 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).
Note: F04FEF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry,  ${\mathbf{N}}<0$, 
or  ${\mathbf{T}}\left(0\right)\le 0.0$. 
 ${\mathbf{IFAIL}}>0$

The principal minor of order (
${\mathbf{IFAIL}}+1$) of the Toeplitz matrix is not positive definite to working accuracy. If, on exit,
${x}_{{\mathbf{IFAIL}}}$ is close to unity, then the principal minor was close to being singular, and the sequence
${\tau}_{0},{\tau}_{1},\dots ,{\tau}_{{\mathbf{IFAIL}}}$ may be a valid sequence nevertheless. The first
IFAIL elements of
X return the solution of the equations
where
${T}_{{\mathbf{IFAIL}}}$ is the
IFAILth principal minor of
$T$. Similarly, if
WANTP and/or
WANTV are true, then
P and/or
V return the first
IFAIL elements of
P and
V respectively and
VLAST returns
${v}_{{\mathbf{IFAIL}}}$. In particular if
${\mathbf{IFAIL}}={\mathbf{N}}$, then the solution of the equations
$Tx=t$ is returned in
X, but
${\tau}_{{\mathbf{N}}}$ is such that
${T}_{{\mathbf{N}}+1}$ would not be positive definite to working accuracy.
 ${\mathbf{IFAIL}}=99$
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
$n$ and
$\epsilon $ being the
machine precision. This bound is almost certainly pessimistic, but it has not yet been established whether or not the method of Durbin is backward stable. If
$\left{p}_{n}\right$ is close to one, then the Toeplitz matrix is probably illconditioned and hence only just positive definite. For further information on stability issues see
Bunch (1985),
Bunch (1987),
Cybenko (1980) and
Golub and Van Loan (1996). The following bounds on
${\Vert {{\mathbf{T}}}^{1}\Vert}_{1}$ hold:
Note: ${v}_{n}<{v}_{n1}$. The norm of
${T}^{1}$ may also be estimated using routine
F04YDF.
8 Parallelism and Performance
F04FEF is not threaded by NAG in any implementation.
F04FEF 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 F04FEF is approximately
$2{n}^{2}$, independent of the values of
WANTP and
WANTV.
The mean square prediction error,
${v}_{i}$, is defined as
where
${y}_{i}$ is the solution of the equations
and the partial correlation coefficient,
${p}_{i}$, is defined as the
$i$th element of
${y}_{i}$. Note that
${v}_{i}=\left(1{p}_{i}^{2}\right){v}_{i1}$.
10 Example
This example finds the solution of the Yule–Walker equations
$Tx=t$, where
10.1 Program Text
Program Text (f04fefe.f90)
10.2 Program Data
Program Data (f04fefe.d)
10.3 Program Results
Program Results (f04fefe.r)