F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF04MEF

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.

## 1  Purpose

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

## 2  Specification

 SUBROUTINE F04MEF ( N, T, X, V, WORK, IFAIL)
 INTEGER N, IFAIL REAL (KIND=nag_wp) T(0:N), X(*), V, WORK(N-1)

## 3  Description

F04MEF solves the equations
 $Tnxn=-tn,$
where ${T}_{n}$ 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 ${t}_{n}$ is the vector
 $tnT =τ1τ2…τn,$
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: ${\mathbf{N}}\ge 0$. When ${\mathbf{N}}=0$, then an immediate return is effected.
2:     T($0:{\mathbf{N}}$) – 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:     X($*$) – 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 ($n-1$) elements of the solution vector ${x}_{n-1}$ 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 $\left(n-1\right)$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:     V – REAL (KIND=nag_wp)Input/Output
On entry: with ${\mathbf{N}}>1$ 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, ${\nu }_{n}$, for the $n$th step. (See Section 8 and the Introduction to Chapter G13.)
5:     WORK(${\mathbf{N}}-1$) – REAL (KIND=nag_wp) arrayWorkspace
6:     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
 $Tnxn=-tn,$
and V returns ${v}_{n}$, but it may not be positive.

## 7  Accuracy

The computed solution of the equations certainly satisfies
 $r=Tnxn+tn,$
where ${‖r‖}_{1}$ is approximately bounded by
 $r1≤cε ∏i=1n1+pi-1 ,$
$c$ 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 ${‖{T}_{n}^{-1}‖}_{1}$ hold:
 $max1vn-1,1∏i=1 n-11-pi ≤Tn-11≤∏i=1 n-1 1+pi 1-pi ,$
where ${v}_{n}$ is the mean square prediction error for the $n$th step. (See Cybenko (1980).) Note that ${v}_{n}<{v}_{n-1}$. The norm of ${T}_{n}^{-1}$ may also be estimated using routine F04YDF.

The number of floating point operations used by this routine is approximately $4n$.
The mean square prediction errors, ${v}_{i}$, is defined as
 $vi=τ0+ ti-1T xi-1/τ0.$
Note that ${v}_{i}=\left(1-{p}_{i}^{2}\right){v}_{i-1}$.

## 9  Example

This example finds the solution of the Yule–Walker equations ${T}_{k}{x}_{k}=-{t}_{k}$, $k=1,2,3,4$ 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)