# NAG FL Interfacef04mef (real_​toeplitz_​yule_​update)

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## 1Purpose

f04mef updates the solution to the Yule–Walker equations for a real symmetric positive definite Toeplitz system.

## 2Specification

Fortran Interface
 Subroutine f04mef ( n, t, x, v, work,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: t(0:n) Real (Kind=nag_wp), Intent (Inout) :: x(*), v Real (Kind=nag_wp), Intent (Out) :: work(n-1)
#include <nag.h>
 void f04mef_ (const Integer *n, const double t[], double x[], double *v, double work[], Integer *ifail)
The routine may be called by the names f04mef or nagf_linsys_real_toeplitz_yule_update.

## 3Description

f04mef solves the equations
 $Tnxn=-tn,$
where ${T}_{n}$ is the $n×n$ 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)).

## 4References

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

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: the order of the Toeplitz matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$. When ${\mathbf{n}}=0$, an immediate return is effected.
2: $\mathbf{t}\left(0:{\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
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, the Toeplitz matrix cannot be positive definite.
3: $\mathbf{x}\left(*\right)$Real (Kind=nag_wp) array Input/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: $|{\mathbf{x}}\left({\mathbf{n}}-1\right)|<1.0$. 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, ${T}_{n}$ 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 $|{\mathbf{x}}\left({\mathbf{n}}\right)|\ge 1.0$, the matrix ${T}_{n+1}$ will not be positive definite to working accuracy.
4: $\mathbf{v}$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 9 and the G13 Chapter Introduction.)
5: $\mathbf{work}\left({\mathbf{n}}-1\right)$Real (Kind=nag_wp) array Workspace
6: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
Matrix of order $⟨\mathit{\text{value}}⟩$ would not be positive definite. Value of the reflection coefficient is $⟨\mathit{\text{value}}⟩$.
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.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{x}}\left({\mathbf{n}}-1\right)=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{n}}>1$, $|{\mathbf{x}}\left({\mathbf{n}}-1\right)|<1.0$.
On entry, ${\mathbf{t}}\left(0\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{t}}\left(0\right)>0.0$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

The computed solution of the equations certainly satisfies
 $r=Tnxn+tn,$
where ${‖r‖}_{1}$ is approximately bounded by
 $‖r‖1≤cε (∏i=1n(1+|pi|)-1) ,$
$c$ 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:
 $max(1vn-1,1∏i=1 n-1(1-pi) ) ≤‖Tn-1‖1≤∏i=1 n-1 (1+|pi| 1-|pi| ) ,$
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.

## 8Parallelism and Performance

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 implementation-specific information.

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

## 10Example

This example finds the solution of the Yule–Walker equations ${T}_{k}{x}_{k}=-{t}_{k}$, $k=1,2,3,4$ where
 $T4=( 4 3 2 1 3 4 3 2 2 3 4 3 1 2 3 4 ) and t4=( 3 2 1 0 ) .$

### 10.1Program Text

Program Text (f04mefe.f90)

### 10.2Program Data

Program Data (f04mefe.d)

### 10.3Program Results

Program Results (f04mefe.r)