# NAG Library Routine Document

## 1Purpose

f04fff solves the equations $Tx=b$, where $T$ is a real symmetric positive definite Toeplitz matrix.

## 2Specification

Fortran Interface
 Subroutine f04fff ( n, t, b, x, p, work,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: t(0:*), b(*) Real (Kind=nag_wp), Intent (Inout) :: p(*) Real (Kind=nag_wp), Intent (Out) :: x(n), work(2*(n-1)) Logical, Intent (In) :: wantp
#include nagmk26.h
 void f04fff_ (const Integer *n, const double t[], const double b[], double x[], const logical *wantp, double p[], double work[], Integer *ifail)

## 3Description

f04fff solves the equations
 $Tx=b,$
where $T$ is the $n$ by $n$ symmetric positive definite Toeplitz matrix
 $T= τ0 τ1 τ2 … τn-1 τ1 τ0 τ1 … τn-2 τ2 τ1 τ0 … τn-3 . . . . τn-1 τn-2 τn-3 … τ0$
and $b$ is an $n$-element vector.
The routine uses the method of Levinson (see Levinson (1947) and Golub and Van Loan (1996)). Optionally, the reflection coefficients for each step may also be returned.

## 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Levinson N (1947) The Weiner RMS error criterion in filter design and prediction J. Math. Phys. 25 261–278

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: the order of the Toeplitz matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$. When ${\mathbf{n}}=0$, an immediate return is effected.
2:     $\mathbf{t}\left(0:*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array t must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: ${\mathbf{t}}\left(\mathit{i}\right)$ must contain the value ${\tau }_{\mathit{i}}$, for $\mathit{i}=0,1,\dots ,{\mathbf{n}}-1$.
Constraint: ${\mathbf{t}}\left(0\right)>0.0$. Note that if this is not true, the Toeplitz matrix cannot be positive definite.
3:     $\mathbf{b}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the right-hand side vector $b$.
4:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the solution vector $x$.
5:     $\mathbf{wantp}$ – LogicalInput
On entry: must be set to .TRUE. if the reflection coefficients are required, and must be set to .FALSE. otherwise.
6:     $\mathbf{p}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$ if ${\mathbf{wantp}}=\mathrm{.TRUE.}$, and at least $1$ otherwise.
On exit: with wantp as .TRUE., the $i$th element of p contains the reflection coefficient, ${p}_{\mathit{i}}$, for the $\mathit{i}$th step, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}-1$. (See Section 9.) If wantp is .FALSE., p is not referenced.
7:     $\mathbf{work}\left(2×\left({\mathbf{n}}-1\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
8:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 arguments 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).

## 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).
Note: f04fff 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 ifail of the Toeplitz matrix is not positive definite to working accuracy. The first (${\mathbf{ifail}}-1$) elements of x return the solution of the equations
 $Tifail-1x=b1,b2,…,bifail-1T,$
where ${T}_{k}$ is the $k$th principal minor of $T$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The computed solution of the equations certainly satisfies
 $r = Tx-b ,$
where $‖r‖$ is approximately bounded by
 $r ≤ cεCT ,$
$c$ being a modest function of $n$, $\epsilon$ being the machine precision and $C\left(T\right)$ being the condition number of $T$ with respect to inversion. This bound is almost certainly pessimistic, but it seems unlikely that the method of Levinson is backward stable, so caution should be exercised when $T$ is ill-conditioned. The following bound on ${T}^{-1}$ holds:
 $max 1 ∏ i=1 n-1 1 - pi2 , 1 ∏ i=1 n-1 1-pi ≤ T-11 ≤ ∏ i=1 n-1 1+pi 1-pi .$
(See Golub and Van Loan (1996).) The norm of ${T}^{-1}$ may also be estimated using routine f04ydf. For further information on stability issues see Bunch (1985), Bunch (1987), Cybenko (1980) and Golub and Van Loan (1996).

## 8Parallelism and Performance

f04fff 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 f04fff is approximately $4{n}^{2}$.
If ${y}_{i}$ is the solution of the equations
 $Tiyi=-τ1τ2…τiT,$
then the partial correlation coefficient ${p}_{i}$ is defined as the $i$th element of ${y}_{i}$.

## 10Example

This example finds the solution of the equations $Tx=b$, where
 $T= 4 3 2 1 3 4 3 2 2 3 4 3 1 2 3 4 and b= 1 1 1 1 .$

### 10.1Program Text

Program Text (f04fffe.f90)

### 10.2Program Data

Program Data (f04fffe.d)

### 10.3Program Results

Program Results (f04fffe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017