NAG Library Routine Document

F04MEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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

T_{n} x_{n} = - t_{n},

where

T_{n}

is the

n

n

symmetric positive definite Toeplitz matrix

T_{n} = (\begin{array}{l} τ_{0} & τ_{1} & τ_{2} & \dots & τ_{n - 1} \\ τ_{1} & τ_{0} & τ_{1} & \dots & τ_{n - 2} \\ τ_{2} & τ_{1} & τ_{0} & \dots & τ_{n - 3} \\ . & . & . & . \\ τ_{n - 1} & τ_{n - 2} & τ_{n - 3} & \dots & τ_{0} \end{array})

and

t_{n}

is the vector

t_{n}^{T} = (τ_{1} τ_{2} \dots τ_{n}),

given the solution of the equations

T_{n - 1} x_{n - 1} = - t_{n - 1} .

The routine will normally be used to successively solve the equations

T_{k} x_{k} = - t_{k}, k = 1, 2, \dots, 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: $N \geq 0$ . When $N = 0$ , then an immediate return is effected.
2: T( $0 : N$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $T (0)$ must contain the value $τ_{0}$ of the diagonal elements of $T$ , and the remaining N elements of T must contain the elements of the vector $t_{n}$ .
Constraint: $T (0) > 0.0$ . 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 $\max (1, N)$ .
On entry: with $N > 1$ the ( $n - 1$ ) elements of the solution vector $x_{n - 1}$ as returned by a previous call to F04MEF. The element $X (N)$ need not be specified.
Constraint: $|X (N - 1)| < 1.0$ . Note that this is the partial (auto)correlation coefficient, or reflection coefficient, for the $(n - 1)$ th step. If the constraint does not hold, then $T_{n}$ cannot be positive definite.

On exit: the solution vector $x_{n}$ . The element $X (N)$ returns the partial (auto)correlation coefficient, or reflection coefficient, for the $n$ th step. If $|X (N)| \geq 1.0$ , 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 $N > 1$ 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, $ν_{n}$ , for the $n$ th step. (See Section 8 and the Introduction to Chapter G13.)
5: WORK( $N - 1$ ) – REAL (KIND=nag_wp) arrayWorkspace
6: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = - 1$

On entry,	$N < 0$ ,
or	$T (0) \leq 0.0$ ,
or	$N > 1$ and $\|X (N - 1)\| \geq 1.0$ .

$IFAIL = 1$

The Toeplitz matrix

T_{n + 1}

is not positive definite to working accuracy. If, on exit,

X (N)

is close to unity, then the principal minor was probably close to being singular, and the sequence

τ_{0}, τ_{1}, \dots, τ_{N}

may be a valid sequence nevertheless. X returns the solution of the equations

T_{n} x_{n} = - t_{n},

and V returns

v_{n}

, but it may not be positive.

7 Accuracy

The computed solution of the equations certainly satisfies

r = T_{n} x_{n} + t_{n},

where

{‖r‖}_{1}

is approximately bounded by

{‖r‖}_{1} \leq c ε (\prod_{i = 1}^{n} (1 + |p_{i}|) - 1),

c

being a modest function of

n

ε

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 (\frac{1}{v_{n - 1}}, \frac{1}{\prod_{i = 1}^{n - 1} (1 - p_{i})}) \leq {‖T_{n}^{- 1}‖}_{1} \leq \prod_{i = 1}^{n - 1} (\frac{1 + |p_{i}|}{1 - |p_{i}|}),

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.

8 Further Comments

The number of floating point operations used by this routine is approximately

4 n

The mean square prediction errors,

v_{i}

, is defined as

v_{i} = (τ_{0} + t_{i - 1}^{T} x_{i - 1}) / τ_{0} .

Note that

v_{i} = (1 - p_{i}^{2}) 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

T_{4} = (\begin{array}{l} 4 & 3 & 2 & 1 \\ 3 & 4 & 3 & 2 \\ 2 & 3 & 4 & 3 \\ 1 & 2 & 3 & 4 \end{array}) and t_{4} = (\begin{array}{l} 3 \\ 2 \\ 1 \\ 0 \end{array}) .

NAG Library Routine DocumentF04MEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F04MEF