NAG Library Routine Document

F04MFF

+− 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

F04MFF updates the solution of the equations

T x = b

, where

T

is a real symmetric positive definite Toeplitz matrix.

2 Specification

SUBROUTINE F04MFF (

N, T, B, X, P, WORK, IFAIL)

INTEGER	N, IFAIL
REAL (KIND=nag_wp)	T(0:), B(), X(), P, WORK()

3 Description

F04MFF solves the equations

T_{n} x_{n} = b_{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

b_{n}

is the

n

-element vector

b_{n} = {(β_{1} β_{2} \dots β_{n})}^{T}

, given the solution of the equations

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

This routine will normally be used to successively solve the equations

T_{k} x_{k} = b_{k}, k = 1, 2, \dots, n .

If it is desired to solve the equations for a single value of

n

, then routine F04FFF may be called. This routine uses the method of Levinson (see Levinson (1947) 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

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

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( $0 : *$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array T must be at least

\max (1, N)

On entry:

T (i)

must contain the values

τ_{i}

i = 0, 1, \dots, N - 1

Constraint:

T (0) > 0.0

. Note that if this is not true, then the Toeplitz matrix cannot be positive definite.

3: B( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array B must be at least

\max (1, N)

On entry: the right-hand side vector

b_{n}

4: 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 F04MFF. The element

X (N)

need not be specified.

On exit: the solution vector

x_{n}

5: P – REAL (KIND=nag_wp)Output

On exit: the reflection coefficient

p_{n - 1}

. (See Section 8.)

6: WORK( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array WORK must be at least

\max (1, 2 \times N - 1)

On entry: with

N > 2

the elements of WORK should be as returned from a previous call to F04MFF with (

N - 1

) as the parameter N.

On exit: the first (

N - 1

) elements of WORK contain the solution to the Yule–Walker equations

T_{n - 1} y_{n - 1} = - t_{n - 1},

where

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

7: 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	$T (0) \leq 0.0$ .

$IFAIL = 1$: The Toeplitz matrix $T_{n}$ is not positive definite to working accuracy. If, on exit, P is close to unity, then $T_{n}$ was probably close to being singular.

7 Accuracy

The computed solution of the equations certainly satisfies

r = T_{n} x_{n} - b_{n},

where

{‖r‖}_{1}

is approximately bounded by

{‖r‖}_{1} \leq c ε C (T_{n}),

c

being a modest function of

n

ε

being the machine precision and

C (T)

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_{n}

is ill-conditioned. The following bound on

T_{n}^{- 1}

holds:

\max (\frac{1}{\prod_{i = 1}^{n - 1} (1 - p_{i}^{2})}, \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}|}) .

(See Golub and Van Loan (1996).) The norm of

T_{n}^{- 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).

8 Further Comments

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

8 n

y_{i}

is the solution of the equations

T_{i} y_{i} = - {(τ_{1} τ_{2} \dots τ_{i})}^{T},

then the reflection coefficient

p_{i}

is defined as the

i

th element of

y_{i}

9 Example

This example finds the solution of the equations

T_{k} x_{k} = b_{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 b_{4} = (\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}) .

NAG Library Routine DocumentF04MFF

+− 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

F04MFF