Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	t(0:), b()
Real (Kind=nag_wp), Intent (Inout)	::	x(), work()
Real (Kind=nag_wp), Intent (Out)	::	p

C Header Interface

#include <nag.h>

void	f04mff_ (const Integer n, const double t[], const double b[], double x[], double p, double work[], Integer *ifail)

The routine may be called by the names f04mff or nagf_linsys_real_toeplitz_update.

3 Description

f04mff solves the equations

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

where

T_{n}

is the

n \times 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 Arguments

1: $n$ – Integer Input

On entry: the order of the Toeplitz matrix

T

Constraint:

n \geq 0

. When

n = 0

, an immediate return is effected.

2: $t (0 : *)$ – Real (Kind=nag_wp) array Input

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

\max (1, n)

On entry:

t (i)

must contain the value

τ_{i}

, for

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

Constraint:

t (0) > 0.0

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

3: $b (*)$ – Real (Kind=nag_wp) array Input

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) array Input/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 9.)

6: $work (*)$ – Real (Kind=nag_wp) array Input/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 argument 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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

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

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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:

$ifail = 1$: The Toeplitz Matrix is not positive definite Value of the reflection coefficient is $⟨ value ⟩$ .
If, on exit, p is close to unity, $T_{n}$ was probably close to being singular.

$ifail = - 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $t (0) = ⟨ value ⟩$ .
Constraint: $t (0) > 0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 Parallelism and Performance

f04mff 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.

9 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}

10 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}) .

f04mf: FL CL CPP AD

NAG FL Interfacef04mff (real_​toeplitz_​update)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
f04mff (real_toeplitz_update)