NAG FL Interface
f04mef (real_​toeplitz_​yule_​update)

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1 Purpose

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

2 Specification

Fortran Interface
Subroutine f04mef ( n, t, x, v, work, ifail)
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)
C Header Interface
#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.

3 Description

f04mef solves the equations
where Tn 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 tn is the vector
tnT =(τ1τ2τn),  
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)).

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 Arguments

1: n Integer Input
On entry: the order of the Toeplitz matrix T.
Constraint: n0. When n=0, an immediate return is effected.
2: t(0:n) Real (Kind=nag_wp) array Input
On entry: t(0) must contain the value τ0 of the diagonal elements of T, and the remaining n elements of t must contain the elements of the vector tn.
Constraint: t(0)>0.0. Note that if this is not true, the Toeplitz matrix cannot be positive definite.
3: 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 xn-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, Tn cannot be positive definite.
On exit: the solution vector xn. The element x(n) returns the partial (auto)correlation coefficient, or reflection coefficient, for the nth step. If |x(n)|1.0, the matrix Tn+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, νn, for the nth step. (See Section 9 and the G13 Chapter Introduction.)
5: work(n-1) Real (Kind=nag_wp) array Workspace
6: 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 -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 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Matrix of order value would not be positive definite. Value of the reflection coefficient is value.
If, on exit, x(n) is close to unity, then the principal minor was probably close to being singular, and the sequence τ0,τ1,,τn may be a valid sequence nevertheless. x returns the solution of the equations
and v returns vn, but it may not be positive.
On entry, n=value.
Constraint: n0.
On entry, n=value and x(n-1)=value.
Constraint: if n>1, |x(n-1)|<1.0.
On entry, t(0)=value.
Constraint: t(0)>0.0.
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.
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.
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
where r1 is approximately bounded by
r1cε (i=1n(1+|pi|)-1) ,  
c being a modest function of n, ε being the machine precision and pk being the kth element of xk. 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 Tn-11 hold:
max(1vn-1,1i=1 n-1(1-pi) ) Tn-11i=1 n-1 (1+|pi| 1-|pi| ) ,  
where vn is the mean square prediction error for the nth step. (See Cybenko (1980).) Note that vn<vn-1. The norm of Tn-1 may also be estimated using routine f04ydf.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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.

9 Further Comments

The number of floating-point operations used by this routine is approximately 4n.
The mean square prediction errors, vi, is defined as
Note that vi=(1-pi2)vi-1.

10 Example

This example finds the solution of the Yule–Walker equations Tkxk=-tk, 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 ) .  

10.1 Program Text

Program Text (f04mefe.f90)

10.2 Program Data

Program Data (f04mefe.d)

10.3 Program Results

Program Results (f04mefe.r)