NAG FL Interface
f04fef (real_​toeplitz_​yule)

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1 Purpose

f04fef solves the Yule–Walker equations for a real symmetric positive definite Toeplitz system.

2 Specification

Fortran Interface
Subroutine f04fef ( n, t, x, wantp, p, wantv, v, vlast, 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) :: p(*), v(*)
Real (Kind=nag_wp), Intent (Out) :: x(n), vlast, work(n-1)
Logical, Intent (In) :: wantp, wantv
C Header Interface
#include <nag.h>
void  f04fef_ (const Integer *n, const double t[], double x[], const logical *wantp, double p[], const logical *wantv, double v[], double *vlast, double work[], Integer *ifail)
The routine may be called by the names f04fef or nagf_linsys_real_toeplitz_yule.

3 Description

f04fef solves the equations
Tx=-t,  
where T is the n×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 t is the vector
tT=(τ1,τ2τn).  
The routine uses the method of Durbin (see Durbin (1960) and Golub and Van Loan (1996)). Optionally the mean square prediction errors and/or the partial correlation coefficients for each step can be returned.

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 t.
Constraint: t(0)>0.0. Note that if this is not true, the Toeplitz matrix cannot be positive definite.
3: x(n) Real (Kind=nag_wp) array Output
On exit: the solution vector x.
4: wantp Logical Input
On entry: must be set to .TRUE. if the partial (auto)correlation coefficients are required, and must be set to .FALSE. otherwise.
5: p(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array p must be at least max(1,n) if wantp=.TRUE., and at least 1 otherwise.
On exit: with wantp as .TRUE., the ith element of p contains the partial (auto)correlation coefficient, or reflection coefficient, pi for the ith step. (See Section 9 and Chapter G13.) If wantp is .FALSE., p is not referenced. Note that in any case, xn=pn.
6: wantv Logical Input
On entry: must be set to .TRUE. if the mean square prediction errors are required, and must be set to .FALSE. otherwise.
7: v(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array v must be at least max(1,n) if wantv=.TRUE., and at least 1 otherwise.
On exit: with wantv as .TRUE., the ith element of v contains the mean square prediction error, or predictor error variance ratio, vi, for the ith step. (See Section 9 and Chapter G13.) If wantv is .FALSE., v is not referenced.
8: vlast Real (Kind=nag_wp) Output
On exit: the value of vn, the mean square prediction error for the final step.
9: work(n-1) Real (Kind=nag_wp) array Workspace
10: 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 −1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. 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:
Note: in some cases f04fef may return useful information.
ifail>0
Principal minor value is not positive definite. Value of the reflection coefficient is value.
If, on exit, xifail is close to unity, the principal minor was close to being singular, and the sequence τ0,τ1,,τifail may be a valid sequence nevertheless. The first ifail elements of x return the solution of the equations
Tifailx=-(τ1,τ2,,τifail)T,  
where Tifail is the ifailth principal minor of T. Similarly, if wantp and/or wantv are true, then p and/or v return the first ifail elements of p and v respectively and vlast returns vifail. In particular if ifail=n, then the solution of the equations Tx=-t is returned in x, but τn is such that Tn+1 would not be positive definite to working accuracy.
ifail=-1
On entry, n=value.
Constraint: n0.
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=Tx+t,  
where r1 is approximately bounded by
r1cε (i=1n(1+|pi|)-1) ,  
c being a modest function of n and ε being the machine precision. This bound is almost certainly pessimistic, but it has not yet been established whether or not the method of Durbin is backward stable. If |pn| is close to one, then the Toeplitz matrix is probably ill-conditioned and hence only just positive definite. For further information on stability issues see Bunch (1985), Bunch (1987), Cybenko (1980) and Golub and Van Loan (1996). The following bounds on t−11 hold:
max( 1 vn-1 , 1 i=1 n-1 (1-pi) ) T-11 i=1 n-1 ( 1+|pi| 1-|pi| ) .  
Note:  vn<vn-1. The norm of T-1 may also be estimated using routine f04ydf.

8 Parallelism and Performance

f04fef 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 f04fef is approximately 2n2, independent of the values of wantp and wantv.
The mean square prediction error, vi, is defined as
vi=(τ0+(τ1τ2τi-1)yi-1)/τ0,  
where yi is the solution of the equations
Tiyi=-(τ1τ2τi)T  
and the partial correlation coefficient, pi, is defined as the ith element of yi. Note that vi=(1-pi2)vi-1.

10 Example

This example finds the solution of the Yule–Walker equations Tx=-t, where
T=( 4 3 2 1 3 4 3 2 2 3 4 3 1 2 3 4 )   and  t=( 3 2 1 0 ) .  

10.1 Program Text

Program Text (f04fefe.f90)

10.2 Program Data

Program Data (f04fefe.d)

10.3 Program Results

Program Results (f04fefe.r)