NAG Library Routine Document
F04MEF updates the solution to the Yule–Walker equations for a real symmetric positive definite Toeplitz system.
||T(0:N), X(*), V, WORK(N-1)
F04MEF solves the equations
symmetric positive definite Toeplitz matrix
is the vector
given the solution of the equations
The routine will normally be used to successively solve the equations
If it is desired to solve the equations for a single value of
, then routine F04FEF
may be called. This routine uses the method of Durbin (see Durbin (1960)
and Golub and Van Loan (1996)
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
- 1: N – INTEGERInput
On entry: the order of the Toeplitz matrix .
. When , then an immediate return is effected.
- 2: T() – REAL (KIND=nag_wp) arrayInput
must contain the value
of the diagonal elements of
, and the remaining N
elements of T
must contain the elements of the vector
. Note that if this is not true, then the Toeplitz matrix cannot be positive definite.
- 3: X() – REAL (KIND=nag_wp) arrayInput/Output
the dimension of the array X
must be at least
On entry: with the () elements of the solution vector as returned by a previous call to F04MEF. The element need not be specified.
. Note that this is the partial (auto)correlation coefficient, or reflection coefficient, for the th step. If the constraint does not hold, then cannot be positive definite.
On exit: the solution vector . The element returns the partial (auto)correlation coefficient, or reflection coefficient, for the th step. If , then the matrix will not be positive definite to working accuracy.
- 4: V – REAL (KIND=nag_wp)Input/Output
On entry: with the mean square prediction error for the ()th step, as returned by a previous call to F04MEF.
: the mean square prediction error, or predictor error variance ratio,
, for the
th step. (See Section 8
and the Introduction to Chapter G13
- 5: WORK() – REAL (KIND=nag_wp) arrayWorkspace
- 6: IFAIL – INTEGERInput/Output
must be set to
. 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
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|or|| and .|
The Toeplitz matrix
is not positive definite to working accuracy. If, on exit,
is close to unity, then the principal minor was probably close to being singular, and the sequence
may be a valid sequence nevertheless. X
returns the solution of the equations
, but it may not be positive.
The computed solution of the equations certainly satisfies
is approximately bounded by
being a modest function of
being the machine precision
th element of
. 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
is the mean square prediction error for the
th step. (See Cybenko (1980)
.) Note that
. The norm of
may also be estimated using routine F04YDF
The number of floating point operations used by this routine is approximately .
The mean square prediction errors,
, is defined as
This example finds the solution of the Yule–Walker equations
9.1 Program Text
Program Text (f04mefe.f90)
9.2 Program Data
Program Data (f04mefe.d)
9.3 Program Results
Program Results (f04mefe.r)