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: – IntegerInput
On entry: the order of the Toeplitz matrix .
. When , an immediate return is effected.
2: – Real (Kind=nag_wp) arrayInput
On entry: 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, the Toeplitz matrix cannot be positive definite.
3: – Real (Kind=nag_wp) arrayInput/Output
Note: 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, 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 , the matrix will not be positive definite to working accuracy.
4: – 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.
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , 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 would not be positive definite. Value of the reflection coefficient is .
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
An unexpected error has been triggered by this routine. Please
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.
The computed solution of the equations certainly satisfies
where is approximately bounded by
being a modest function of , being the machine precision and being the 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 hold:
where 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.
8Parallelism 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.
The number of floating-point operations used by this routine is approximately .
The mean square prediction errors, , is defined as
Note that .
This example finds the solution of the Yule–Walker equations , where