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
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1: – IntegerInput
On entry: the order of the Toeplitz matrix .
. When , an immediate return is effected.
2: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array t
must be at least
On entry: must contain the value , for .
. Note that if this is not true, the Toeplitz matrix cannot be positive definite.
3: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array b
must be at least
On entry: the right-hand side vector .
4: – Real (Kind=nag_wp) arrayOutput
On exit: the solution vector .
5: – LogicalInput
On entry: must be set to .TRUE. if the reflection coefficients are required, and must be set to .FALSE. otherwise.
6: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array p
must be at least
if , and at least otherwise.
On exit: with wantp as .TRUE., the th element of p contains the reflection coefficient,
, for the th step, for . (See Section 9.) If wantp is .FALSE., p is not referenced.
7: – Real (Kind=nag_wp) arrayWorkspace
8: – IntegerInput/Output
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 since useful values can be provided in some output arguments even when on exit. 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:
Note: in some cases f04fff may return useful information.
Principal minor is not positive definite. Value of the reflection coefficient is .
The first () elements of x return the solution of the equations
where is the th principal minor of .
On entry, .
On entry, .
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 condition number of 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 is ill-conditioned. The following bound on holds:
Background information to multithreading can be found in the Multithreading documentation.
f04fff 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 f04fff is approximately .
If is the solution of the equations
then the partial correlation coefficient is defined as the th element of .
This example finds the solution of the equations , where