G13DXF calculates the zeros of a vector autoregressive (or moving average) operator.
This routine is likely to be used in conjunction with
G05PJF,
G13ASF,
G13DDF or
G13DSF.
Consider the vector autoregressive moving average (VARMA) model
where
denotes a vector of
time series and
is a vector of
residual series having zero mean and a constant variance-covariance matrix. The components of
are also assumed to be uncorrelated at non-simultaneous lags.
denotes a sequence of
by
matrices of autoregressive (AR) parameters and
denotes a sequence of
by
matrices of moving average (MA) parameters.
is a vector of length
containing the series means. Let
where
denotes the
by
identity matrix.
The model
(1) is said to be stationary if the eigenvalues of
lie inside the unit circle. Similarly let
Then the model is said to be invertible if the eigenvalues of
lie inside the unit circle.
- 1: – INTEGERInput
-
On entry: , the dimension of the multivariate time series.
Constraint:
.
- 2: – INTEGERInput
-
On entry: the number of AR (or MA) parameter matrices, (or ).
Constraint:
.
- 3: – REAL (KIND=nag_wp) arrayInput
-
On entry: the AR (or MA) parameter matrices read in row by row in the order (or ). That is,
must be set equal to the th element of , for (or the
th element of , for ).
- 4: – REAL (KIND=nag_wp) arrayOutput
-
On exit: the real parts of the eigenvalues.
- 5: – REAL (KIND=nag_wp) arrayOutput
-
On exit: the imaginary parts of the eigenvalues.
- 6: – REAL (KIND=nag_wp) arrayOutput
-
On exit: the moduli of the eigenvalues.
- 7: – REAL (KIND=nag_wp) arrayWorkspace
- 8: – INTEGER arrayWorkspace
-
- 9: – INTEGERInput/Output
-
On entry:
IFAIL 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.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
The accuracy of the results depends on the original matrix and the multiplicity of the roots.
G13DXF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
G13DXF 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.