NAG FL Interface
g13dxf (uni_​arma_​roots)

Settings help

FL Name Style:

FL Specification Language:

1 Purpose

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.

2 Specification

Fortran Interface
Subroutine g13dxf ( k, ip, par, rr, ri, rmod, work, iwork, ifail)
Integer, Intent (In) :: k, ip
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: iwork(k*ip)
Real (Kind=nag_wp), Intent (In) :: par(ip*k*k)
Real (Kind=nag_wp), Intent (Out) :: rr(k*ip), ri(k*ip), rmod(k*ip), work(k*k*ip*ip)
C Header Interface
#include <nag.h>
void  g13dxf_ (const Integer *k, const Integer *ip, const double par[], double rr[], double ri[], double rmod[], double work[], Integer iwork[], Integer *ifail)
The routine may be called by the names g13dxf or nagf_tsa_uni_arma_roots.

3 Description

Consider the vector autoregressive moving average (VARMA) model
Wt-μ=ϕ1(Wt-1-μ)+ϕ2(Wt-2-μ)++ϕp(Wt-p-μ)+εt-θ1εt-1-θ2εt-2--θqεt-q, (1)
where Wt denotes a vector of k time series and εt is a vector of k residual series having zero mean and a constant variance-covariance matrix. The components of εt are also assumed to be uncorrelated at non-simultaneous lags. ϕ1,ϕ2,,ϕp denotes a sequence of k×k matrices of autoregressive (AR) parameters and θ1,θ2,,θq denotes a sequence of k×k matrices of moving average (MA) parameters. μ is a vector of length k containing the series means. Let
A(ϕ)= [ ϕ1 I 0 . . . 0 ϕ2 0 I 0 . . 0 . . . . . . ϕp-1 0 . . . 0 I ϕp 0 . . . 0 0 ] pk×pk  
where I denotes the k×k identity matrix.
The model (1) is said to be stationary if the eigenvalues of A(ϕ) lie inside the unit circle. Similarly let
B(θ)= [ θ1 I 0 . . . 0 θ2 0 I 0 . . 0 . . . . . . θq-1 0 . . . 0 I θq 0 . . . 0 0 ] qk×qk .  
Then the model is said to be invertible if the eigenvalues of B(θ) lie inside the unit circle.
g13dxf returns the pk eigenvalues of A(ϕ) (or the qk eigenvalues of B(θ)) along with their moduli, in descending order of magnitude. Thus to check for stationarity or invertibility you should check whether the modulus of the largest eigenvalue is less than 1.

4 References

Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley

5 Arguments

1: k Integer Input
On entry: k, the dimension of the multivariate time series.
Constraint: k1.
2: ip Integer Input
On entry: the number of AR (or MA) parameter matrices, p (or q).
Constraint: ip1.
3: par(ip×k×k) Real (Kind=nag_wp) array Input
On entry: the AR (or MA) parameter matrices read in row by row in the order ϕ1,ϕ2,,ϕp (or θ1,θ2,,θq). That is, par((l-1)×k×k+(i-1)×k+j) must be set equal to the (i,j)th element of ϕl, for l=1,2,,p (or the (i,j)th element of θl, for l=1,2,,q).
4: rr(k×ip) Real (Kind=nag_wp) array Output
On exit: the real parts of the eigenvalues.
5: ri(k×ip) Real (Kind=nag_wp) array Output
On exit: the imaginary parts of the eigenvalues.
6: rmod(k×ip) Real (Kind=nag_wp) array Output
On exit: the moduli of the eigenvalues.
7: work(k×k×ip×ip) Real (Kind=nag_wp) array Workspace
8: iwork(k×ip) Integer array Workspace
9: 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 0 is recommended. 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:
On entry, ip=value.
Constraint: ip1.
On entry, k=value.
Constraint: k1.
An excessive number of iterations have been required to calculate the eigenvalues.
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.
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.

7 Accuracy

The accuracy of the results depends on the original matrix and the multiplicity of the roots.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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.

9 Further Comments

The time taken is approximately proportional to kp3 (or kq3).

10 Example

This example finds the eigenvalues of A(ϕ) where k=2 and p=1 and ϕ1=[ 0.802 0.065 0.000 0.575 ] .

10.1 Program Text

Program Text (g13dxfe.f90)

10.2 Program Data

Program Data (g13dxfe.d)

10.3 Program Results

Program Results (g13dxfe.r)