PDF version (NAG web site
, 64-bit version, 64-bit version)
NAG Toolbox: nag_linsys_real_toeplitz_yule (f04fe)
Purpose
nag_linsys_real_toeplitz_yule (f04fe) solves the Yule–Walker equations for a real symmetric positive definite Toeplitz system.
Syntax
Description
nag_linsys_real_toeplitz_yule (f04fe) solves the equations
where
is the
by
symmetric positive definite Toeplitz matrix
and
is the vector
The function uses the method of Durbin (see
Durbin (1960) and
Golub and Van Loan (1996)). Optionally the mean square prediction errors and/or the partial correlation coefficients for each step can be returned.
References
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
Parameters
Compulsory Input Parameters
- 1:
– int64int32nag_int scalar
-
The order of the Toeplitz matrix .
Constraint:
. When , then an immediate return is effected.
- 2:
– double array
-
must contain the value
of the diagonal elements of
, and the remaining
n elements of
t must contain the elements of the vector
.
Constraint:
. Note that if this is not true, then the Toeplitz matrix cannot be positive definite.
- 3:
– logical scalar
-
Must be set to true if the partial (auto)correlation coefficients are required, and must be set to false otherwise.
- 4:
– logical scalar
-
Must be set to true if the mean square prediction errors are required, and must be set to false otherwise.
Optional Input Parameters
None.
Output Parameters
- 1:
– double array
-
The solution vector .
- 2:
– double array
-
The dimension of the array
p will be
if
and
otherwise
With
wantp as
true, the
th element of
p contains the partial (auto)correlation coefficient, or reflection coefficient,
for the
th step. (See
Further Comments and
Chapter G13.) If
wantp is
false, then
p is not referenced. Note that in any case,
.
- 3:
– double array
-
The dimension of the array
v will be
if
and
otherwise
With
wantv as
true, the
th element of
v contains the mean square prediction error, or predictor error variance ratio,
, for the
th step. (See
Further Comments and
Chapter G13.) If
wantv is
false, then
v is not referenced.
- 4:
– double scalar
-
The value of , the mean square prediction error for the final step.
- 5:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Note: nag_linsys_real_toeplitz_yule (f04fe) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
Cases prefixed with W are classified as warnings and
do not generate an error of type NAG:error_n. See nag_issue_warnings.
-
-
- W
-
The principal minor of order (
) of the Toeplitz matrix is not positive definite to working accuracy. If, on exit,
is close to unity, then the principal minor was close to being singular, and the sequence
may be a valid sequence nevertheless. The first
ifail elements of
x return the solution of the equations
where
is the
ifailth principal minor of
. Similarly, if
wantp and/or
wantv are true, then
p and/or
v return the first
ifail elements of
p and
v respectively and
vlast returns
. In particular if
, then the solution of the equations
is returned in
x, but
is such that
would not be positive definite to working accuracy.
-
An unexpected error has been triggered by this routine. Please
contact
NAG.
-
Your licence key may have expired or may not have been installed correctly.
-
Dynamic memory allocation failed.
Accuracy
The computed solution of the equations certainly satisfies
where
is approximately bounded by
being a modest function of
and
being the
machine precision. This bound is almost certainly pessimistic, but it has not yet been established whether or not the method of Durbin is backward stable. If
is close to one, then the Toeplitz matrix is probably ill-conditioned and hence only just positive definite. For further information on stability issues see
Bunch (1985),
Bunch (1987),
Cybenko (1980) and
Golub and Van Loan (1996). The following bounds on
hold:
Note: . The norm of
may also be estimated using function
nag_linsys_real_gen_norm_rcomm (f04yd).
Further Comments
The number of floating-point operations used by
nag_linsys_real_toeplitz_yule (f04fe) is approximately
, independent of the values of
wantp and
wantv.
The mean square prediction error,
, is defined as
where
is the solution of the equations
and the partial correlation coefficient,
, is defined as the
th element of
. Note that
.
Example
This example finds the solution of the Yule–Walker equations
, where
Open in the MATLAB editor:
f04fe_example
function f04fe_example
fprintf('f04fe example results\n\n');
n = int64(4);
t = [4 3 2 1 0];
wantp = true;
wantv = true;
[x, p, v, vlast, ifail] = ...
f04fe(n, t, wantp, wantv);
disp('Solution vector');
disp(x');
disp('Reflection coefficients');
disp(p');
disp('Mean square prediction errors');
disp(v');
f04fe example results
Solution vector
-0.8000 0.0000 -0.0000 0.2000
Reflection coefficients
-0.7500 0.1429 0.1667 0.2000
Mean square prediction errors
0.4375 0.4286 0.4167 0.4000
PDF version (NAG web site
, 64-bit version, 64-bit version)
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015