naginterfaces.library.linsys.real_​toeplitz_​update

naginterfaces.library.linsys.real_toeplitz_update(t, b, x, work)[source]

real_toeplitz_update updates the solution of the equations , where is a real symmetric positive definite Toeplitz matrix.

For full information please refer to the NAG Library document for f04mf

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f04/f04mff.html

Parameters
tfloat, array-like, shape

must contain the value , for .

bfloat, array-like, shape

The right-hand side vector .

xfloat, array-like, shape

With the () elements of the solution vector as returned by a previous call to real_toeplitz_update. The element need not be specified.

workfloat, array-like, shape

With the elements of should be as returned from a previous call to real_toeplitz_update with () as the argument .

Returns
xfloat, ndarray, shape

The solution vector .

pfloat

The reflection coefficient . (See Further Comments.)

workfloat, ndarray, shape

The first () elements of contain the solution to the Yule–Walker equations

where .

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

The Toeplitz Matrix is not positive definite Value of the reflection coefficient is .

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

real_toeplitz_update solves the equations

where is the symmetric positive definite Toeplitz matrix

and is the -element vector , given the solution of the equations

This function will normally be used to successively solve the equations

If it is desired to solve the equations for a single value of , then function real_toeplitz_solve() may be called. This function uses the method of Levinson (see Levinson (1947) and Golub and Van Loan (1996)).

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

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore

Levinson, N, 1947, The Weiner RMS error criterion in filter design and prediction, J. Math. Phys. (25), 261–278