naginterfaces.library.linsys.real_​toeplitz_​solve

naginterfaces.library.linsys.real_toeplitz_solve(n, t, b, wantp)[source]

real_toeplitz_solve solves the equations , where is a real symmetric positive definite Toeplitz matrix.

For full information please refer to the NAG Library document for f04ff

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f04/f04fff.html

Parameters
nint

The order of the Toeplitz matrix .

tfloat, array-like, shape

must contain the value , for .

bfloat, array-like, shape

The right-hand side vector .

wantpbool

Must be set to if the reflection coefficients are required, and must be set to otherwise.

Returns
xfloat, ndarray, shape

The solution vector .

pfloat, ndarray, shape

With as , the th element of contains the reflection coefficient, , for the th step, for . (See Further Comments.) If is , is not referenced.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

Principal minor 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_solve solves the equations

where is the symmetric positive definite Toeplitz matrix

and is an -element vector.

The function uses the method of Levinson (see Levinson (1947) and Golub and Van Loan (1996)). Optionally, the reflection coefficients for each step may also 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

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