naginterfaces.library.linsys.real_​toeplitz_​solve

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

real_toeplitz_solve solves the equations $Tx=b$, where $T$ is a real symmetric positive definite Toeplitz matrix.

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

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/f04/f04fff.html

Parameters

nint

The order of the Toeplitz matrix $T$.

tfloat, array-like, shape $\left(n\right)$

$t\left[\text{i}\right]$ must contain the value ${\tau }_{\text{i}}$, for $\text{i}=0,1,\dots ,n-1$.

bfloat, array-like, shape $\left(n\right)$

The right-hand side vector $b$.

wantpbool

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

Returns

xfloat, ndarray, shape $\left(n\right)$

The solution vector $x$.

pfloat, ndarray, shape $(:)$

With $wantp$ as $True$, the $i$th element of $p$ contains the reflection coefficient, $p_{\text{i}}$, for the $\text{i}$th step, for $\text{i}=1,2,\dots ,n-1$. (See Further Comments.) If $wantp$ is $False$, $p$ is not referenced.

Raises

NagValueError

(errno $-1$)

On entry, $t\left[0\right]=⟨value⟩$.

Constraint: $t\left[0\right]>0.0$.

(errno $-1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

Principal minor $⟨value⟩$ is not positive definite. Value of the reflection coefficient is $⟨value⟩$.

Notes

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

real_toeplitz_solve solves the equations

$$Tx=b\text{,}$$

where $T$ is the $n\times n$ symmetric positive definite Toeplitz matrix

$$\begin{array}{c}T=\left(\begin{array}{ccccc}{\tau }_0& {\tau }_1& {\tau }_2& \dots & {\tau }_{n-1}\\ {\tau }_1& {\tau }_0& {\tau }_1& \dots & {\tau }_{n-2}\\ {\tau }_2& {\tau }_1& {\tau }_0& \dots & {\tau }_{n-3}\\ .& .& .& & .\\ {\tau }_{n-1}& {\tau }_{n-2}& {\tau }_{n-3}& \dots & {\tau }_0\end{array}\right)\end{array}$$

and $b$ is an $n$-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