naginterfaces.library.linsys.real_​toeplitz_​update

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

real_toeplitz_update updates the solution of 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 f04mf

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f04/f04mff.html

Parameters

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 $b_n$.

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

With $n>1$ the ($n-1$) elements of the solution vector $x_{n-1}$ as returned by a previous call to real_toeplitz_update. The element $x[n-1]$ need not be specified.

workfloat, array-like, shape $(2\times n-1)$

With $n>2$ the elements of $work$ should be as returned from a previous call to real_toeplitz_update with ($n-1$) as the argument $\text{n}$.

Returns

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

The solution vector $x_n$.

pfloat

The reflection coefficient $p_{n-1}$. (See Further Comments.)

workfloat, ndarray, shape $(2\times n-1)$

The first ($n-1$) elements of $work$ contain the solution to the Yule–Walker equations

$$T_{n-1}{y}_{n-1}=-t_{n-1}\text{,}$$

where $t_{n-1}={({\tau }_1{\tau }_2\dots {\tau }_{n-1})}^t$.

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 $1$)

The Toeplitz Matrix 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_update solves the equations

$$T_n{x}_n=b_n\text{,}$$

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

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

and $b_n$ is the $n$-element vector $b_n={({\beta }_1{\beta }_2\dots {\beta }_n)}^T$, given the solution of the equations

$$T_{n-1}{x}_{n-1}=b_{n-1}\text{.}$$

This function will normally be used to successively solve the equations

$$T_k{x}_k=b_k\text{,}k=1,2,\dots ,n\text{.}$$

If it is desired to solve the equations for a single value of $n$, 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