naginterfaces.library.linsys.real_​toeplitz_​yule_​update

naginterfaces.library.linsys.real_toeplitz_yule_update(t, x, v)

real_toeplitz_yule_update updates the solution to the Yule–Walker equations for a real symmetric positive definite Toeplitz system.

For full information please refer to the NAG Library document for f04me

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f04/f04mef.html

Parameters

tfloat, array-like, shape $(n+1)$

$t\left[0\right]$ must contain the value ${\tau }_0$ of the diagonal elements of $T$, and the remaining $\text{n}$ elements of $t$ must contain the elements of the vector $t_n$.

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_yule_update. The element $x[n-1]$ need not be specified.

vfloat

With $n>1$ the mean square prediction error for the ($n-1$)th step, as returned by a previous call to real_toeplitz_yule_update.

Returns

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

The solution vector $x_n$. The element $x[n-1]$ returns the partial (auto)correlation coefficient, or reflection coefficient, for the $n$th step. If $\left|x\right[n-1\left]\right|\ge 1.0$, the matrix $T_{n+1}$ will not be positive definite to working accuracy.

vfloat

The mean square prediction error, or predictor error variance ratio, ${\nu }_n$, for the $n$th step. (See Further Comments and the G13 Introduction.)

Raises

NagValueError

(errno $-1$)

On entry, $n=⟨value⟩$ and $x[n-2]=⟨value⟩$.

Constraint: if $n>1$, $\left|x\right[n-2\left]\right|<1.0$.

(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 $1$)

Matrix of order $⟨value⟩$ would not be 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_yule_update solves the equations

$$T_n{x}_n=-t_n\text{,}$$

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 $t_n$ is the vector

$$t_n^T=({\tau }_1{\tau }_2\dots {\tau }_n)\text{,}$$

given the solution of the equations

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

The function will normally be used to successively solve the equations

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

If it is desired to solve the equations for a single value of $n$, then function real_toeplitz_yule() may be called. This function uses the method of Durbin (see Durbin (1960) 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

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