The routine uses the method of Durbin (see Durbin (1960) and Golub and Van Loan (1996)). Optionally the mean square prediction errors and/or the partial correlation coefficients for each step can be returned.
4References
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
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: the order of the Toeplitz matrix $T$.
Constraint:
${\mathbf{n}}\ge 0$. When ${\mathbf{n}}=0$, an immediate return is effected.
2: $\mathbf{t}\left(0:{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{t}}\left(0\right)$ must contain the value ${\tau}_{0}$ of the diagonal elements of $T$, and the remaining n elements of t must contain the elements of the vector $t$.
Constraint:
${\mathbf{t}}\left(0\right)>0.0$. Note that if this is not true, the Toeplitz matrix cannot be positive definite.
3: $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the solution vector $x$.
4: $\mathbf{wantp}$ – LogicalInput
On entry: must be set to .TRUE. if the partial (auto)correlation coefficients are required, and must be set to .FALSE. otherwise.
5: $\mathbf{p}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array p
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$ if ${\mathbf{wantp}}=\mathrm{.TRUE.}$, and at least $1$ otherwise.
On exit: with wantp as .TRUE., the $i$th element of p contains the partial (auto)correlation coefficient, or reflection coefficient, ${p}_{i}$ for the $i$th step. (See Section 9 and Chapter G13.) If wantp is .FALSE., p is not referenced. Note that in any case, ${x}_{n}={p}_{n}$.
6: $\mathbf{wantv}$ – LogicalInput
On entry: must be set to .TRUE. if the mean square prediction errors are required, and must be set to .FALSE. otherwise.
7: $\mathbf{v}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array v
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$ if ${\mathbf{wantv}}=\mathrm{.TRUE.}$, and at least $1$ otherwise.
On exit: with wantv as .TRUE., the $i$th element of v contains the mean square prediction error, or predictor error variance ratio, ${v}_{i}$, for the $i$th step. (See Section 9 and Chapter G13.) If wantv is .FALSE., v is not referenced.
8: $\mathbf{vlast}$ – Real (Kind=nag_wp)Output
On exit: the value of ${v}_{n}$, the mean square prediction error for the final step.
9: $\mathbf{work}\left({\mathbf{n}}-1\right)$ – Real (Kind=nag_wp) arrayWorkspace
10: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $-1$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases f04fef may return useful information.
${\mathbf{ifail}}>0$
Principal minor $\u27e8\mathit{\text{value}}\u27e9$ is not positive definite. Value of the reflection coefficient is $\u27e8\mathit{\text{value}}\u27e9$.
If, on exit, ${x}_{{\mathbf{ifail}}}$ is close to unity, the principal minor was close to being singular, and the sequence ${\tau}_{0},{\tau}_{1},\dots ,{\tau}_{{\mathbf{ifail}}}$ may be a valid sequence nevertheless. The first ifail elements of x return the solution of the equations
where ${T}_{{\mathbf{ifail}}}$ is the ifailth principal minor of $T$. Similarly, if wantp and/or wantv are true, then p and/or v return the first ifail elements of p and v respectively and vlast returns ${v}_{{\mathbf{ifail}}}$. In particular if ${\mathbf{ifail}}={\mathbf{n}}$, then the solution of the equations $Tx=-t$ is returned in x, but ${\tau}_{{\mathbf{n}}}$ is such that ${T}_{{\mathbf{n}}+1}$ would not be positive definite to working accuracy.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{t}}\left(0\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{t}}\left(0\right)>0.0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The computed solution of the equations certainly satisfies
$$r=Tx+t\text{,}$$
where ${\Vert r\Vert}_{1}$ is approximately bounded by
$c$ being a modest function of $n$ and $\epsilon $ being the machine precision. This bound is almost certainly pessimistic, but it has not yet been established whether or not the method of Durbin is backward stable. If $\left|{p}_{n}\right|$ is close to one, then the Toeplitz matrix is probably ill-conditioned and hence only just positive definite. For further information on stability issues see Bunch (1985), Bunch (1987), Cybenko (1980) and Golub and Van Loan (1996). The following bounds on ${\Vert {{\mathbf{t}}}^{-1}\Vert}_{1}$ hold:
Note:${v}_{n}<{v}_{n-1}$. The norm of ${T}^{-1}$ may also be estimated using routine f04ydf.
8Parallelism and Performance
f04fef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The number of floating-point operations used by f04fef is approximately $2{n}^{2}$, independent of the values of wantp and wantv.
The mean square prediction error, ${v}_{i}$, is defined as