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NAG Toolbox: nag_lapack_dsptri (f07pj)

Purpose

nag_lapack_dsptri (f07pj) computes the inverse of a real symmetric indefinite matrix A$A$, where A$A$ has been factorized by nag_lapack_dsptrf (f07pd), using packed storage.

Syntax

[ap, info] = f07pj(uplo, ap, ipiv, 'n', n)
[ap, info] = nag_lapack_dsptri(uplo, ap, ipiv, 'n', n)

Description

nag_lapack_dsptri (f07pj) is used to compute the inverse of a real symmetric indefinite matrix A$A$, the function must be preceded by a call to nag_lapack_dsptrf (f07pd), which computes the Bunch–Kaufman factorization of A$A$, using packed storage.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, A = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ and A1${A}^{-1}$ is computed by solving UTPTXPU = D1${U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU={D}^{-1}$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ and A1${A}^{-1}$ is computed by solving LTPTXPL = D1${L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL={D}^{-1}$.

References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how A$A$ has been factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The factorization of A$A$ stored in packed form, as returned by nag_lapack_dsptrf (f07pd).
3:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the interchanges and the block structure of D$D$, as returned by nag_lapack_dsptrf (f07pd).

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array ipiv.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

work

Output Parameters

1:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The factorization stores the n$n$ by n$n$ matrix A1${A}^{-1}$.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A1${A}^{-1}$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A1${A}^{-1}$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: ipiv, 5: work, 6: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, d(i,i)$d\left(i,i\right)$ is exactly zero; D$D$ is singular and the inverse of A$A$ cannot be computed.

Accuracy

The computed inverse X$X$ satisfies a bound of the form
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, |DUTPTXPUI|c(n)ε(|D||UT|PT|X|P|U| + |D||D1|)$|D{U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU-I|\le c\left(n\right)\epsilon \left(|D||{U}^{\mathrm{T}}|{P}^{\mathrm{T}}|X|P|U|+|D||{D}^{-1}|\right)$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, |DLTPTXPLI|c(n)ε(|D||LT|PT|X|P|L| + |D||D1|)$|D{L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL-I|\le c\left(n\right)\epsilon \left(|D||{L}^{\mathrm{T}}|{P}^{\mathrm{T}}|X|P|L|+|D||{D}^{-1}|\right)$,
c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision

Further Comments

The total number of floating point operations is approximately (2/3)n3$\frac{2}{3}{n}^{3}$.
The complex analogues of this function are nag_lapack_zhptri (f07pw) for Hermitian matrices and nag_lapack_zsptri (f07qw) for symmetric matrices.

Example

```function nag_lapack_dsptri_example
uplo = 'L';
ap = [2.07;
4.2;
0.2230413840558341;
0.6536583767489105;
1.15;
0.8115010321439103;
-0.5959697237786296;
-2.59067708640519;
0.3030846795506181;
0.4073851981348882];
ipiv = [int64(-3);-3;3;4];
[apOut, info] = nag_lapack_dsptri(uplo, ap, ipiv)
```
```

apOut =

0.7485
0.5221
-1.0058
-1.4386
-0.1605
-0.3131
-0.7440
1.3501
2.0667
2.4547

info =

0

```
```function f07pj_example
uplo = 'L';
ap = [2.07;
4.2;
0.2230413840558341;
0.6536583767489105;
1.15;
0.8115010321439103;
-0.5959697237786296;
-2.59067708640519;
0.3030846795506181;
0.4073851981348882];
ipiv = [int64(-3);-3;3;4];
[apOut, info] = f07pj(uplo, ap, ipiv)
```
```

apOut =

0.7485
0.5221
-1.0058
-1.4386
-0.1605
-0.3131
-0.7440
1.3501
2.0667
2.4547

info =

0

```

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