NAG FL Interfacef07pjf (dsptri)

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1Purpose

f07pjf computes the inverse of a real symmetric indefinite matrix $A$, where $A$ has been factorized by f07pdf, using packed storage.

2Specification

Fortran Interface
 Subroutine f07pjf ( uplo, n, ap, ipiv, work, info)
 Integer, Intent (In) :: n, ipiv(*) Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: ap(*) Real (Kind=nag_wp), Intent (Out) :: work(n) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07pjf_ (const char *uplo, const Integer *n, double ap[], const Integer ipiv[], double work[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f07pjf, nagf_lapacklin_dsptri or its LAPACK name dsptri.

3Description

f07pjf is used to compute the inverse of a real symmetric indefinite matrix $A$, the routine must be preceded by a call to f07pdf, which computes the Bunch–Kaufman factorization of $A$, using packed storage.
If ${\mathbf{uplo}}=\text{'U'}$, $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU={D}^{-1}$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL={D}^{-1}$.

4References

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

5Arguments

1: $\mathbf{uplo}$Character(1) Input
On entry: specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{ap}\left(*\right)$Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the factorization of $A$ stored in packed form, as returned by f07pdf.
On exit: the factorization is overwritten by the $n×n$ matrix ${A}^{-1}$.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of ${A}^{-1}$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of ${A}^{-1}$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
4: $\mathbf{ipiv}\left(*\right)$Integer array Input
Note: the dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: details of the interchanges and the block structure of $D$, as returned by f07pdf.
5: $\mathbf{work}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
6: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
Element $⟨\mathit{\text{value}}⟩$ of the diagonal is exactly zero. $D$ is singular and the inverse of $A$ cannot be computed.

7Accuracy

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

8Parallelism and Performance

f07pjf 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.

The total number of floating-point operations is approximately $\frac{2}{3}{n}^{3}$.
The complex analogues of this routine are f07pwf for Hermitian matrices and f07qwf for symmetric matrices.

10Example

This example computes the inverse of the matrix $A$, where
 $A= ( 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ) .$
Here $A$ is symmetric indefinite, stored in packed form, and must first be factorized by f07pdf.

10.1Program Text

Program Text (f07pjfe.f90)

10.2Program Data

Program Data (f07pjfe.d)

10.3Program Results

Program Results (f07pjfe.r)