NAG FL Interfacef07mjf (dsytri)

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1Purpose

f07mjf computes the inverse of a real symmetric indefinite matrix $A$, where $A$ has been factorized by f07mdf.

2Specification

Fortran Interface
 Subroutine f07mjf ( uplo, n, a, lda, ipiv, work, info)
 Integer, Intent (In) :: n, lda, ipiv(*) Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*) Real (Kind=nag_wp), Intent (Out) :: work(n) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07mjf_ (const char *uplo, const Integer *n, double a[], const Integer *lda, const Integer ipiv[], double work[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f07mjf, nagf_lapacklin_dsytri or its LAPACK name dsytri.

3Description

f07mjf is used to compute the inverse of a real symmetric indefinite matrix $A$, the routine must be preceded by a call to f07mdf, which computes the Bunch–Kaufman factorization of $A$.
If ${\mathbf{uplo}}=\text{'U'}$, $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${U}^{\mathrm{T}}{P}^{\mathrm{T}}XPU={D}^{-1}$ for $X$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ and ${A}^{-1}$ is computed by solving ${L}^{\mathrm{T}}{P}^{\mathrm{T}}XPL={D}^{-1}$ for $X$.

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: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: details of the factorization of $A$, as returned by f07mdf.
On exit: the factorization is overwritten by the $n×n$ symmetric matrix ${A}^{-1}$.
If ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of ${A}^{-1}$ is stored in the upper triangular part of the array.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of ${A}^{-1}$ is stored in the lower triangular part of the array.
4: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07mjf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
5: $\mathbf{ipiv}\left(*\right)$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 f07mdf.
6: $\mathbf{work}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
7: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

$-999<{\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
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

f07mjf 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 f07mwf for Hermitian matrices and f07nwf 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 and must first be factorized by f07mdf.

10.1Program Text

Program Text (f07mjfe.f90)

10.2Program Data

Program Data (f07mjfe.d)

10.3Program Results

Program Results (f07mjfe.r)