NAG FL Interface
f07pjf (dsptri)

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1 Purpose

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

2 Specification

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
C Header Interface
#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.

3 Description

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 uplo='U', A=PUDUTPT and A-1 is computed by solving UTPTXPU=D-1.
If uplo='L', A=PLDLTPT and A-1 is computed by solving LTPTXPL=D-1.

4 References

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

5 Arguments

1: uplo Character(1) Input
On entry: specifies how A has been factorized.
A=PUDUTPT, where U is upper triangular.
A=PLDLTPT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: ap(*) Real (Kind=nag_wp) array Input/Output
Note: the dimension of the array ap must be at least max(1,n×(n+1)/2).
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 uplo='U', the upper triangle of A-1 must be stored with element Aij in ap(i+j(j-1)/2) for ij;
  • if uplo='L', the lower triangle of A-1 must be stored with element Aij in ap(i+(2n-j)(j-1)/2) for ij.
4: ipiv(*) Integer array Input
Note: the dimension of the array ipiv must be at least max(1,n).
On entry: details of the interchanges and the block structure of D, as returned by f07pdf.
5: work(n) Real (Kind=nag_wp) array Workspace
6: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
Element value of the diagonal is exactly zero. D is singular and the inverse of A cannot be computed.

7 Accuracy

The computed inverse X satisfies a bound of the form c(n) is a modest linear function of n, and ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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.

9 Further Comments

The total number of floating-point operations is approximately 23n3.
The complex analogues of this routine are f07pwf for Hermitian matrices and f07qwf for symmetric matrices.

10 Example

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.

10.1 Program Text

Program Text (f07pjfe.f90)

10.2 Program Data

Program Data (f07pjfe.d)

10.3 Program Results

Program Results (f07pjfe.r)