NAG CL Interface
f07pjc (dsptri)

1 Purpose

f07pjc computes the inverse of a real symmetric indefinite matrix A, where A has been factorized by f07pdc, using packed storage.

2 Specification

#include <nag.h>
void  f07pjc (Nag_OrderType order, Nag_UploType uplo, Integer n, double ap[], const Integer ipiv[], NagError *fail)
The function may be called by the names: f07pjc, nag_lapacklin_dsptri or nag_dsptri.

3 Description

f07pjc is used to compute the inverse of a real symmetric indefinite matrix A, the function must be preceded by a call to f07pdc, which computes the Bunch–Kaufman factorization of A, using packed storage.
If uplo=Nag_Upper, A=PUDUTPT and A-1 is computed by solving UTPTXPU=D-1.
If uplo=Nag_Lower, 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: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
On entry: specifies how A has been factorized.
uplo=Nag_Upper
A=PUDUTPT, where U is upper triangular.
uplo=Nag_Lower
A=PLDLTPT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: ap[dim] double Input/Output
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the factorization of A stored in packed form, as returned by f07pdc.
On exit: the factorization is overwritten by the n by n matrix A-1.
The storage of elements Aij depends on the order and uplo arguments as follows:
if order=Nag_ColMajor and uplo=Nag_Upper,
Aij is stored in ap[j-1×j/2+i-1], for ij;
if order=Nag_ColMajor and uplo=Nag_Lower,
Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Upper,
Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
if order=Nag_RowMajor and uplo=Nag_Lower,
Aij is stored in ap[i-1×i/2+j-1], for ij.
5: ipiv[dim] const Integer Input
Note: the dimension, dim, of the array ipiv must be at least max1,n.
On entry: details of the interchanges and the block structure of D, as returned by f07pdc.
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
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 cn is a modest linear function of n, and ε is the machine precision.

8 Parallelism and Performance

f07pjc 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 function. 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 function are f07pwc for Hermitian matrices and f07qwc 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 f07pdc.

10.1 Program Text

Program Text (f07pjce.c)

10.2 Program Data

Program Data (f07pjce.d)

10.3 Program Results

Program Results (f07pjce.r)