NAG Library Function Document

nag_dsptri (f07pjc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

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

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dsptri (Nag_OrderType order, Nag_UploType uplo, Integer n, double ap[], const Integer ipiv[], NagError *fail)

3
Description

nag_dsptri (f07pjc) is used to compute the inverse of a real symmetric indefinite matrix A, the function must be preceded by a call to nag_dsptrf (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_OrderTypeInput
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uplo Nag_UploTypeInput
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 IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     ap[dim] doubleInput/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 nag_dsptrf (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 IntegerInput
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 nag_dsptrf (f07pdc).
6:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_dsptri (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 nag_zhptri (f07pwc) for Hermitian matrices and nag_zsptri (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 nag_dsptrf (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)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017