NAG Library Routine Document

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 <nagmk26.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 its LAPACK name dsptri.

3

Description

f07pjf (dsptri) is used to compute the inverse of a real symmetric indefinite matrix

A

, the routine must be preceded by a call to f07pdf (dsptrf), which computes the Bunch–Kaufman factorization of

A

, using packed storage.

uplo ='U'

A = P U D U^{T} P^{T}

and

A^{- 1}

is computed by solving

U^{T} P^{T} X P U = D^{- 1}

uplo ='L'

A = P L D L^{T} P^{T}

and

A^{- 1}

is computed by solving

L^{T} P^{T} X P L = 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.

$uplo ='U'$: $A = P U D U^{T} P^{T}$ , where $U$ is upper triangular.
$uplo ='L'$: $A = P L D L^{T} P^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $ap (*)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

On entry: the factorization of

A

stored in packed form, as returned by f07pdf (dsptrf).

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 $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A^{- 1}$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

4: $ipiv (*)$ – Integer arrayInput

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 (dsptrf).

5: $work (n)$ – Real (Kind=nag_wp) arrayWorkspace

6: $info$ – IntegerOutput

On exit:

info = 0

unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: 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

if $uplo ='U'$ , $|D U^{T} P^{T} X P U - I| \leq c (n) ε (|D| |U^{T}| P^{T} |X| P |U| + |D| |D^{- 1}|)$ ;
if $uplo ='L'$ , $|D L^{T} P^{T} X P L - I| \leq c (n) ε (|D| |L^{T}| P^{T} |X| P |L| + |D| |D^{- 1}|)$ ,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

8

Parallelism and Performance

f07pjf (dsptri) 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

\frac{2}{3} n^{3}

The complex analogues of this routine are f07pwf (zhptri) for Hermitian matrices and f07qwf (zsptri) for symmetric matrices.

10

Example

This example computes the inverse of the matrix

A

, where

A = (\begin{matrix} 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 \end{matrix}) .

Here

A

is symmetric indefinite, stored in packed form, and must first be factorized by f07pdf (dsptrf).

NAG Library Routine Document

f07pjf (dsptri)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification