NAG Library Routine Document

F07PJF (DSPTRI)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07PJF (DSPTRI) computes the inverse of a real symmetric indefinite matrix

A

, where

A

has been factorized by F07PDF (DSPTRF), using packed storage.

2 Specification

SUBROUTINE F07PJF (

UPLO, N, AP, IPIV, WORK, INFO)

INTEGER	N, IPIV(*), INFO
REAL (KIND=nag_wp)	AP(*), WORK(N)
CHARACTER(1)	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 Parameters

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

Errors or warnings detected by the routine:

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

$INFO > 0$: If $INFO = i$ , $d (i, i)$ 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 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.

9 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 DocumentF07PJF (DSPTRI)