Integer, Intent (In)	::	n, lda, ipiv(*)
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,*)
Real (Kind=nag_wp), Intent (Out)	::	work(n)
Character (1), Intent (In)	::	uplo

C Header Interface

#include nagmk26.h

void	f07mjf_ ( const char uplo, const Integer n, double a[], const Integer lda, const Integer ipiv[], double work[], Integer info, const Charlen length_uplo)

The routine may be called by its LAPACK name dsytri.

3

Description

f07mjf (dsytri) is used to compute the inverse of a real symmetric indefinite matrix

A

, the routine must be preceded by a call to f07mdf (dsytrf), which computes the Bunch–Kaufman factorization of

A

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}

for

X

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}

for

X

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: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array a must be at least

\max (1, n)

On entry: details of the factorization of

A

, as returned by f07mdf (dsytrf).

On exit: the factorization is overwritten by the

n

n

symmetric matrix

A^{- 1}

uplo ='U'

, the upper triangle of

A^{- 1}

is stored in the upper triangular part of the array.

uplo ='L'

, the lower triangle of

A^{- 1}

is stored in the lower triangular part of the array.

4: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f07mjf (dsytri) is called.

Constraint:

lda \geq \max (1, n)

5: $ipiv (*)$ – 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 f07mdf (dsytrf).

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

7: $info$ – IntegerOutput

On exit:

info = 0

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

6

Error Indicators and Warnings

$- 999 < info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info = - 999$: Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
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

f07mjf (dsytri) 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 f07mwf (zhetri) for Hermitian matrices and f07nwf (zsytri) 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 and must first be factorized by f07mdf (dsytrf).

NAG Library Routine Document

f07mjf (dsytri)

▸▿ 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

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