NAG Library Routine Document

F07PDF (DSPTRF)

+− 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

F07PDF (DSPTRF) computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.

2 Specification

SUBROUTINE F07PDF (

UPLO, N, AP, IPIV, INFO)

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

The routine may be called by its LAPACK name dsptrf.

3 Description

F07PDF (DSPTRF) factorizes a real symmetric matrix

A

, using the Bunch–Kaufman diagonal pivoting method and packed storage.

A

is factorized as either

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

UPLO ='U'

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

UPLO ='L'

, where

P

is a permutation matrix,

U

(or

L

) is a unit upper (or lower) triangular matrix and

D

is a symmetric block diagonal matrix with

1

1

and

2

2

diagonal blocks;

U

(or

L

) has

2

2

unit diagonal blocks corresponding to the

2

2

blocks of

D

. Row and column interchanges are performed to ensure numerical stability while preserving symmetry.

This method is suitable for symmetric matrices which are not known to be positive definite. If

A

is in fact positive definite, no interchanges are performed and no

2

2

blocks occur in

D

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

1: UPLO – CHARACTER(1)Input

On entry: specifies whether the upper or lower triangular part of

A

is stored and how

A

is to be factorized.

$UPLO ='U'$: The upper triangular part of $A$ is stored and $A$ is factorized as $P U D U^{T} P^{T}$ , where $U$ is upper triangular.
$UPLO ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $P L D L^{T} P^{T}$ , 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

n

n

symmetric matrix

A

, packed by columns.

More precisely,

if $UPLO ='U'$ , the upper triangle of $A$ 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 $A$ must be stored with element $A_{i j}$ in $AP (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

On exit:

A

is overwritten by details of the block diagonal matrix

D

and the multipliers used to obtain the factor

U

L

as specified by UPLO.

4: IPIV(N) – INTEGER arrayOutput

On exit: details of the interchanges and the block structure of

D

. More precisely,

if $IPIV (i) = k > 0$ , $d_{i i}$ is a $1$ by $1$ pivot block and the $i$ th row and column of $A$ were interchanged with the $k$ th row and column;
if $UPLO ='U'$ and $IPIV (i - 1) = IPIV (i) = - l < 0$ , $(\begin{matrix} d_{i - 1, i - 1} & {\bar{d}}_{i, i - 1} \\ {\bar{d}}_{i, i - 1} & d_{i i} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i - 1)$ th row and column of $A$ were interchanged with the $l$ th row and column;
if $UPLO ='L'$ and $IPIV (i) = IPIV (i + 1) = - m < 0$ , $(\begin{matrix} d_{i i} & d_{i + 1, i} \\ d_{i + 1, i} & d_{i + 1, i + 1} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i + 1)$ th row and column of $A$ were interchanged with the $m$ th row and column.

5: 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. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

7 Accuracy

UPLO ='U'

, the computed factors

U

and

D

are the exact factors of a perturbed matrix

A + E

, where

|E| \leq c (n) ε P |U| |D| |U^{T}| P^{T},

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

UPLO ='L'

, a similar statement holds for the computed factors

L

and

D

8 Further Comments

The elements of

D

overwrite the corresponding elements of

A

; if

D

has

2

2

blocks, only the upper or lower triangle is stored, as specified by UPLO.

The unit diagonal elements of

U

L

and the

2

2

unit diagonal blocks are not stored. The remaining elements of

U

L

overwrite elements in the corresponding columns of

A

, but additional row interchanges must be applied to recover

U

L

explicitly (this is seldom necessary). If

IPIV (i) = i

, for

i = 1, 2, \dots, n

(as is the case when

A

is positive definite), then

U

L

are stored explicitly in packed form (except for their unit diagonal elements which are equal to

1

The total number of floating point operations is approximately

\frac{1}{3} n^{3}

A call to F07PDF (DSPTRF) may be followed by calls to the routines:

F07PEF (DSPTRS) to solve $A X = B$ ;
F07PGF (DSPCON) to estimate the condition number of $A$ ;
F07PJF (DSPTRI) to compute the inverse of $A$ .

The complex analogues of this routine are F07PRF (ZHPTRF) for Hermitian matrices and F07QRF (ZSPTRF) for symmetric matrices.

9 Example

This example computes the Bunch–Kaufman factorization 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}),

using packed storage.

NAG Library Routine DocumentF07PDF (DSPTRF)

+− 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

NAG Library Routine Document

F07PDF (DSPTRF)