NAG Library Routine Document

F07KDF (DPSTRF)

+− 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

F07KDF (DPSTRF) computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.

2 Specification

SUBROUTINE F07KDF (

UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)

INTEGER	N, LDA, PIV(N), RANK, INFO
REAL (KIND=nag_wp)	A(LDA,), TOL, WORK(2N)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name dpstrf.

3 Description

F07KDF (DPSTRF) forms the Cholesky factorization of a real symmetric positive semidefinite matrix

A

either as

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

UPLO ='U'

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

UPLO ='L'

, where

P

is a permutation matrix,

U

is an upper triangular matrix and

L

is lower triangular.

This algorithm does not attempt to check that

A

is positive semidefinite.

4 References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Lucas C (2004) LAPACK-style codes for Level 2 and 3 pivoted Cholesky factorizations LAPACK Working Note 161. Technical Report CS-04-522 Department of Computer Science, University of Tennessee, 107 Ayres Hall, Knoxville, TN 37996-1301, USA

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

n

n

symmetric positive semidefinite matrix

A

If $UPLO ='U'$ , the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If $UPLO ='L'$ , the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: if

UPLO ='U'

, the first RANK rows of the upper triangle of

A

are overwritten with the nonzero elements of the Cholesky factor

U

, and the remaining rows of the triangle are destroyed.

UPLO ='L'

, the first RANK columns of the lower triangle of

A

are overwritten with the nonzero elements of the Cholesky factor

L

, and the remaining columns of the triangle are destroyed.

4: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F07KDF (DPSTRF) is called.

Constraint:

LDA \geq \max (1, N)

5: PIV(N) – INTEGER arrayOutput

On exit: PIV is such that the nonzero entries of

P

are

P (PIV (k), k) = 1

, for

k = 1, 2, \dots, n

6: RANK – INTEGEROutput

On exit: the computed rank of

A

given by the number of steps the algorithm completed.

7: TOL – REAL (KIND=nag_wp)Input

On entry: user defined tolerance. If

TOL < 0

, then

n \times max_{k = 1, n} |A_{k k}| \times machine precision

will be used. The algorithm terminates at the

r

th step if the

(r + 1)

th step pivot

< TOL

8: WORK( $2 * N$ ) – REAL (KIND=nag_wp) arrayWorkspace

9: 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.

$INFO = 1$: The matrix $A$ is either rank deficient with computed rank as returned in RANK, or is indefinite, see Section 8.

7 Accuracy

UPLO ='L'

and

RANK = r

, the computed Cholesky factor

L

and permutation matrix

P

satisfy the following upper bound

\frac{{‖A - P L L^{T} P^{T}‖}_{2}}{{‖A‖}_{2}} \leq 2 r c (r) ε {({‖W‖}_{2} + 1)}^{2} + O (ε^{2}),

where

W = L_{11}^{- 1} L_{12}, L = (\begin{matrix} L_{11} & 0 \\ L_{12} & 0 \end{matrix}), L_{11} \in ℝ^{r \times r},

c (r)

is a modest linear function of

r

ε

is machine epsilon, and

{‖W‖}_{2} \leq \sqrt{\frac{1}{3} (n - r) (4^{r} - 1)} .

So there is no guarantee of stability of the algorithm for large

n

and

r

, although

{‖W‖}_{2}

is generally small in practice.

8 Further Comments

The total number of floating point operations is approximately

n r^{2} - 2 / 3 r^{3}

, where

r

is the computed rank of

A

This algorithm does not attempt to check that

A

is positive semidefinite, and in particular the rank detection criterion in the algorithm is based on

A

being positive semidefinite. If there is doubt over semidefiniteness then you should use the indefinite factorization F07MDF (DSYTRF). See Lucas (2004) for further information.

The complex analogue of this routine is F07KRF (ZPSTRF).

9 Example

This example computes the Cholesky factorization of the matrix

A

, where

A = (\begin{matrix} 2.51 & 4.04 & 3.34 & 1.34 & 1.29 \\ 4.04 & 8.22 & 7.38 & 2.68 & 2.44 \\ 3.34 & 7.38 & 7.06 & 2.24 & 2.14 \\ 1.34 & 2.68 & 2.24 & 0.96 & 0.80 \\ 1.29 & 2.44 & 2.14 & 0.80 & 0.74 \end{matrix}) .

NAG Library Routine DocumentF07KDF (DPSTRF)

+− 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

F07KDF (DPSTRF)