NAG Library Routine Document

F07KRF (ZPSTRF)

Lucas C (2004) LAPACK-style codes for Level 2 and 3 pivoted Cholesky factorizations LAPACK Working Note No. 161. Technical Report CS-04-522 Department of Computer Science, University of Tennessee, 107 Ayres Hall, Knoxville, TN 37996-1301, USA http://www.netlib.org/lapack/lawnspdf/lawn161.pdf

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 $U^{H} U$ , where $U$ is upper triangular.
$UPLO ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $L L^{H}$ , 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: $A (LDA *)$ – COMPLEX (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

Hermitian 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 F07KRF (ZPSTRF) 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

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

$INFO = 1$: The matrix $A$ is not positive definite. It is either positive semidefinite with computed rank as returned in RANK and less than $n$ , or it may be indefinite, see Section 9.

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^{H} 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 ℂ^{r \times r},

c (r)

is a modest linear function of

r

ε

is machine precision, 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 Parallelism and Performance

F07KRF (ZPSTRF) is not threaded by NAG in any implementation.

F07KRF (ZPSTRF) 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 real floating-point operations is approximately

4 n r^{2} - 8 / 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 F07MRF (ZHETRF). See Lucas (2004) for further information.

The real analogue of this routine is F07KDF (DPSTRF).

10 Example

This example computes the Cholesky factorization of the matrix

A

, where

A = (\begin{array}{r} 12.40 + 0.00 i & 2.39 + 0.00 i & 5.50 + 0.05 i & 4.47 + 0.00 i & 11.89 + 0.00 i \\ 2.39 + 0.00 i & 1.63 + 0.00 i & 1.04 + 0.10 i & 1.14 + 0.00 i & 1.81 + 0.00 i \\ 5.50 + 0.05 i & 1.04 + 0.10 i & 2.45 + 0.00 i & 1.98 - 0.03 i & 5.28 - 0.02 i \\ 4.47 + 0.00 i & 1.14 + 0.00 i & 1.98 - 0.03 i & 1.71 + 0.00 i & 4.14 + 0.00 i \\ 11.89 + 0.00 i & 1.81 + 0.00 i & 5.28 - 0.02 i & 4.14 + 0.00 i & 11.63 + 0.00 i \end{array}) .

NAG Library Routine DocumentF07KRF (ZPSTRF)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

NAG Library Routine Document

F07KRF (ZPSTRF)