Integer, Intent (In)	::	n, lda
Integer, Intent (Out)	::	piv(n), rank, info
Real (Kind=nag_wp), Intent (In)	::	tol
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,*)
Real (Kind=nag_wp), Intent (Out)	::	work(2*n)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f07kdf_ (const char uplo, const Integer n, double a[], const Integer lda, Integer piv[], Integer rank, const double tol, double work[], Integer info, const Charlen length_uplo)

The routine may be called by the names f07kdf, nagf_lapacklin_dpstrf or its LAPACK name dpstrf.

3 Description

f07kdf 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 No. 161. Technical Report CS-04-522 Department of Computer Science, University of Tennessee, 107 Ayres Hall, Knoxville, TN 37996-1301, USA https://www.netlib.org/lapack/lawnspdf/lawn161.pdf

5 Arguments

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: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $a (lda, *)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: the

n \times 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: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f07kdf is called.

Constraint:

lda \geq \max (1, n)

5: $piv (n)$ – Integer array Output

On exit: piv is such that the nonzero entries of

P

are

P (piv (k), k) = 1

, for

k = 1, 2, \dots, n

6: $rank$ – Integer Output

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

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) array Workspace

9: $info$ – Integer Output

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^{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 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

Background information to multithreading can be found in the Multithreading documentation.

f07kdf 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

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. See Lucas (2004) for further information.

The complex analogue of this routine is f07krf.

10 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}) .

f07kd: FL CL CPP AD PY MB

NAG FL Interfacef07kdf (dpstrf)

▸▿ 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

NAG FL Interface
f07kdf (dpstrf)