NAG CL Interface
f07kdc (dpstrf)

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1 Purpose

f07kdc computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.

2 Specification

#include <nag.h>
void  f07kdc (Nag_OrderType order, Nag_UploType uplo, Integer n, double a[], Integer pda, Integer piv[], Integer *rank, double tol, NagError *fail)
The function may be called by the names: f07kdc, nag_lapacklin_dpstrf or nag_dpstrf.

3 Description

f07kdc forms the Cholesky factorization of a real symmetric positive semidefinite matrix A either as PTAP=UTU if uplo=Nag_Upper or PTAP=LLT if uplo=Nag_Lower, 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: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
On entry: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo=Nag_Upper
The upper triangular part of A is stored and A is factorized as UTU, where U is upper triangular.
uplo=Nag_Lower
The lower triangular part of A is stored and A is factorized as LLT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
4: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least max(1,pda×n).
On entry: the n×n symmetric positive semidefinite matrix A.
If order=Nag_ColMajor, Aij is stored in a[(j-1)×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[(i-1)×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, 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=Nag_Upper, 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.
If uplo=Nag_Lower, 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.
5: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax(1,n).
6: piv[n] Integer Output
On exit: piv is such that the nonzero entries of P are P (piv[k-1],k) =1, for k=1,2,,n.
7: rank Integer * Output
On exit: the computed rank of A given by the number of steps the algorithm completed.
8: tol double Input
On entry: user defined tolerance. If tol<0, n×maxk=1,n|Akk|×machine precision will be used. The algorithm terminates at the rth step if the (r+1)th step pivot <tol.
9: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_NOT_POS_DEF
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

If uplo=Nag_Lower and rank=r, the computed Cholesky factor L and permutation matrix P satisfy the following upper bound
A-PLLTPT 2 A2 2r c(r) ε ( W 2 +1) 2 + O(ε2) ,  
where
W = L 11 −1 L12 ,   L = ( L11 0 L12 0 ) ,   L11 r×r ,  
c(r) is a modest linear function of r, ε is machine precision, and
W2 13 (n-r) (4r-1) .  
So there is no guarantee of stability of the algorithm for large n and r, although W2 is generally small in practice.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07kdc 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 function. 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 nr2-2/3r3, 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 f07mdc. See Lucas (2004) for further information.
The complex analogue of this function is f07krc.

10 Example

This example computes the Cholesky factorization of the matrix A, where
A= ( 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 ) .  

10.1 Program Text

Program Text (f07kdce.c)

10.2 Program Data

Program Data (f07kdce.d)

10.3 Program Results

Program Results (f07kdce.r)