NAG FL Interface
f07krf (zpstrf)

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1 Purpose

f07krf computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix.

2 Specification

Fortran Interface
Subroutine f07krf ( uplo, n, a, lda, piv, rank, tol, work, info)
Integer, Intent (In) :: n, lda
Integer, Intent (Out) :: piv(n), rank, info
Real (Kind=nag_wp), Intent (In) :: tol
Real (Kind=nag_wp), Intent (Out) :: work(2*n)
Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07krf_ (const char *uplo, const Integer *n, Complex 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 f07krf, nagf_lapacklin_zpstrf or its LAPACK name zpstrf.

3 Description

f07krf forms the Cholesky factorization of a complex Hermitian positive semidefinite matrix A either as PTAP=UHU if uplo='U' or PTAP=LLH if 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

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.
The upper triangular part of A is stored and A is factorized as UHU, where U is upper triangular.
The lower triangular part of A is stored and A is factorized as LLH, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: a(lda,*) Complex (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×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.
If 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 f07krf is called.
Constraint: ldamax(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,,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×maxk=1,n|Akk|×machine precision will be used. The algorithm terminates at the rth 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

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
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='L' and rank=r, the computed Cholesky factor L and permutation matrix P satisfy the following upper bound
A-PLLHPT 2 A2 2r c(r) ε ( W 2 +1) 2 + O(ε2) ,  
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.
f07krf 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 4nr2-8/3r3, 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 f07mrf. See Lucas (2004) for further information.
The real analogue of this routine is f07kdf.

10 Example

This example computes the Cholesky factorization of the matrix A, where
A= ( 12.40+0.00i 2.39+0.00i 5.50+0.05i 4.47+0.00i 11.89+0.00i 2.39+0.00i 1.63+0.00i 1.04+0.10i 1.14+0.00i 1.81+0.00i 5.50+0.05i 1.04+0.10i 2.45+0.00i 1.98-0.03i 5.28-0.02i 4.47+0.00i 1.14+0.00i 1.98-0.03i 1.71+0.00i 4.14+0.00i 11.89+0.00i 1.81+0.00i 5.28-0.02i 4.14+0.00i 11.63+0.00i ) .  

10.1 Program Text

Program Text (f07krfe.f90)

10.2 Program Data

Program Data (f07krfe.d)

10.3 Program Results

Program Results (f07krfe.r)