NAG Library Routine Document
F07KRF (ZPSTRF)
1 Purpose
F07KRF (ZPSTRF) computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix.
2 Specification
INTEGER |
N, LDA, PIV(N), RANK, INFO |
REAL (KIND=nag_wp) |
TOL, WORK(2*N) |
COMPLEX (KIND=nag_wp) |
A(LDA,*) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name zpstrf.
3 Description
F07KRF (ZPSTRF) forms the Cholesky factorization of a complex Hermitian positive semidefinite matrix either as if or if , where is a permutation matrix, is an upper triangular matrix and is lower triangular.
This algorithm does not attempt to check that 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
is stored and how
is to be factorized.
- The upper triangular part of is stored and is factorized as , where is upper triangular.
- The lower triangular part of is stored and is factorized as , where is lower triangular.
Constraint:
or .
- 2: N – INTEGERInput
On entry: , the order of the matrix .
Constraint:
.
- 3: A(LDA,) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
.
On entry: the
by
Hermitian positive semidefinite matrix
.
- If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
- If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: if
, the first
RANK rows of the upper triangle of
are overwritten with the nonzero elements of the Cholesky factor
, and the remaining rows of the triangle are destroyed.
If
, the first
RANK columns of the lower triangle of
are overwritten with the nonzero elements of the Cholesky factor
, 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:
.
- 5: PIV(N) – INTEGER arrayOutput
On exit:
PIV is such that the nonzero entries of
are
, for
.
- 6: RANK – INTEGEROutput
On exit: the computed rank of given by the number of steps the algorithm completed.
- 7: TOL – REAL (KIND=nag_wp)Input
On entry: user defined tolerance. If , then will be used. The algorithm terminates at the th step if the th step pivot .
- 8: WORK() – REAL (KIND=nag_wp) arrayWorkspace
- 9: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , the th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
The matrix
is either rank deficient with computed rank as returned in
RANK, or is indefinite, see
Section 8.
7 Accuracy
If
and
, the computed Cholesky factor
and permutation matrix
satisfy the following upper bound
where
is a modest linear function of
,
is
machine precision, and
So there is no guarantee of stability of the algorithm for large and , although is generally small in practice.
The total number of real floating point operations is approximately , where is the computed rank of .
This algorithm does not attempt to check that
is positive semidefinite, and in particular the rank detection criterion in the algorithm is based on
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).
9 Example
This example computes the Cholesky factorization of the matrix
, where
9.1 Program Text
Program Text (f07krfe.f90)
9.2 Program Data
Program Data (f07krfe.d)
9.3 Program Results
Program Results (f07krfe.r)