NAG Library Routine Document
F07KDF (DPSTRF)
1 Purpose
F07KDF (DPSTRF) computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
2 Specification
INTEGER |
N, LDA, PIV(N), RANK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), TOL, WORK(2*N) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name dpstrf.
3 Description
F07KDF (DPSTRF) forms the Cholesky factorization of a real symmetric positive semidefinite matrix A either as PTAP=UTU if UPLO='U' or PTAP=LLT 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 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
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 UTU, where U is upper triangular.
- UPLO='L'
- The lower triangular part of A is stored and A is factorized as LLT, where L is lower triangular.
Constraint:
UPLO='U' or 'L'.
- 2: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 3: A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the
n by
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.
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 – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F07KDF (DPSTRF) is called.
Constraint:
LDA≥max1,N.
- 5: PIV(N) – INTEGER arrayOutput
On exit:
PIV is such that the nonzero entries of
P are
PPIVk,k=1, for
k=1,2,…,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×maxk=1,nAkk×machine precision will be used. The algorithm terminates at the rth step if the r+1th 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
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO=1
The matrix
A is either rank deficient with computed rank as returned in
RANK, or is indefinite, see
Section 8.
7 Accuracy
If
UPLO='L' and
RANK=r, the computed Cholesky factor
L and permutation matrix
P satisfy the following upper bound
where
cr is a modest linear function of
r,
ε is machine epsilon, and
So there is no guarantee of stability of the algorithm for large n and r, although W2 is generally small in practice.
8 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
F07MDF (DSYTRF). See
Lucas (2004) for further information.
The complex analogue of this routine is
F07KRF (ZPSTRF).
9 Example
This example computes the Cholesky factorization of the matrix
A, where
9.1 Program Text
Program Text (f07kdfe.f90)
9.2 Program Data
Program Data (f07kdfe.d)
9.3 Program Results
Program Results (f07kdfe.r)