The routine may be called by the names f07kdf, nagf_lapacklin_dpstrf or its LAPACK name dpstrf.
f07kdf forms the Cholesky factorization of a real symmetric 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.
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
1: – 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.
2: – IntegerInput
On entry: , the order of the matrix .
3: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the symmetric 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: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07kdf is called.
5: – Integer arrayOutput
On exit: piv is such that the nonzero entries of
are , for .
6: – IntegerOutput
On exit: the computed rank of given by the number of steps the algorithm completed.
7: – Real (Kind=nag_wp)Input
On entry: user defined tolerance. If , will be used. The algorithm terminates at the th step if the th step pivot .
8: – Real (Kind=nag_wp) arrayWorkspace
9: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The matrix is not positive definite. It is either positive semidefinite with computed rank as returned in rank and less than , or it may be indefinite, see Section 9.
If and , the computed Cholesky factor and permutation matrix satisfy the following upper bound
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.
8Parallelism and Performance
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.
The total number of 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 f07mdf. See Lucas (2004) for further information.