The routine may be called by the names f07wrf, nagf_lapacklin_zpftrf or its LAPACK name zpftrf.
f07wrf forms the Cholesky factorization of a complex Hermitian positive definite matrix either as if or if , where is an upper triangular matrix and is a lower triangular, stored in RFP format.
The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software37, 2
1: – Character(1)Input
On entry: specifies whether the normal RFP representation of or its conjugate transpose is stored.
The matrix is stored in normal RFP format.
The conjugate transpose of the RFP representation of the matrix is stored.
2: – Character(1)Input
On entry: specifies whether the upper or lower triangular part of is stored.
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.
3: – IntegerInput
On entry: , the order of the matrix .
4: – Complex (Kind=nag_wp) arrayInput/Output
On entry: the upper or lower triangular part (as specified by uplo) of the Hermitian matrix , in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Section 3.3.3 in the F07 Chapter Introduction.
On exit: if , the factor or from the Cholesky factorization or , in the same storage format as .
5: – 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 leading minor of order is not positive definite and the factorization could not be completed. Hence itself is not positive definite. This may indicate an error in forming the matrix .
There is no routine specifically designed to factorize a Hermitian matrix stored in RFP format which is not positive definite; the matrix must be treated as a full Hermitian matrix, by calling f07mrf.
If , the computed factor is the exact factor of a perturbed matrix , where
is a modest linear function of , and is the machine precision.
If , a similar statement holds for the computed factor . It follows that .
8Parallelism and Performance
f07wrf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07wrf 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 real floating-point operations is approximately .
A call to f07wrf may be followed by calls to the routines: