NAG FL Interface
f07wrf computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in Rectangular Full Packed (RFP) format.
|Integer, Intent (In)
|Integer, Intent (Out)
|Complex (Kind=nag_wp), Intent (Inout)
|Character (1), Intent (In)
The routine may be called by the names f07wrf, nagf_lapacklin_zpftrf or its LAPACK name zpftrf.
forms the Cholesky factorization of a complex Hermitian positive definite matrix
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
Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14
University of Tennessee, Knoxville https://www.netlib.org/lapack/lawnspdf/lawn14.pdf
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. Software 37, 2
: 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.
: specifies whether the upper or lower triangular part of
- 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.
On entry: , the order of the matrix .
– Complex (Kind=nag_wp) array
: the upper or lower triangular part (as specified by uplo
) of the
, 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
On exit: if , the factor or from the Cholesky factorization or , in the same storage format as .
unless the routine detects an error (see Section 6
Error 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
, the computed factor
is the exact factor of a perturbed matrix
is a modest linear function of
is the machine precision
If , a similar statement holds for the computed factor . It follows that .
Parallelism 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:
The real analogue of this routine is f07wdf
This example computes the Cholesky factorization of the matrix
and is stored using RFP format.