The function may be called by the names: f07wrc, nag_lapacklin_zpftrf or nag_zpftrf.
f07wrc 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 3.4.3 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: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
2: – Nag_RFP_StoreInput
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.
3: – Nag_UploTypeInput
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.
4: – IntegerInput
On entry: , the order of the matrix .
5: – ComplexInput/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 3.4.3 in the F07 Chapter Introduction.
On exit: if NE_NOERROR, the factor or from the Cholesky factorization or , in the same storage format as .
6: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
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 function 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 f07mrc.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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
Background information to multithreading can be found in the Multithreading documentation.
f07wrc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07wrc 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 function. 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 f07wrc may be followed by calls to the functions: