NAG Library Function Document
nag_zpftrf (f07wrc)
1
Purpose
nag_zpftrf (f07wrc) computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in Rectangular Full Packed (RFP) format.
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_zpftrf (Nag_OrderType order,
Nag_RFP_Store transr,
Nag_UploType uplo,
Integer n,
Complex ar[],
NagError *fail) |
|
3
Description
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 3.3.3 in the f07 Chapter Introduction.
4
References
Demmel J W (1989) On floating-point errors in Cholesky
LAPACK Working Note No. 14 University of Tennessee, Knoxville
http://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
5
Arguments
- 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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 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.
Constraint:
or .
- 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.
Constraint:
or .
- 4:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 5:
– ComplexInput/Output
-
On entry: the upper or lower triangular part (as specified by
uplo) of the
by
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 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 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
- NE_INTERNAL_ERROR
-
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_MAT_NOT_POS_DEF
-
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
nag_zhetrf (f07mrc).
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
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 .
8
Parallelism and Performance
nag_zpftrf (f07wrc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zpftrf (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
nag_zpftrf (f07wrc) may be followed by calls to the functions:
The real analogue of this function is
nag_dpftrf (f07wdc).
10
Example
This example computes the Cholesky factorization of the matrix
, where
and is stored using RFP format.
10.1
Program Text
Program Text (f07wrce.c)
10.2
Program Data
Program Data (f07wrce.d)
10.3
Program Results
Program Results (f07wrce.r)