NAG Library Routine Document
f07wdf (dpftrf) computes the Cholesky factorization of a real symmetric positive definite matrix stored in Rectangular Full Packed (RFP) format.
|Integer, Intent (In)||:: ||
|Integer, Intent (Out)||:: ||
|Real (Kind=nag_wp), Intent (Inout)||:: ||
|Character (1), Intent (In)||:: ||
The routine may be called by its
forms the Cholesky factorization of a real symmetric 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 Chapter Introduction.
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
- 1: – Character(1)Input
: specifies whether the RFP representation of
is normal or transposed.
- The matrix is stored in normal RFP format.
- The matrix is stored in transposed RFP format.
- 2: – Character(1)Input
: 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.
- 3: – IntegerInput
On entry: , the order of the matrix .
- 4: – Real (Kind=nag_wp) arrayInput/Output
: 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 Chapter Introduction.
On exit: if , the factor or from the Cholesky factorization or , in the same storage format as .
- 5: – IntegerOutput
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 symmetric matrix stored in
RFP format which is not positive definite; the matrix must be treated as a
full symmetric matrix, by calling f07mdf (dsytrf)
, 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
f07wdf (dpftrf) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07wdf (dpftrf) 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 .
A call to f07wdf (dpftrf)
may be followed by calls to the routines:
The complex analogue of this routine is f07wrf (zpftrf)
This example computes the Cholesky factorization of the matrix
and is stored using RFP format.
Program Text (f07wdfe.f90)
Program Data (f07wdfe.d)
Program Results (f07wdfe.r)