NAG Library Routine Document

f07wrf  (zpftrf)


    1  Purpose
    7  Accuracy


f07wrf (zpftrf) computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in Rectangular Full Packed (RFP) format.


Fortran Interface
Subroutine f07wrf ( transr, uplo, n, ar, info)
Integer, Intent (In):: n
Integer, Intent (Out):: info
Complex (Kind=nag_wp), Intent (Inout):: ar(n*(n+1)/2)
Character (1), Intent (In):: transr, uplo
C Header Interface
#include nagmk26.h
void  f07wrf_ ( const char *transr, const char *uplo, const Integer *n, Complex ar[], Integer *info, const Charlen length_transr, const Charlen length_uplo)
The routine may be called by its LAPACK name zpftrf.


f07wrf (zpftrf) forms the Cholesky factorization of a complex Hermitian positive definite matrix A either as A=UHU if uplo='U' or A=LLH if uplo='L', where U is an upper triangular matrix and L 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
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:     transr – Character(1)Input
On entry: specifies whether the normal RFP representation of A or its conjugate transpose is stored.
The matrix A is stored in normal RFP format.
The conjugate transpose of the RFP representation of the matrix A is stored.
Constraint: transr='N' or 'C'.
2:     uplo – Character(1)Input
On entry: specifies whether the upper or lower triangular part of A is stored.
The upper triangular part of A is stored, and A is factorized as UHU, where U is upper triangular.
The lower triangular part of A is stored, and A is factorized as LLH, where L is lower triangular.
Constraint: uplo='U' or 'L'.
3:     n – IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     arn×n+1/2 – Complex (Kind=nag_wp) arrayInput/Output
On entry: the upper or lower triangular part (as specified by uplo) of the n by n Hermitian matrix A, 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 info=0, the factor U or L from the Cholesky factorization A=UHU or A=LLH, in the same storage format as A.
5:     info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
The leading minor of order value is not positive definite and the factorization could not be completed. Hence A itself is not positive definite. This may indicate an error in forming the matrix A. 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 (zhetrf).


If uplo='U', the computed factor U is the exact factor of a perturbed matrix A+E, where
EcnεUHU ,  
cn is a modest linear function of n, and ε is the machine precision.
If uplo='L', a similar statement holds for the computed factor L. It follows that eijcnεaiiajj.

Parallelism and Performance

f07wrf (zpftrf) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07wrf (zpftrf) 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.

Further Comments

The total number of real floating-point operations is approximately 43n3.
A call to f07wrf (zpftrf) may be followed by calls to the routines:
The real analogue of this routine is f07wdf (dpftrf).


This example computes the Cholesky factorization of the matrix A, where
A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .  
and is stored using RFP format.

Program Text

Program Text (f07wrfe.f90)

Program Data

Program Data (f07wrfe.d)

Program Results

Program Results (f07wrfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017