F07WRF (ZPFTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07WRF (ZPFTRF)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F07WRF (ZPFTRF) computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section 3.3.3 in the F07 Chapter Introduction.

2  Specification

SUBROUTINE F07WRF ( TRANSR, UPLO, N, A, INFO)
INTEGER  N, INFO
COMPLEX (KIND=nag_wp)  A(N*(N+1)/2)
CHARACTER(1)  TRANSR, UPLO
The routine may be called by its LAPACK name zpftrf.

3  Description

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 an lower triangular, stored using RFP format.

4  References

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

5  Parameters

1:     TRANSR – CHARACTER(1)Input
On entry: specifies whether the normal RFP representation of A or its conjugate transpose is stored.
TRANSR='N'
The matrix A is stored in normal RFP format.
TRANSR='C'
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.
UPLO='U'
The upper triangular part of A is stored, and A is factorized as UHU, where U is upper triangular.
UPLO='L'
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:     A(N×N+1/2) – COMPLEX (KIND=nag_wp) arrayInput/Output
On entry: the n by n Hermitian matrix A, stored in RFP format, as described 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).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
INFO<0
If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0
If INFO=i, the leading minor of order i 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.

7  Accuracy

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.

8  Further Comments

The total number of real floating point operations is approximately 43n2.
A call to F07WRF (ZPFTRF) may be followed by calls to the routines:
The real analogue of this routine is F07WDF (DPFTRF).

9  Example

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.

9.1  Program Text

Program Text (f07wrfe.f90)

9.2  Program Data

Program Data (f07wrfe.d)

9.3  Program Results

Program Results (f07wrfe.r)


F07WRF (ZPFTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012