F07GDF (DPPTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F07GDF (DPPTRF)

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

F07GDF (DPPTRF) computes the Cholesky factorization of a real symmetric positive definite matrix, using packed storage.

2  Specification

SUBROUTINE F07GDF ( UPLO, N, AP, INFO)
INTEGER  N, INFO
REAL (KIND=nag_wp)  AP(*)
CHARACTER(1)  UPLO
The routine may be called by its LAPACK name dpptrf.

3  Description

F07GDF (DPPTRF) forms the Cholesky factorization of a real symmetric positive definite matrix A either as A=UTU if UPLO='U' or A=LLT if UPLO='L', where U is an upper triangular matrix and L is lower triangular, using packed storage.

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     UPLO – CHARACTER(1)Input
On entry: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
UPLO='U'
The upper triangular part of A is stored and A is factorized as UTU, where U is upper triangular.
UPLO='L'
The lower triangular part of A is stored and A is factorized as LLT, where L is lower triangular.
Constraint: UPLO='U' or 'L'.
2:     N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint: N0.
3:     AP* – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array AP must be at least max1,N×N+1/2.
On entry: the n by n symmetric matrix A, packed by columns.
More precisely,
  • if UPLO='U', the upper triangle of A must be stored with element Aij in APi+jj-1/2 for ij;
  • if UPLO='L', the lower triangle of A must be stored with element Aij in APi+2n-jj-1/2 for ij.
On exit: if INFO=0, the factor U or L from the Cholesky factorization A=UTU or A=LLT, in the same storage format as A.
4:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
INFO>0
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. To factorize a symmetric matrix which is not positive definite, call F07PDF (DSPTRF) instead.

7  Accuracy

If UPLO='U', the computed factor U is the exact factor of a perturbed matrix A+E, where
EcnεUTU ,  
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  Parallelism and Performance

F07GDF (DPPTRF) 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.

9  Further Comments

The total number of floating-point operations is approximately 13n3.
A call to F07GDF (DPPTRF) may be followed by calls to the routines:
The complex analogue of this routine is F07GRF (ZPPTRF).

10  Example

This example computes the Cholesky factorization of the matrix A, where
A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 ,  
using packed storage.

10.1  Program Text

Program Text (f07gdfe.f90)

10.2  Program Data

Program Data (f07gdfe.d)

10.3  Program Results

Program Results (f07gdfe.r)


F07GDF (DPPTRF) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016