NAG Library Function Document

nag_dpotrf (f07fdc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_dpotrf (f07fdc) computes the Cholesky factorization of a real symmetric positive definite matrix.

2
Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dpotrf (Nag_OrderType order, Nag_UploType uplo, Integer n, double a[], Integer pda, NagError *fail)

3
Description

nag_dpotrf (f07fdc) forms the Cholesky factorization of a real symmetric positive definite matrix A either as A=UTU if uplo=Nag_Upper or A=LLT if uplo=Nag_Lower, where U is an upper triangular matrix and L is lower triangular.

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:     order 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 order=Nag_RowMajor. 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: order=Nag_RowMajor or Nag_ColMajor.
2:     uplo Nag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo=Nag_Upper
The upper triangular part of A is stored and A is factorized as UTU, where U is upper triangular.
uplo=Nag_Lower
The lower triangular part of A is stored and A is factorized as LLT, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     a[dim] doubleInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the n by n symmetric positive definite matrix A.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: the upper or lower triangle of A is overwritten by the Cholesky factor U or L as specified by uplo.
5:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax1,n.
6:     fail 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 value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
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_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.
NE_POS_DEF
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 nag_dsytrf (f07mdc) instead.

7
Accuracy

If uplo=Nag_Upper, 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=Nag_Lower, a similar statement holds for the computed factor L. It follows that eijcnεaiiajj.

8
Parallelism and Performance

nag_dpotrf (f07fdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dpotrf (f07fdc) 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.

9
Further Comments

The total number of floating-point operations is approximately 13n3.
A call to nag_dpotrf (f07fdc) may be followed by calls to the functions:
The complex analogue of this function is nag_zpotrf (f07frc).

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 .  

10.1
Program Text

Program Text (f07fdce.c)

10.2
Program Data

Program Data (f07fdce.d)

10.3
Program Results

Program Results (f07fdce.r)

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