NAG CL Interface
f07wdc (dpftrf)

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1 Purpose

f07wdc computes the Cholesky factorization of a real symmetric positive definite matrix stored in Rectangular Full Packed (RFP) format.

2 Specification

#include <nag.h>
void  f07wdc (Nag_OrderType order, Nag_RFP_Store transr, Nag_UploType uplo, Integer n, double ar[], NagError *fail)
The function may be called by the names: f07wdc, nag_lapacklin_dpftrf or nag_dpftrf.

3 Description

f07wdc 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 a lower triangular, stored in RFP format. The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction.

4 References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville https://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

5 Arguments

1: order Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: transr Nag_RFP_Store Input
On entry: specifies whether the RFP representation of A is normal or transposed.
transr=Nag_RFP_Normal
The matrix A is stored in normal RFP format.
transr=Nag_RFP_Trans
The matrix A is stored in transposed RFP format.
Constraint: transr=Nag_RFP_Normal or Nag_RFP_Trans.
3: uplo Nag_UploType Input
On entry: specifies whether the upper or lower triangular part of A is stored.
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.
4: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
5: ar[n×(n+1)/2] double Input/Output
On entry: the upper or lower triangular part (as specified by uplo) of the n×n symmetric matrix A, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Section 3.4.3 in the F07 Chapter Introduction.
On exit: if fail.code= NE_NOERROR, the factor U or L from the Cholesky factorization A=UTU or A=LLT, in the same storage format as A.
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MAT_NOT_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. There is no function 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 f07mdc.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

If uplo=Nag_Upper, the computed factor U is the exact factor of a perturbed matrix A+E, where
|E|c(n)ε|UT||U| ,  
c(n) 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 |eij|c(n)εaiiajj.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f07wdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07wdc 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 f07wdc may be followed by calls to the functions:
The complex analogue of this function is f07wrc.

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 ) ,  
and is stored using RFP format.

10.1 Program Text

Program Text (f07wdce.c)

10.2 Program Data

Program Data (f07wdce.d)

10.3 Program Results

Program Results (f07wdce.r)