NAG CL Interface
f07wdc (dpftrf)
1
Purpose
f07wdc computes the Cholesky factorization of a real symmetric positive definite matrix stored in Rectangular Full Packed (RFP) format.
2
Specification
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
$A={U}^{\mathrm{T}}U$ if
${\mathbf{uplo}}=\mathrm{Nag\_Upper}$ or
$A=L{L}^{\mathrm{T}}$ if
${\mathbf{uplo}}=\mathrm{Nag\_Lower}$, 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 floatingpoint 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:
$\mathbf{order}$ – Nag_OrderType
Input

On entry: the
order argument specifies the twodimensional storage scheme being used, i.e., rowmajor ordering or columnmajor ordering. C language defined storage is specified by
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. 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:
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:
$\mathbf{transr}$ – Nag_RFP_Store
Input

On entry: specifies whether the RFP representation of
$A$ is normal or transposed.
 ${\mathbf{transr}}=\mathrm{Nag\_RFP\_Normal}$
 The matrix $A$ is stored in normal RFP format.
 ${\mathbf{transr}}=\mathrm{Nag\_RFP\_Trans}$
 The matrix $A$ is stored in transposed RFP format.
Constraint:
${\mathbf{transr}}=\mathrm{Nag\_RFP\_Normal}$ or $\mathrm{Nag\_RFP\_Trans}$.

3:
$\mathbf{uplo}$ – Nag_UploType
Input

On entry: specifies whether the upper or lower triangular part of
$A$ is stored.
 ${\mathbf{uplo}}=\mathrm{Nag\_Upper}$
 The upper triangular part of $A$ is stored, and $A$ is factorized as ${U}^{\mathrm{T}}U$, where $U$ is upper triangular.
 ${\mathbf{uplo}}=\mathrm{Nag\_Lower}$
 The lower triangular part of $A$ is stored, and $A$ is factorized as $L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint:
${\mathbf{uplo}}=\mathrm{Nag\_Upper}$ or $\mathrm{Nag\_Lower}$.

4:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

5:
$\mathbf{ar}\left[{\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2\right]$ – double
Input/Output

On entry: the upper or lower triangular part (as specified by
uplo) of the
$n$ by
$n$ 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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$, in the same storage format as $A$.

6:
$\mathbf{fail}$ – 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
 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
$\u2329\mathit{\text{value}}\u232a$ 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
${\mathbf{uplo}}=\mathrm{Nag\_Upper}$, the computed factor
$U$ is the exact factor of a perturbed matrix
$A+E$, where
$c\left(n\right)$ is a modest linear function of
$n$, and
$\epsilon $ is the
machine precision.
If ${\mathbf{uplo}}=\mathrm{Nag\_Lower}$, a similar statement holds for the computed factor $L$. It follows that $\left{e}_{ij}\right\le c\left(n\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.
8
Parallelism and Performance
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 implementationspecific information.
The total number of floatingpoint operations is approximately $\frac{1}{3}{n}^{3}$.
A call to
f07wdc may be followed by calls to the functions:
 f07wec to solve $AX=B$;
 f07wjc to compute the inverse of $A$.
The complex analogue of this function is
f07wrc.
10
Example
This example computes the Cholesky factorization of the matrix
$A$, where
and is stored using RFP format.
10.1
Program Text
10.2
Program Data
10.3
Program Results