# NAG Library Routine Document

## 1Purpose

f07wdf (dpftrf) computes the Cholesky factorization of a real symmetric positive definite matrix stored in Rectangular Full Packed (RFP) format.

## 2Specification

Fortran Interface
 Subroutine f07wdf ( uplo, n, ar, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: ar(n*(n+1)/2) Character (1), Intent (In) :: transr, uplo
#include nagmk26.h
 void f07wdf_ ( const char *transr, const char *uplo, const Integer *n, double ar[], Integer *info, const Charlen length_transr, const Charlen length_uplo)
The routine may be called by its LAPACK name dpftrf.

## 3Description

f07wdf (dpftrf) forms the Cholesky factorization of a real symmetric positive definite matrix $A$ either as $A={U}^{\mathrm{T}}U$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=L{L}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'L'}$, 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 3.3.3 in the F07 Chapter Introduction.

## 4References

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
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

## 5Arguments

1:     $\mathbf{transr}$ – Character(1)Input
On entry: specifies whether the RFP representation of $A$ is normal or transposed.
${\mathbf{transr}}=\text{'N'}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\text{'T'}$
The matrix $A$ is stored in transposed RFP format.
Constraint: ${\mathbf{transr}}=\text{'N'}$ or $\text{'T'}$.
2:     $\mathbf{uplo}$ – Character(1)Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored, and $A$ is factorized as ${U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored, and $A$ is factorized as $L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – Real (Kind=nag_wp) arrayInput/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 3.3.3 in the F07 Chapter Introduction.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, 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 $A$.
5:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
The leading minor of order $〈\mathit{\text{value}}〉$ 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 routine 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 f07mdf (dsytrf).

## 7Accuracy

If ${\mathbf{uplo}}=\text{'U'}$, the computed factor $U$ is the exact factor of a perturbed matrix $A+E$, where
 $E≤cnεUTU ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.
If ${\mathbf{uplo}}=\text{'L'}$, a similar statement holds for the computed factor $L$. It follows that $\left|{e}_{ij}\right|\le c\left(n\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

## 8Parallelism and Performance

f07wdf (dpftrf) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07wdf (dpftrf) 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.

The total number of floating-point operations is approximately $\frac{1}{3}{n}^{3}$.
A call to f07wdf (dpftrf) may be followed by calls to the routines:
• f07wef (dpftrs) to solve $AX=B$;
• f07wjf (dpftri) to compute the inverse of $A$.
The complex analogue of this routine is f07wrf (zpftrf).

## 10Example

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.

### 10.1Program Text

Program Text (f07wdfe.f90)

### 10.2Program Data

Program Data (f07wdfe.d)

### 10.3Program Results

Program Results (f07wdfe.r)

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