# NAG FL Interfacef07wrf (zpftrf)

## 1Purpose

f07wrf computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in Rectangular Full Packed (RFP) format.

## 2Specification

Fortran Interface
 Subroutine f07wrf ( uplo, n, ar, info)
 Integer, Intent (In) :: n Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (Inout) :: ar(n*(n+1)/2) Character (1), Intent (In) :: transr, uplo
#include <nag.h>
 void f07wrf_ (const char *transr, const char *uplo, const Integer *n, Complex ar[], Integer *info, const Charlen length_transr, const Charlen length_uplo)
The routine may be called by the names f07wrf, nagf_lapacklin_zpftrf or its LAPACK name zpftrf.

## 3Description

f07wrf forms the Cholesky factorization of a complex Hermitian positive definite matrix $A$ either as $A={U}^{\mathrm{H}}U$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=L{L}^{\mathrm{H}}$ 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 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

## 5Arguments

1: $\mathbf{transr}$Character(1) Input
On entry: specifies whether the normal RFP representation of $A$ or its conjugate transpose is stored.
${\mathbf{transr}}=\text{'N'}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\text{'C'}$
The conjugate transpose of the RFP representation of the matrix $A$ is stored.
Constraint: ${\mathbf{transr}}=\text{'N'}$ or $\text{'C'}$.
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 ${U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored, and $A$ is factorized as $L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$Complex (Kind=nag_wp) array Input/Output
On entry: the upper or lower triangular part (as specified by uplo) of the $n$ by $n$ Hermitian 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 $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$, in the same storage format as $A$.
5: $\mathbf{info}$Integer Output
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 Hermitian matrix stored in RFP format which is not positive definite; the matrix must be treated as a full Hermitian matrix, by calling f07mrf.

## 7Accuracy

If ${\mathbf{uplo}}=\text{'U'}$, the computed factor $U$ is the exact factor of a perturbed matrix $A+E$, where
 $E≤cnεUHU ,$
$c\left(n\right)$ 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

f07wrf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07wrf 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 real floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
A call to f07wrf may be followed by calls to the routines:
• f07wsf to solve $AX=B$;
• f07wwf to compute the inverse of $A$.
The real analogue of this routine is f07wdf.

## 10Example

This example computes the Cholesky factorization of the matrix $A$, where
 $A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .$
and is stored using RFP format.

### 10.1Program Text

Program Text (f07wrfe.f90)

### 10.2Program Data

Program Data (f07wrfe.d)

### 10.3Program Results

Program Results (f07wrfe.r)