# NAG CL Interfacef07grc (zpptrf)

## 1Purpose

f07grc computes the Cholesky factorization of a complex Hermitian positive definite matrix, using packed storage.

## 2Specification

 #include
 void f07grc (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex ap[], NagError *fail)
The function may be called by the names: f07grc, nag_lapacklin_zpptrf or nag_zpptrf.

## 3Description

f07grc forms the Cholesky factorization of a complex Hermitian positive definite matrix $A$ either as $A={U}^{\mathrm{H}}U$ if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $A=L{L}^{\mathrm{H}}$ if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, where $U$ is an upper triangular matrix and $L$ is lower triangular, using packed storage.

## 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. 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: $\mathbf{uplo}$Nag_UploType Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored and $A$ is factorized as ${U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{ap}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the $n$ by $n$ Hermitian matrix $A$, packed by rows or columns.
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, 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: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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_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.
NE_POS_DEF
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 $A$. To factorize a Hermitian matrix which is not positive definite, call f07prc instead.

## 7Accuracy

If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, 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 ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, a similar statement holds for the computed factor $L$. It follows that $\left|{e}_{ij}\right|\le c\left(n\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

## 8Parallelism and Performance

f07grc 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.

The total number of real floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
A call to f07grc may be followed by calls to the functions:
• f07gsc to solve $AX=B$;
• f07guc to estimate the condition number of $A$;
• f07gwc to compute the inverse of $A$.
The real analogue of this function is f07gdc.

## 10Example

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

### 10.1Program Text

Program Text (f07grce.c)

### 10.2Program Data

Program Data (f07grce.d)

### 10.3Program Results

Program Results (f07grce.r)