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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpftrf (f07wr)

## Purpose

nag_lapack_zpftrf (f07wr) computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in Rectangular Full Packed (RFP) format.

## Syntax

[ar, info] = f07wr(transr, uplo, n, ar)
[ar, info] = nag_lapack_zpftrf(transr, uplo, n, ar)

## Description

nag_lapack_zpftrf (f07wr) 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 Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{transr}$ – string (length ≥ 1)
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:     $\mathrm{uplo}$ – string (length ≥ 1)
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:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathrm{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – complex array
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 Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction.

None.

### Output Parameters

1:     $\mathrm{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – complex array
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 $A$.
2:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error 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 $_$ 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 Hermitian matrix stored in RFP format which is not positive definite; the matrix must be treated as a full Hermitian matrix, by calling nag_lapack_zhetrf (f07mr).

## Accuracy

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 $\left|{e}_{ij}\right|\le c\left(n\right)\epsilon \sqrt{{a}_{ii}{a}_{jj}}$.

The total number of real floating-point operations is approximately $\frac{4}{3}{n}^{3}$.
A call to nag_lapack_zpftrf (f07wr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dpftrf (f07wd).

## Example

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.
```function f07wr_example

fprintf('f07wr example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
ar = [ 4.09 + 0.00i  2.33 + 0.14i;
3.23 + 0.00i  4.29 + 0.00i;
1.51 + 1.92i  3.58 + 0.00i;
1.90 - 0.84i -0.23 - 1.11i;
0.42 - 2.50i -1.18 - 1.37i];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

% Factorize a
[ar, info] = f07wr(transr, uplo, n, ar);

if info == 0
% Convert factor to full array form for display
[a, info] = f01vh(transr, uplo, n, ar);
fprintf('\n');
ncols  = int64(80);
indent = int64(0);
form   = 'f7.4';
title  = 'Factor L:';
diag   = 'n';
[ifail] = x04db( ...
uplo, diag, a, 'brackets', form, title, ...
'int', 'int', ncols, indent);
else
fprintf('\na is not positive definite.\n');
end

```
```f07wr example results

Factor L:
1                 2                 3                 4
1  ( 1.7972, 0.0000)
2  ( 0.8402, 1.0683) ( 1.3164, 0.0000)
3  ( 1.0572,-0.4674) (-0.4702, 0.3131) ( 1.5604,-0.0000)
4  ( 0.2337,-1.3910) ( 0.0834, 0.0368) ( 0.9360, 0.8105) ( 0.8713,-0.0000)
```