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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. The RFP storage format is described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

## Syntax

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

## Description

nag_lapack_zpftrf (f07wr) forms the Cholesky factorization of a complex Hermitian positive definite matrix A$A$ either as A = UHU$A={U}^{\mathrm{H}}U$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or A = LLH$A=L{L}^{\mathrm{H}}$ if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, where U$U$ is an upper triangular matrix and L$L$ is an lower triangular, stored using RFP format.

## References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
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:     transr – string (length ≥ 1)
Specifies whether the normal RFP representation of A$A$ or its conjugate transpose is stored.
transr = 'N'${\mathbf{transr}}=\text{'N'}$
The matrix A$A$ is stored in normal RFP format.
transr = 'C'${\mathbf{transr}}=\text{'C'}$
The conjugate transpose of the RFP representation of the matrix A$A$ is stored.
Constraint: transr = 'N'${\mathbf{transr}}=\text{'N'}$ or 'C'$\text{'C'}$.
2:     uplo – string (length ≥ 1)
Specifies whether the upper or lower triangular part of A$A$ is stored.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of A$A$ is stored, and A$A$ is factorized as UHU${U}^{\mathrm{H}}U$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of A$A$ is stored, and A$A$ is factorized as LLH$L{L}^{\mathrm{H}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
3:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
4:     a(n × (n + 1) / 2${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – complex array
The n$n$ by n$n$ Hermitian matrix A$A$, stored in RFP format, as described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

None.

None.

### Output Parameters

1:     a(n × (n + 1) / 2${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – complex array
If ${\mathbf{INFO}}={\mathbf{0}}$, the factor U$U$ or L$L$ from the Cholesky factorization A = UHU$A={U}^{\mathrm{H}}U$ or A = LLH$A=L{L}^{\mathrm{H}}$, in the same storage format as A$A$.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

INFO > 0${\mathbf{INFO}}>0$
The leading minor of order _$_$ is not positive definite and the factorization could not be completed. Hence A$A$ itself is not positive definite. This may indicate an error in forming the matrix A$A$. There is no function specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling nag_lapack_zgbtrf (f07br) or as a full Hermitian matrix, by calling nag_lapack_zhetrf (f07mr).

## Accuracy

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

The total number of real floating point operations is approximately (4/3)n2$\frac{4}{3}{n}^{2}$.
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

```function nag_lapack_zpftrf_example
a = [ 4.09 + 0.00i;
3.23 + 0.00i;
1.51 + 1.92i;
1.90 - 0.84i;
0.42 - 2.50i;
2.33 + 0.14i;
4.29 + 0.00i;
3.58 + 0.00i;
-0.23 - 1.11i;
-1.18 - 1.37i];
transr = 'n';
uplo   = 'l';
n      = int64(4);

% Factorize a
[aOut, info] = nag_lapack_zpftrf(transr, uplo, n, a);

if info == 0
% Convert factor to full array form, and print it
[f, info] = nag_matop_ztfttr(transr, uplo, n, aOut);
fprintf('\n');
[ifail] = ...
nag_file_print_matrix_complex_gen_comp(uplo, 'n', f, 'b', 'f7.4', 'Factor', 'i', 'i', int64(80), int64(0));
else
fprintf('\na is not positive definite.\n');
end
```
```

Factor
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)

```
```function f07wr_example
a = [ 4.09 + 0.00i;
3.23 + 0.00i;
1.51 + 1.92i;
1.90 - 0.84i;
0.42 - 2.50i;
2.33 + 0.14i;
4.29 + 0.00i;
3.58 + 0.00i;
-0.23 - 1.11i;
-1.18 - 1.37i];
transr = 'n';
uplo   = 'l';
n      = int64(4);

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

if info == 0
% Convert factor to full array form, and print it
[f, info] = f01vh(transr, uplo, n, aOut);
fprintf('\n');
[ifail] = x04db(uplo, 'n', f, 'b', 'f7.4', 'Factor', 'i', 'i', int64(80), int64(0));
else
fprintf('\na is not positive definite.\n');
end
```
```

Factor
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)

```