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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpptrf (f07gr)

## Purpose

nag_lapack_zpptrf (f07gr) computes the Cholesky factorization of a complex Hermitian positive definite matrix, using packed storage.

## Syntax

[ap, info] = f07gr(uplo, n, ap)
[ap, info] = nag_lapack_zpptrf(uplo, n, ap)

## Description

nag_lapack_zpptrf (f07gr) 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 lower triangular, using packed storage.

## References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether the upper or lower triangular part of A$A$ is stored and how A$A$ is to be factorized.
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'$\text{'L'}$.
2:     n – int64int32nag_int scalar
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The n$n$ by n$n$ Hermitian matrix A$A$, packed by columns.
More precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + j(j1) / 2)${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for ij$i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of A$A$ must be stored with element Aij${A}_{ij}$ in ap(i + (2nj)(j1) / 2)${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for ij$i\ge j$.

None.

None.

### Output Parameters

1:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
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 = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: info.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the leading minor of order i$i$ 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$A$. To factorize a Hermitian matrix which is not positive definite, call nag_lapack_zhptrf (f07pr) instead.

## 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)n3$\frac{4}{3}{n}^{3}$.
A call to nag_lapack_zpptrf (f07gr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dpptrf (f07gd).

## Example

```function nag_lapack_zpptrf_example
uplo = 'L';
n = int64(4);
ap = [3.23;
1.51 + 1.92i;
1.9 - 0.84i;
0.42 - 2.5i;
3.58 + 0i;
-0.23 - 1.11i;
-1.18 - 1.37i;
4.09 + 0i;
2.33 + 0.14i;
4.29 + 0i];
[apOut, info] = nag_lapack_zpptrf(uplo, n, ap)
```
```

apOut =

1.7972 + 0.0000i
0.8402 + 1.0683i
1.0572 - 0.4674i
0.2337 - 1.3910i
1.3164 + 0.0000i
-0.4702 + 0.3131i
0.0834 + 0.0368i
1.5604 + 0.0000i
0.9360 + 0.9900i
0.6603 + 0.0000i

info =

0

```
```function f07gr_example
uplo = 'L';
n = int64(4);
ap = [3.23;
1.51 + 1.92i;
1.9 - 0.84i;
0.42 - 2.5i;
3.58 + 0i;
-0.23 - 1.11i;
-1.18 - 1.37i;
4.09 + 0i;
2.33 + 0.14i;
4.29 + 0i];
[apOut, info] = f07gr(uplo, n, ap)
```
```

apOut =

1.7972 + 0.0000i
0.8402 + 1.0683i
1.0572 - 0.4674i
0.2337 - 1.3910i
1.3164 + 0.0000i
-0.4702 + 0.3131i
0.0834 + 0.0368i
1.5604 + 0.0000i
0.9360 + 0.9900i
0.6603 + 0.0000i

info =

0

```