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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhptrf (f07pr)

## Purpose

nag_lapack_zhptrf (f07pr) computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix, using packed storage.

## Syntax

[ap, ipiv, info] = f07pr(uplo, n, ap)
[ap, ipiv, info] = nag_lapack_zhptrf(uplo, n, ap)

## Description

nag_lapack_zhptrf (f07pr) factorizes a complex Hermitian matrix A$A$, using the Bunch–Kaufman diagonal pivoting method and packed storage. A$A$ is factorized as either A = PUDUHPT$A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or A = PLDLHPT$A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$ if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, where P$P$ is a permutation matrix, U$U$ (or L$L$) is a unit upper (or lower) triangular matrix and D$D$ is an Hermitian block diagonal matrix with 1$1$ by 1$1$ and 2$2$ by 2$2$ diagonal blocks; U$U$ (or L$L$) has 2$2$ by 2$2$ unit diagonal blocks corresponding to the 2$2$ by 2$2$ blocks of D$D$. Row and column interchanges are performed to ensure numerical stability while keeping the matrix Hermitian.
This method is suitable for Hermitian matrices which are not known to be positive definite. If A$A$ is in fact positive definite, no interchanges are performed and no 2$2$ by 2$2$ blocks occur in D$D$.

## References

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 PUDUHPT$PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$, 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 PLDLHPT$PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
A$A$ stores details of the block diagonal matrix D$D$ and the multipliers used to obtain the factor U$U$ or L$L$ as specified by uplo.
2:     ipiv(n) – int64int32nag_int array
Details of the interchanges and the block structure of D$D$. More precisely,
• if ipiv(i) = k > 0${\mathbf{ipiv}}\left(i\right)=k>0$, dii${d}_{ii}$ is a 1$1$ by 1$1$ pivot block and the i$i$th row and column of A$A$ were interchanged with the k$k$th row and column;
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ and ipiv(i1) = ipiv(i) = l < 0${\mathbf{ipiv}}\left(i-1\right)={\mathbf{ipiv}}\left(i\right)=-l<0$,  ( di − 1,i − 1 di,i − 1 ) di,i − 1 dii
$\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ is a 2$2$ by 2$2$ pivot block and the (i1)$\left(i-1\right)$th row and column of A$A$ were interchanged with the l$l$th row and column;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$ and ipiv(i) = ipiv(i + 1) = m < 0${\mathbf{ipiv}}\left(i\right)={\mathbf{ipiv}}\left(i+1\right)=-m<0$,  ( dii di + 1,i ) di + 1,i di + 1,i + 1
$\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ is a 2$2$ by 2$2$ pivot block and the (i + 1)$\left(i+1\right)$th row and column of A$A$ were interchanged with the m$m$th row and column.
3:     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

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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: ipiv, 5: info.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, d(i,i)$d\left(i,i\right)$ is exactly zero. The factorization has been completed, but the block diagonal matrix D$D$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

## Accuracy

If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the computed factors U$U$ and D$D$ are the exact factors of a perturbed matrix A + E$A+E$, where
 |E| ≤ c(n)εP|U||D||UH|PT , $|E|≤c(n)εP|U||D||UH|PT ,$
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 factors L$L$ and D$D$.

The elements of D$D$ overwrite the corresponding elements of A$A$; if D$D$ has 2$2$ by 2$2$ blocks, only the upper or lower triangle is stored, as specified by uplo.
The unit diagonal elements of U$U$ or L$L$ and the 2$2$ by 2$2$ unit diagonal blocks are not stored. The remaining elements of U$U$ and L$L$ are stored in the corresponding columns of the array ap, but additional row interchanges must be applied to recover U$U$ or L$L$ explicitly (this is seldom necessary). If ipiv(i) = i${\mathbf{ipiv}}\left(\mathit{i}\right)=\mathit{i}$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$ (as is the case when A$A$ is positive definite), then U$U$ or L$L$ are stored explicitly in packed form (except for their unit diagonal elements which are equal to 1$1$).
The total number of real floating point operations is approximately (4/3)n3$\frac{4}{3}{n}^{3}$.
A call to nag_lapack_zhptrf (f07pr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dsptrf (f07pd).

## Example

```function nag_lapack_zhptrf_example
uplo = 'L';
n = int64(4);
ap = [-1.36;
1.58 - 0.9i;
2.21 + 0.21i;
3.91 - 1.5i;
-8.87 + 0i;
-1.84 + 0.03i;
-1.78 - 1.18i;
-4.63 + 0i;
0.11 - 0.11i;
-1.84 + 0i];
[apOut, ipiv, info] = nag_lapack_zhptrf(uplo, n, ap)
```
```

apOut =

-1.3600 + 0.0000i
3.9100 - 1.5000i
0.3100 + 0.0433i
-0.1518 + 0.3743i
-1.8400 + 0.0000i
0.5637 + 0.2850i
0.3397 + 0.0303i
-5.4176 + 0.0000i
0.2997 + 0.1578i
-7.1028 + 0.0000i

ipiv =

-4
-4
3
4

info =

0

```
```function f07pr_example
uplo = 'L';
n = int64(4);
ap = [-1.36;
1.58 - 0.9i;
2.21 + 0.21i;
3.91 - 1.5i;
-8.87 + 0i;
-1.84 + 0.03i;
-1.78 - 1.18i;
-4.63 + 0i;
0.11 - 0.11i;
-1.84 + 0i];
[apOut, ipiv, info] = f07pr(uplo, n, ap)
```
```

apOut =

-1.3600 + 0.0000i
3.9100 - 1.5000i
0.3100 + 0.0433i
-0.1518 + 0.3743i
-1.8400 + 0.0000i
0.5637 + 0.2850i
0.3397 + 0.0303i
-5.4176 + 0.0000i
0.2997 + 0.1578i
-7.1028 + 0.0000i

ipiv =

-4
-4
3
4

info =

0

```