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# NAG Toolbox: nag_lapack_zsptrf (f07qr)

## Purpose

nag_lapack_zsptrf (f07qr) computes the Bunch–Kaufman factorization of a complex symmetric matrix, using packed storage.

## Syntax

[ap, ipiv, info] = f07qr(uplo, n, ap)
[ap, ipiv, info] = nag_lapack_zsptrf(uplo, n, ap)

## Description

nag_lapack_zsptrf (f07qr) factorizes a complex symmetric matrix A$A$, using the Bunch–Kaufman diagonal pivoting method and packed storage. A$A$ is factorized as either A = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$ 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 a symmetric 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 preserving symmetry.

## 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 PUDUTPT$PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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 PLDLTPT$PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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$n$ symmetric 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||UT|PT , $|E|≤c(n)εP|U||D||UT|PT ,$
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$U$ or L$L$ overwrite elements in the corresponding columns of A$A$, 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$\mathit{i}=1,2,\dots ,n$, 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_zsptrf (f07qr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dsptrf (f07pd).

## Example

```function nag_lapack_zsptrf_example
uplo = 'L';
n = int64(4);
ap = [ -0.39 - 0.71i;
5.14 - 0.64i;
-7.86 - 2.96i;
3.8 + 0.92i;
8.86 + 1.81i;
-3.52 + 0.58i;
5.32 - 1.59i;
-2.83 - 0.03i;
-1.54 - 2.86i;
-0.56 + 0.12i];
[apOut, ipiv, info] = nag_lapack_zsptrf(uplo, n, ap)
```
```

apOut =

-0.3900 - 0.7100i
-7.8600 - 2.9600i
0.5279 - 0.3715i
0.4426 + 0.1936i
-2.8300 - 0.0300i
-0.6078 + 0.2811i
-0.4823 + 0.0150i
4.4079 + 5.3991i
-0.1071 - 0.3157i
-2.0954 - 2.2011i

ipiv =

-3
-3
3
4

info =

0

```
```function f07qr_example
uplo = 'L';
n = int64(4);
ap = [ -0.39 - 0.71i;
5.14 - 0.64i;
-7.86 - 2.96i;
3.8 + 0.92i;
8.86 + 1.81i;
-3.52 + 0.58i;
5.32 - 1.59i;
-2.83 - 0.03i;
-1.54 - 2.86i;
-0.56 + 0.12i];
[apOut, ipiv, info] = f07qr(uplo, n, ap)
```
```

apOut =

-0.3900 - 0.7100i
-7.8600 - 2.9600i
0.5279 - 0.3715i
0.4426 + 0.1936i
-2.8300 - 0.0300i
-0.6078 + 0.2811i
-0.4823 + 0.0150i
4.4079 + 5.3991i
-0.1071 - 0.3157i
-2.0954 - 2.2011i

ipiv =

-3
-3
3
4

info =

0

```

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