# NAG FL Interfacef06zrf (zher2k)

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## 1Purpose

f06zrf performs one of the Hermitian rank-$2k$ update operations
 $C←αABH + α¯BAH + βC or C←αAHB + α¯BHA+βC ,$
where $A$ and $B$ are complex matrices, $C$ is an $n×n$ complex Hermitian matrix, $\alpha$ is a complex scalar, and $\beta$ is a real scalar.

## 2Specification

Fortran Interface
 Subroutine f06zrf ( uplo, n, k, a, lda, b, ldb, beta, c, ldc)
 Integer, Intent (In) :: n, k, lda, ldb, ldc Real (Kind=nag_wp), Intent (In) :: beta Complex (Kind=nag_wp), Intent (In) :: alpha, a(lda,*), b(ldb,*) Complex (Kind=nag_wp), Intent (Inout) :: c(ldc,*) Character (1), Intent (In) :: uplo, trans
#include <nag.h>
 void f06zrf_ (const char *uplo, const char *trans, const Integer *n, const Integer *k, const Complex *alpha, const Complex a[], const Integer *lda, const Complex b[], const Integer *ldb, const double *beta, Complex c[], const Integer *ldc, const Charlen length_uplo, const Charlen length_trans)
The routine may be called by the names f06zrf, nagf_blas_zher2k or its BLAS name zher2k.

None.
None.

## 5Arguments

1: $\mathbf{uplo}$Character(1) Input
On entry: specifies whether the upper or lower triangular part of $C$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $C$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $C$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{trans}$Character(1) Input
On entry: specifies the operation to be performed.
${\mathbf{trans}}=\text{'N'}$
$C←\alpha A{B}^{\mathrm{H}}+\overline{\alpha }B{A}^{\mathrm{H}}+\beta C$.
${\mathbf{trans}}=\text{'C'}$
$C←\alpha {A}^{\mathrm{H}}B+\overline{\alpha }{B}^{\mathrm{H}}A+\beta C$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'C'}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $C$; the number of rows of $A$ if ${\mathbf{trans}}=\text{'N'}$, or the number of columns of $A$ if ${\mathbf{trans}}=\text{'C'}$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{k}$Integer Input
On entry: $k$, the number of columns of $A$ if ${\mathbf{trans}}=\text{'N'}$, or the number of rows of $A$ if ${\mathbf{trans}}=\text{'C'}$.
Constraint: ${\mathbf{k}}\ge 0$.
5: $\mathbf{alpha}$Complex (Kind=nag_wp) Input
On entry: the scalar $\alpha$.
6: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$ if ${\mathbf{trans}}=\text{'N'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{trans}}=\text{'C'}$.
On entry: the matrix $A$; $A$ is $n×k$ if ${\mathbf{trans}}=\text{'N'}$, or $k×n$ if ${\mathbf{trans}}=\text{'C'}$.
7: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f06zrf is called.
Constraints:
• if ${\mathbf{trans}}=\text{'N'}$, ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{trans}}=\text{'C'}$, ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
8: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Complex (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$ if ${\mathbf{trans}}=\text{'N'}$ and at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{trans}}=\text{'C'}$.
On entry: the matrix $B$; $B$ is $n×k$ if ${\mathbf{trans}}=\text{'N'}$, or $k×n$ if ${\mathbf{trans}}=\text{'C'}$.
9: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f06zrf is called.
Constraints:
• if ${\mathbf{trans}}=\text{'N'}$, ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{trans}}=\text{'C'}$, ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
10: $\mathbf{beta}$Real (Kind=nag_wp) Input
On entry: the scalar $\beta$.
11: $\mathbf{c}\left({\mathbf{ldc}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array c must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n×n$ Hermitian matrix $C$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $C$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $C$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the updated matrix $C$. The imaginary parts of the diagonal elements are set to zero.
12: $\mathbf{ldc}$Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f06zrf is called.
Constraint: ${\mathbf{ldc}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

None.

Not applicable.