# NAG Library Routine Document

## 1Purpose

f06tmf performs a Unitary similarity transformation (as a sequence of plane rotations) of a complex Hermitian matrix.

## 2Specification

Fortran Interface
 Subroutine f06tmf ( uplo, n, k1, k2, c, s, a, lda)
 Integer, Intent (In) :: n, k1, k2, lda Real (Kind=nag_wp), Intent (In) :: c(*) Complex (Kind=nag_wp), Intent (In) :: s(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (1), Intent (In) :: uplo, pivot, direct
#include nagmk26.h
 void f06tmf_ (const char *uplo, const char *pivot, const char *direct, const Integer *n, const Integer *k1, const Integer *k2, const double c[], const Complex s[], Complex a[], const Integer *lda, const Charlen length_uplo, const Charlen length_pivot, const Charlen length_direct)

## 3Description

f06tmf performs the transformation
 $A←PAPH$
where $A$ is an $n$ by $n$ complex Hermitian matrix, and $P$ is a complex unitary matrix defined as a sequence of plane rotations, ${P}_{k}$, applied in planes ${k}_{1}$ to ${k}_{2}$.
The $2$ by $2$ plane rotation part of ${P}_{k}$ is assumed to have the form
 $ck s-k -sk ck$
with ${c}_{k}$ real.

None.

## 5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{pivot}$ – Character(1)Input
On entry: specifies the plane rotated by ${P}_{k}$.
${\mathbf{pivot}}=\text{'V'}$ (variable pivot)
${P}_{k}$ rotates the $\left(k,k+1\right)$ plane.
${\mathbf{pivot}}=\text{'T'}$ (top pivot)
${P}_{k}$ rotates the $\left({k}_{1},k+1\right)$ plane.
${\mathbf{pivot}}=\text{'B'}$ (bottom pivot)
${P}_{k}$ rotates the $\left(k,{k}_{2}\right)$ plane.
Constraint: ${\mathbf{pivot}}=\text{'V'}$, $\text{'T'}$ or $\text{'B'}$.
3:     $\mathbf{direct}$ – Character(1)Input
On entry: specifies the sequence direction.
${\mathbf{direct}}=\text{'F'}$ (forward sequence)
$P={P}_{{k}_{2}-1}\cdots {P}_{{k}_{1}+1}{P}_{{k}_{1}}$.
${\mathbf{direct}}=\text{'B'}$ (backward sequence)
$P={P}_{{k}_{1}}{P}_{{k}_{1}+1}\cdots {P}_{{k}_{2}-1}$.
Constraint: ${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{k1}$ – IntegerInput
6:     $\mathbf{k2}$ – IntegerInput
On entry: the values ${k}_{1}$ and ${k}_{2}$.
If ${\mathbf{k1}}<1$ or ${\mathbf{k2}}\le {\mathbf{k1}}$ or ${\mathbf{k2}}>{\mathbf{n}}$, an immediate return is effected.
7:     $\mathbf{c}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array c must be at least ${\mathbf{k2}}-1$.
On entry: ${\mathbf{c}}\left(\mathit{k}\right)$ must hold ${c}_{\mathit{k}}$, the cosine of the rotation ${P}_{\mathit{k}}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$.
8:     $\mathbf{s}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array s must be at least ${\mathbf{k2}}-1$.
On entry: ${\mathbf{s}}\left(\mathit{k}\right)$ must hold ${s}_{\mathit{k}}$, the sine of the rotation ${P}_{\mathit{k}}$, for $\mathit{k}={\mathit{k}}_{1},\dots ,{\mathit{k}}_{2}-1$.
9:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ Hermitian matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $A$ 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 $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the transformed matrix $A$. The imaginary parts of the diagonal elements are set to zero.
10:   $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06tmf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

None.

Not applicable.

## 8Parallelism and Performance

f06tmf is not threaded in any implementation.