# NAG Library Routine Document

## 1Purpose

f06tbf performs the symmetric rank-1 update operation
 $A←αxxT + A ,$
where $A$ is an $n$ by $n$ complex symmetric matrix, $x$ is an $n$-element complex vector, and $\alpha$ is a complex scalar.

## 2Specification

Fortran Interface
 Subroutine f06tbf ( uplo, n, x, incx, a, lda)
 Integer, Intent (In) :: n, incx, lda Complex (Kind=nag_wp), Intent (In) :: alpha, x(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Character (1), Intent (In) :: uplo
C Header Interface
#include nagmk26.h
 void f06tbf_ (const char *uplo, const Integer *n, const Complex *alpha, const Complex x[], const Integer *incx, Complex a[], const Integer *lda, const Charlen length_uplo)

None.

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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{alpha}$ – Complex (Kind=nag_wp)Input
On entry: the scalar $\alpha$.
4:     $\mathbf{x}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incx}}\right|\right)$.
On entry: the $n$-element vector $x$.
If ${\mathbf{incx}}>0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1+\left(\mathit{i}-1\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{incx}}<0$, ${x}_{\mathit{i}}$ must be stored in ${\mathbf{x}}\left(1-\left({\mathbf{n}}-\mathit{i}\right)×{\mathbf{incx}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Intermediate elements of x are not referenced.
5:     $\mathbf{incx}$ – IntegerInput
On entry: the increment in the subscripts of x between successive elements of $x$.
Constraint: ${\mathbf{incx}}\ne 0$.
6:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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 $n$ symmetric 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 updated matrix $A$.
7:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f06tbf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

None.

Not applicable.

## 8Parallelism and Performance

f06tbf is not threaded in any implementation.

None.

## 10Example

None.
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017