NAG Library Routine Document

F06AAF (DROTG)

+− Contents

1 Purpose

9 Example

F06AAF (DROTG) generates a real Givens plane rotation.

SUBROUTINE F06AAF (

A, B, C, S)

REAL (KIND=nag_wp)

A, B, C, S

The routine may be called by its BLAS name drotg.

F06AAF (DROTG) generates a real Givens plane rotation with parameters

c

and

s

, such that, given real

a

and

b

(\begin{array}{rc} c & s \\ - s & c \end{array}) (\begin{matrix} a \\ b \end{matrix}) = (\begin{matrix} d \\ 0 \end{matrix}) .

The routine computes

c

s

and

d

as follows:

d = σ \sqrt{a^{2} + b^{2}};

c = \{\begin{cases} a / d, & if ​ d \neq 0, \\ 1, & if ​ d = 0, \end{cases} s = \{\begin{cases} b / d, & if ​ d \neq 0, \\ 0, & if ​ d = 0, \end{cases}

where σ = \{\begin{cases} sign a, & if ​ |a| > |b|, \\ sign b, & if ​ |a| \leq |b| . \end{cases}

The routine also computes the value

z

defined as

z = \{\begin{cases} s, & if ​ |s| < c ​ or ​ c = 0, \\ 1 / c, & if ​ 0 < |c| \leq s . \end{cases}

This enables

c

and

s

to be reconstructed from the single value

z

c = \{\begin{cases} \sqrt{1 - z^{2}}, & if ​ |z| \leq 1, \\ 1 / z, & if ​ |z| > 1, \end{cases} s = \{\begin{cases} z, & if ​ |z| \leq 1, \\ \sqrt{1 - c^{2}}, & if ​ |z| > 1 . \end{cases}

To apply the plane rotation to a pair of real vectors, call F06EPF (DROT); to apply it to a pair of complex vectors, call F06KPF.

None.

1: A – REAL (KIND=nag_wp)Input/Output: On entry: the value $a$ , the first element of the vector which determines the rotation.

On exit: the value $d$ .
2: B – REAL (KIND=nag_wp)Input/Output: On entry: the value $b$ , the second element of the vector which determines the rotation.

On exit: the value $z$ , from which $c$ and $s$ can be reconstructed.
3: C – REAL (KIND=nag_wp)Output: On exit: the value $c$ , the cosine of the rotation.
4: S – REAL (KIND=nag_wp)Output: On exit: the value $s$ , the sine of the rotation.