There are a number of methods in the chapter concerned with setting up and applying plane rotations. This section discusses the real case and the next section looks at the complex case. For further background information see Golub and Van Loan (1996).

A plane rotation matrix for the $\left(i,j\right)$ plane, $R_{ij}$, is an orthogonal matrix that is different from the unit matrix only in the elements $r_{ii}$, $r_{jj}$, $r_{ij}$ and $r_{ji}$. If we put then, in the real case, it is usual to choose $R_{ij}$ so that An exception is method (F06FPF not in this release) which applies the so-called symmetric rotation for which The application of plane rotations is straightforward and needs no further elaboration, so further comment is made only on the construction of plane rotations.

The most common use of plane rotations is to choose $c$ and $s$ so that for given $a$ and $b$, In such an application the matrix $R$ is often termed a Givens rotation matrix. There are two approaches to the construction of real Givens rotations in F06 class.

The BLAS method (F06AAF not in this release), see Lawson et al. (1979) and Dodson and Grimes (1982), computes $c$, $s$ and $d$ as where $\sigma =\left\{\begin{array}{cc}\mathrm{sign} a\text{,}& \left|a\right|>\left|b\right|\\ \mathrm{sign} b\text{,}& \left|a\right|\le \left|b\right|\end{array}\right.\text{.}$

The value $z$ defined as is also computed and this enables $c$ and $s$ to be reconstructed from the single value $z$ as The other F06 class methods for constructing Givens rotations are based on the computation of the tangent, $t=\tan \theta$. $t$ is computed as where $\mathit{flmax}=1/\mathit{flmin}$ and $\mathit{flmin}$ is the small positive value returned by x02am. The values of $c$ and $s$ are then computed or reconstructed via $t$ as where $\mathrm{eps}$ is the machine precision. Note that $c$ is always non-negative in this scheme and that the same expressions are used in the initial computation of $c$ and $s$ from $a$ and $b$ as in any subsequent recovery of $c$ and $s$ via $t$. This is the approach used by many of the NAG Library methods that require plane rotations. $d$ is computed simply as You need not be too concerned with the above detail, since methods are provided for setting up, recovering and applying such rotations.

Another use of plane rotations is to choose $c$ and $s$ so that for given $x$, $y$ and $z$  In such an application the matrix $R$ is often termed a Jacobi rotation matrix. The method that generates a Jacobi rotation ( (F06BEF not in this release)) first computes the tangent $t$ and then computes $c$ and $s$ via $t$ as described above for the Givens rotation.

In the complex case a plane rotation matrix for the $\left(i,j\right)$ plane, $R_{ij}$ is a unitary matrix and, analogously to the real case, it is usual to choose $R_{ij}$ so that where $\stackrel{-}{a}$ denotes the complex conjugate of $a$.

The BLAS (see Lawson et al. (1979)) do not contain a method for the generation of complex rotations, and so the methods in F06 class are all based upon computing $c$ and $s$ via $t=b/a$ in an analogous manner to the real case. $R$ can be chosen to have either $c$ real, or $s$ real and there are methods for both cases.

When $c$ is real then it is non-negative and the transformation is such that if $a$ is real then $d$ is also real.

When $s$ is real then the transformation is such that if $b$ is real then $d$ is also real.

There are a number of methods in the chapter concerned with setting up and applying Householder transformations. This section discusses the real case and the next section looks at the complex case. For further background information see Golub and Van Loan (1996).

A real elementary reflector, $P$, is a matrix of the form where $\mu$ is a scalar and $u$ is a vector, and $P$ is both symmetric and orthogonal. In the methods in F06 class, $u$ is expressed in the form because in many applications $\zeta$ and $z$ are not contiguous elements. The usual use of elementary reflectors is to choose $\mu$ and $u$ so that for given $\alpha$ and $x$  Such a transformation is often termed a Householder transformation. There are two choices of $\mu$ and $u$ available in F06 class.

The first form of the Householder transformation is compatible with that used by LINPACK (see Dongarra et al. (1979)) and has

This choice makes $\zeta$ satisfy The second form, and the form used by many of the NAG Library methods, has which makes In both cases the special setting is used by the methods to flag the case where $P=I$.

Note that while there are methods to apply an elementary reflector to a vector, there are no methods available in F06 class to apply an elementary reflector to a matrix. This is because such transformations can readily and efficiently be achieved by calls to the matrix-vector Level 2 BLAS methods. For example, to form $PA$ for a given matrix and so we can call a matrix-vector product method to form $b=A^{\mathrm{T}}u$ and then call a rank-one update method to form $\left(A-\mu u{b}^{\mathrm{T}}\right)$. Of course, we must skip the transformation when $\zeta$ has been set to zero.

A complex elementary reflector, $P$, is a matrix of the form where $u^{\mathrm{H}}$ denotes the complex conjugate of $u^{\mathrm{T}}$, and $P$ is both Hermitian and unitary. For convenience in a number of applications this definition can be generalized slightly by allowing $\mu$ to be complex and so defining the generalized elementary reflector as for which $P$ is still unitary, but is no longer Hermitian.

The F06 class methods choose $\mu$ and $\zeta$ so that and this reduces to (12) with the choice (16) when $\mu$ and $u$ are real. This choice is used because $\mu$ and $u$ can now be chosen so that in the Householder transformation (14) we can make and, as in the real case, Rather than returning $\mu$ and $\zeta$ as separate parameters the F06 class methods return the single complex value $\theta$ defined as Obviously $\zeta$ and $\mu$ can be recovered as The special setting is used to flag the case where $P=I$, and is used to flag the case where and in this case $\theta$ actually contains the value of $\gamma$. Notice that with both (18) and (21) we merely have to supply $\stackrel{-}{\theta }$ rather than $\theta$ in order to represent $P^{\mathrm{H}}$.