 CLASR(3) applies a sequence of real plane rotations to a complex matrix A, from either the left or the right

## SYNOPSIS

SUBROUTINE CLASR(
SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )

CHARACTER DIRECT, PIVOT, SIDE

INTEGER LDA, M, N

REAL C( * ), S( * )

COMPLEX A( LDA, * )

## PURPOSE

CLASR applies a sequence of real plane rotations to a complex matrix A, from either the left or the right. When SIDE = 'L', the transformation takes the form

A := P*A
and when SIDE = 'R', the transformation takes the form

A := A*P**T
where P is an orthogonal matrix consisting of a sequence of z plane rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R', and P**T is the transpose of P.

When DIRECT = 'F' (Forward sequence), then

P = P(z-1) * ... * P(2) * P(1)

and when DIRECT = 'B' (Backward sequence), then

P = P(1) * P(2) * ... * P(z-1)

where P(k) is a plane rotation matrix defined by the 2-by-2 rotation

R(k) = (  c(k)  s(k) )

= ( -s(k)  c(k) ).

When PIVOT = 'V' (Variable pivot), the rotation is performed for the plane (k,k+1), i.e., P(k) has the form

P(k) = (  1                                            )
(       ...                                     )
(              1                                )
(                   c(k)  s(k)                  )
(                  -s(k)  c(k)                  )
(                                1              )
(                                     ...       )
(                                            1  )
where R(k) appears as a rank-2 modification to the identity matrix in rows and columns k and k+1.

When PIVOT = 'T' (Top pivot), the rotation is performed for the plane (1,k+1), so P(k) has the form

P(k) = (  c(k)                    s(k)                 )
(         1                                     )
(              ...                              )
(                     1                         )
( -s(k)                    c(k)                 )
(                                 1             )
(                                      ...      )
(                                             1 )
where R(k) appears in rows and columns 1 and k+1.

Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is performed for the plane (k,z), giving P(k) the form

P(k) = ( 1                                             )
(      ...                                      )
(             1                                 )
(                  c(k)                    s(k) )
(                         1                     )
(                              ...              )
(                                     1         )
(                 -s(k)                    c(k) )
where R(k) appears in rows and columns k and z. The rotations are performed without ever forming P(k) explicitly.

## ARGUMENTS

SIDE (input) CHARACTER*1
Specifies whether the plane rotation matrix P is applied to A on the left or the right. = 'L': Left, compute A := P*A
= 'R': Right, compute A:= A*P**T
PIVOT (input) CHARACTER*1
Specifies the plane for which P(k) is a plane rotation matrix. = 'V': Variable pivot, the plane (k,k+1)
= 'T': Top pivot, the plane (1,k+1)
= 'B': Bottom pivot, the plane (k,z)
DIRECT (input) CHARACTER*1
Specifies whether P is a forward or backward sequence of plane rotations. = 'F': Forward, P = P(z-1)*...*P(2)*P(1)
= 'B': Backward, P = P(1)*P(2)*...*P(z-1)
M (input) INTEGER
The number of rows of the matrix A. If m <= 1, an immediate return is effected.
N (input) INTEGER
The number of columns of the matrix A. If n <= 1, an immediate return is effected.
C (input) REAL array, dimension
(M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the plane rotations.
S (input) REAL array, dimension
(M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The sines s(k) of the plane rotations. The 2-by-2 plane rotation part of the matrix P(k), R(k), has the form R(k) = ( c(k) s(k) ) ( -s(k) c(k) ).
A (input/output) COMPLEX array, dimension (LDA,N)
The M-by-N matrix A. On exit, A is overwritten by P*A if SIDE = 'R' or by A*P**T if SIDE = 'L'.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).