DLASR applies a sequence of plane rotations to a real 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.

SIDE

          SIDE is 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

          PIVOT is 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

          DIRECT is 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

          M is INTEGER


          The number of rows of the matrix A.  If m <= 1, an immediate


          return is effected.

N

          N is INTEGER


          The number of columns of the matrix A.  If n <= 1, an


          immediate return is effected.

C

          C is DOUBLE PRECISION array, dimension


                  (M-1) if SIDE = 'L'


                  (N-1) if SIDE = 'R'


          The cosines c(k) of the plane rotations.

S

          S is DOUBLE PRECISION 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

          A is DOUBLE PRECISION 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

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,M).

Univ. of Tennessee

Univ. of California Berkeley

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NAG Ltd.

December 2016