lasr(3)

NAME
lasr - lasr: apply series of plane rotations

SYNOPSIS

  Functions

subroutine clasr (side, pivot, direct, m, n, c, s, a, lda)
CLASR applies a sequence of plane rotations to a general rectangular matrix.
subroutine dlasr (side, pivot, direct, m, n, c, s, a, lda)
DLASR applies a sequence of plane rotations to a general rectangular matrix.
subroutine slasr (side, pivot, direct, m, n, c, s, a, lda)
SLASR applies a sequence of plane rotations to a general rectangular matrix.
subroutine zlasr (side, pivot, direct, m, n, c, s, a, lda)
ZLASR applies a sequence of plane rotations to a general rectangular matrix.

Detailed Description

Function Documentation

  subroutine clasr (character side, character pivot, character direct, integer m, integer n, real, dimension( * ) c, real, dimension( * ) s, complex, dimension( lda, * ) a, integer lda)

CLASR applies a sequence of plane rotations to a general rectangular matrix.

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.
    !>

Parameters
    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 REAL array, dimension
    !>                  (M-1) if SIDE = 'L'
    !>                  (N-1) if SIDE = 'R'
    !>          The cosines c(k) of the plane rotations.
    !>

    S

    !>          S is 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

    !>          A is 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

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine dlasr (character side, character pivot, character direct, integer m, integer n, double precision, dimension( * ) c, double precision, dimension( * ) s, double precision, dimension( lda, * ) a, integer lda)

DLASR applies a sequence of plane rotations to a general rectangular matrix.

Purpose:

    !>
    !> 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.
    !>

Parameters
    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 = 'L' or by A*P**T if SIDE = 'R'.
    !>

    LDA

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

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine slasr (character side, character pivot, character direct, integer m, integer n, real, dimension( * ) c, real, dimension( * ) s, real, dimension( lda, * ) a, integer lda)

SLASR applies a sequence of plane rotations to a general rectangular matrix.

Purpose:

    !>
    !> SLASR 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.
    !>

Parameters
    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 REAL array, dimension
    !>                  (M-1) if SIDE = 'L'
    !>                  (N-1) if SIDE = 'R'
    !>          The cosines c(k) of the plane rotations.
    !>

    S

    !>          S is 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

    !>          A is REAL 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).
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine zlasr (character side, character pivot, character direct, integer m, integer n, double precision, dimension( * ) c, double precision, dimension( * ) s, complex*16, dimension( lda, * ) a, integer lda)

ZLASR applies a sequence of plane rotations to a general rectangular matrix.

Purpose:

    !>
    !> ZLASR 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.
    !>

Parameters
    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 COMPLEX*16 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).
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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