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.

Definition at line 199 of file clasr.f.

  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.

Definition at line 198 of file dlasr.f.

  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.

Definition at line 198 of file slasr.f.

  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.

Definition at line 199 of file zlasr.f.

Author

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