table of contents
| lasr(3) | LAPACK | 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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