.TH "lasr" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME lasr \- lasr: apply series of plane rotations .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBclasr\fP (side, pivot, direct, m, n, c, s, a, lda)" .br .RI "\fBCLASR\fP applies a sequence of plane rotations to a general rectangular matrix\&. " .ti -1c .RI "subroutine \fBdlasr\fP (side, pivot, direct, m, n, c, s, a, lda)" .br .RI "\fBDLASR\fP applies a sequence of plane rotations to a general rectangular matrix\&. " .ti -1c .RI "subroutine \fBslasr\fP (side, pivot, direct, m, n, c, s, a, lda)" .br .RI "\fBSLASR\fP applies a sequence of plane rotations to a general rectangular matrix\&. " .ti -1c .RI "subroutine \fBzlasr\fP (side, pivot, direct, m, n, c, s, a, lda)" .br .RI "\fBZLASR\fP applies a sequence of plane rotations to a general rectangular matrix\&. " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine clasr (character side, character pivot, character direct, integer m, integer n, real, dimension( * ) c, real, dimension( * ) s, complex, dimension( lda, * ) a, integer lda)" .PP \fBCLASR\fP applies a sequence of plane rotations to a general rectangular matrix\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fISIDE\fP .PP .nf 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 .fi .PP .br \fIPIVOT\fP .PP .nf 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) .fi .PP .br \fIDIRECT\fP .PP .nf 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) .fi .PP .br \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. If m <= 1, an immediate return is effected\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. If n <= 1, an immediate return is effected\&. .fi .PP .br \fIC\fP .PP .nf C is REAL array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the plane rotations\&. .fi .PP .br \fIS\fP .PP .nf 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) )\&. .fi .PP .br \fIA\fP .PP .nf 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'\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBDLASR\fP applies a sequence of plane rotations to a general rectangular matrix\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fISIDE\fP .PP .nf 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 .fi .PP .br \fIPIVOT\fP .PP .nf 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) .fi .PP .br \fIDIRECT\fP .PP .nf 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) .fi .PP .br \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. If m <= 1, an immediate return is effected\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. If n <= 1, an immediate return is effected\&. .fi .PP .br \fIC\fP .PP .nf C is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the plane rotations\&. .fi .PP .br \fIS\fP .PP .nf 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) )\&. .fi .PP .br \fIA\fP .PP .nf 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'\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "subroutine slasr (character side, character pivot, character direct, integer m, integer n, real, dimension( * ) c, real, dimension( * ) s, real, dimension( lda, * ) a, integer lda)" .PP \fBSLASR\fP applies a sequence of plane rotations to a general rectangular matrix\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fISIDE\fP .PP .nf 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 .fi .PP .br \fIPIVOT\fP .PP .nf 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) .fi .PP .br \fIDIRECT\fP .PP .nf 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) .fi .PP .br \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. If m <= 1, an immediate return is effected\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. If n <= 1, an immediate return is effected\&. .fi .PP .br \fIC\fP .PP .nf C is REAL array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the plane rotations\&. .fi .PP .br \fIS\fP .PP .nf 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) )\&. .fi .PP .br \fIA\fP .PP .nf 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'\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBZLASR\fP applies a sequence of plane rotations to a general rectangular matrix\&. .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fISIDE\fP .PP .nf 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 .fi .PP .br \fIPIVOT\fP .PP .nf 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) .fi .PP .br \fIDIRECT\fP .PP .nf 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) .fi .PP .br \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. If m <= 1, an immediate return is effected\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. If n <= 1, an immediate return is effected\&. .fi .PP .br \fIC\fP .PP .nf C is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the plane rotations\&. .fi .PP .br \fIS\fP .PP .nf 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) )\&. .fi .PP .br \fIA\fP .PP .nf 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'\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.