.TH CLASR 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) " .SH NAME CLASR - a sequence of real plane rotations to a complex matrix A, from either the left or the right .SH SYNOPSIS .TP 18 SUBROUTINE CLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA ) .TP 18 .ti +4 CHARACTER DIRECT, PIVOT, SIDE .TP 18 .ti +4 INTEGER LDA, M, N .TP 18 .ti +4 REAL C( * ), S( * ) .TP 18 .ti +4 COMPLEX A( LDA, * ) .SH PURPOSE CLASR applies a sequence of real plane rotations to a complex matrix A, from either the left or the right. When SIDE = \(aqL\(aq, the transformation takes the form .br A := P*A .br and when SIDE = \(aqR\(aq, the transformation takes the form .br A := A*P**T .br where P is an orthogonal matrix consisting of a sequence of z plane rotations, with z = M when SIDE = \(aqL\(aq and z = N when SIDE = \(aqR\(aq, and P**T is the transpose of P. .br .br When DIRECT = \(aqF\(aq (Forward sequence), then .br .br P = P(z-1) * ... * P(2) * P(1) .br .br and when DIRECT = \(aqB\(aq (Backward sequence), then .br .br P = P(1) * P(2) * ... * P(z-1) .br .br where P(k) is a plane rotation matrix defined by the 2-by-2 rotation .br R(k) = ( c(k) s(k) ) .br = ( -s(k) c(k) ). .br .br When PIVOT = \(aqV\(aq (Variable pivot), the rotation is performed for the plane (k,k+1), i.e., P(k) has the form .br .br P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) .br where R(k) appears as a rank-2 modification to the identity matrix in rows and columns k and k+1. .br .br When PIVOT = \(aqT\(aq (Top pivot), the rotation is performed for the plane (1,k+1), so P(k) has the form .br .br P(k) = ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) ( 1 ) ( ... ) ( 1 ) .br where R(k) appears in rows and columns 1 and k+1. .br .br Similarly, when PIVOT = \(aqB\(aq (Bottom pivot), the rotation is performed for the plane (k,z), giving P(k) the form .br .br P(k) = ( 1 ) ( ... ) ( 1 ) ( c(k) s(k) ) ( 1 ) ( ... ) ( 1 ) ( -s(k) c(k) ) .br where R(k) appears in rows and columns k and z. The rotations are performed without ever forming P(k) explicitly. .br .SH ARGUMENTS .TP 8 SIDE (input) CHARACTER*1 Specifies whether the plane rotation matrix P is applied to A on the left or the right. = \(aqL\(aq: Left, compute A := P*A .br = \(aqR\(aq: Right, compute A:= A*P**T .TP 8 PIVOT (input) CHARACTER*1 Specifies the plane for which P(k) is a plane rotation matrix. = \(aqV\(aq: Variable pivot, the plane (k,k+1) .br = \(aqT\(aq: Top pivot, the plane (1,k+1) .br = \(aqB\(aq: Bottom pivot, the plane (k,z) .TP 8 DIRECT (input) CHARACTER*1 Specifies whether P is a forward or backward sequence of plane rotations. = \(aqF\(aq: Forward, P = P(z-1)*...*P(2)*P(1) .br = \(aqB\(aq: Backward, P = P(1)*P(2)*...*P(z-1) .TP 8 M (input) INTEGER The number of rows of the matrix A. If m <= 1, an immediate return is effected. .TP 8 N (input) INTEGER The number of columns of the matrix A. If n <= 1, an immediate return is effected. .TP 8 C (input) REAL array, dimension (M-1) if SIDE = \(aqL\(aq (N-1) if SIDE = \(aqR\(aq The cosines c(k) of the plane rotations. .TP 8 S (input) REAL array, dimension (M-1) if SIDE = \(aqL\(aq (N-1) if SIDE = \(aqR\(aq 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) ). .TP 8 A (input/output) COMPLEX array, dimension (LDA,N) The M-by-N matrix A. On exit, A is overwritten by P*A if SIDE = \(aqR\(aq or by A*P**T if SIDE = \(aqL\(aq. .TP 8 LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M).