1CLASR(1) LAPACK auxiliary routine (version 3.1) CLASR(1)
2
6 CLASR - a sequence of real plane rotations to a complex matrix A, from
7 either the left or the right
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10 SUBROUTINE CLASR( SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA )
11 CHARACTER DIRECT, PIVOT, SIDE
13 INTEGER LDA, M, N
15 REAL C( * ), S( * )
17 COMPLEX A( LDA, * )
19
21 CLASR applies a sequence of real plane rotations to a complex matrix A,
22 from either the left or the right.
23 When SIDE = 'L', the transformation takes the form
25 A := P*A
27 and when SIDE = 'R', the transformation takes the form
29 A := A*P**T
31 where P is an orthogonal matrix consisting of a sequence of z plane
33 rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R', and
34 P**T is the transpose of P.
35 When DIRECT = 'F' (Forward sequence), then
37 P = P(z-1) * ... * P(2) * P(1)
39 and when DIRECT = 'B' (Backward sequence), then
41 P = P(1) * P(2) * ... * P(z-1)
43 where P(k) is a plane rotation matrix defined by the 2-by-2 rotation
45 R(k) = ( c(k) s(k) )
47 = ( -s(k) c(k) ).
48 When PIVOT = 'V' (Variable pivot), the rotation is performed for the
50 plane (k,k+1), i.e., P(k) has the form
51 P(k) = ( 1 )
53 ( ... )
54 ( 1 )
55 ( c(k) s(k) )
56 ( -s(k) c(k) )
57 ( 1 )
58 ( ... )
59 ( 1 )
60 where R(k) appears as a rank-2 modification to the identity matrix in
62 rows and columns k and k+1.
63 When PIVOT = 'T' (Top pivot), the rotation is performed for the plane
65 (1,k+1), so P(k) has the form
66 P(k) = ( c(k) s(k) )
68 ( 1 )
69 ( ... )
70 ( 1 )
71 ( -s(k) c(k) )
72 ( 1 )
73 ( ... )
74 ( 1 )
75 where R(k) appears in rows and columns 1 and k+1.
77 Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is performed
79 for the plane (k,z), giving P(k) the form
80 P(k) = ( 1 )
82 ( ... )
83 ( 1 )
84 ( c(k) s(k) )
85 ( 1 )
86 ( ... )
87 ( 1 )
88 ( -s(k) c(k) )
89 where R(k) appears in rows and columns k and z. The rotations are per‐
91 formed without ever forming P(k) explicitly.
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95 SIDE (input) CHARACTER*1
96 Specifies whether the plane rotation matrix P is applied to A
97 on the left or the right. = 'L': Left, compute A := P*A
98 = 'R': Right, compute A:= A*P**T
99 PIVOT (input) CHARACTER*1
101 Specifies the plane for which P(k) is a plane rotation matrix.
102 = 'V': Variable pivot, the plane (k,k+1)
103 = 'T': Top pivot, the plane (1,k+1)
104 = 'B': Bottom pivot, the plane (k,z)
105 DIRECT (input) CHARACTER*1
107 Specifies whether P is a forward or backward sequence of plane
108 rotations. = 'F': Forward, P = P(z-1)*...*P(2)*P(1)
109 = 'B': Backward, P = P(1)*P(2)*...*P(z-1)
110 M (input) INTEGER
112 The number of rows of the matrix A. If m <= 1, an immediate
113 return is effected.
114 N (input) INTEGER
116 The number of columns of the matrix A. If n <= 1, an immediate
117 return is effected.
118 C (input) REAL array, dimension
120 (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the
121 plane rotations.
122 S (input) REAL array, dimension
124 (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The sines s(k) of the
125 plane rotations. The 2-by-2 plane rotation part of the matrix
126 P(k), R(k), has the form R(k) = ( c(k) s(k) ) ( -s(k) c(k)
127 ).
128 A (input/output) COMPLEX array, dimension (LDA,N)
130 The M-by-N matrix A. On exit, A is overwritten by P*A if SIDE
131 = 'R' or by A*P**T if SIDE = 'L'.
132 LDA (input) INTEGER
134 The leading dimension of the array A. LDA >= max(1,M).
135 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006 CLASR(1)