1CLASR(1) LAPACK auxiliary routine (version 3.2) CLASR(1)
2
6 CLASR - applies a sequence of real plane rotations to a complex matrix
7 A, from either the left or the right
8
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. When SIDE = 'L', the transformation
23 takes the form
24 A := P*A
25 and when SIDE = 'R', the transformation takes the form
26 A := A*P**T
27 where P is an orthogonal matrix consisting of a sequence of z plane
28 rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R', and
29 P**T is the transpose of P.
30 When DIRECT = 'F' (Forward sequence), then
31 P = P(z-1) * ... * P(2) * P(1)
32 and when DIRECT = 'B' (Backward sequence), then
33 P = P(1) * P(2) * ... * P(z-1)
34 where P(k) is a plane rotation matrix defined by the 2-by-2 rotation
35 R(k) = ( c(k) s(k) )
36 = ( -s(k) c(k) ).
37 When PIVOT = 'V' (Variable pivot), the rotation is performed for the
38 plane (k,k+1), i.e., P(k) has the form
39 P(k) = ( 1 )
40 ( ... )
41 ( 1 )
42 ( c(k) s(k) )
43 ( -s(k) c(k) )
44 ( 1 )
45 ( ... )
46 ( 1 )
47 where R(k) appears as a rank-2 modification to the identity matrix in
48 rows and columns k and k+1.
49 When PIVOT = 'T' (Top pivot), the rotation is performed for the plane
50 (1,k+1), so P(k) has the form
51 P(k) = ( c(k) s(k) )
52 ( 1 )
53 ( ... )
54 ( 1 )
55 ( -s(k) c(k) )
56 ( 1 )
57 ( ... )
58 ( 1 )
59 where R(k) appears in rows and columns 1 and k+1.
60 Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is performed
61 for the plane (k,z), giving P(k) the form
62 P(k) = ( 1 )
63 ( ... )
64 ( 1 )
65 ( c(k) s(k) )
66 ( 1 )
67 ( ... )
68 ( 1 )
69 ( -s(k) c(k) )
70 where R(k) appears in rows and columns k and z. The rotations are per‐
71 formed without ever forming P(k) explicitly.
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74 SIDE (input) CHARACTER*1
75 Specifies whether the plane rotation matrix P is applied to A
76 on the left or the right. = 'L': Left, compute A := P*A
77 = 'R': Right, compute A:= A*P**T
78 PIVOT (input) CHARACTER*1
80 Specifies the plane for which P(k) is a plane rotation matrix.
81 = 'V': Variable pivot, the plane (k,k+1)
82 = 'T': Top pivot, the plane (1,k+1)
83 = 'B': Bottom pivot, the plane (k,z)
84 DIRECT (input) CHARACTER*1
86 Specifies whether P is a forward or backward sequence of plane
87 rotations. = 'F': Forward, P = P(z-1)*...*P(2)*P(1)
88 = 'B': Backward, P = P(1)*P(2)*...*P(z-1)
89 M (input) INTEGER
91 The number of rows of the matrix A. If m <= 1, an immediate
92 return is effected.
93 N (input) INTEGER
95 The number of columns of the matrix A. If n <= 1, an immediate
96 return is effected.
97 C (input) REAL array, dimension
99 (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The cosines c(k) of the
100 plane rotations.
101 S (input) REAL array, dimension
103 (M-1) if SIDE = 'L' (N-1) if SIDE = 'R' The sines s(k) of the
104 plane rotations. The 2-by-2 plane rotation part of the matrix
105 P(k), R(k), has the form R(k) = ( c(k) s(k) ) ( -s(k) c(k)
106 ).
107 A (input/output) COMPLEX array, dimension (LDA,N)
109 The M-by-N matrix A. On exit, A is overwritten by P*A if SIDE
110 = 'R' or by A*P**T if SIDE = 'L'.
111 LDA (input) INTEGER
113 The leading dimension of the array A. LDA >= max(1,M).
114 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 CLASR(1)