1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
|
SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
*
* -- LAPACK routine (version 3.3.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* -- April 2011 --
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
REAL D( * ), E( * )
COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* CGEBD2 reduces a complex general m by n matrix A to upper or lower
* real bidiagonal form B by a unitary transformation: Q**H * A * P = B.
*
* If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows in the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns in the matrix A. N >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the m by n general matrix to be reduced.
* On exit,
* if m >= n, the diagonal and the first superdiagonal are
* overwritten with the upper bidiagonal matrix B; the
* elements below the diagonal, with the array TAUQ, represent
* the unitary matrix Q as a product of elementary
* reflectors, and the elements above the first superdiagonal,
* with the array TAUP, represent the unitary matrix P as
* a product of elementary reflectors;
* if m < n, the diagonal and the first subdiagonal are
* overwritten with the lower bidiagonal matrix B; the
* elements below the first subdiagonal, with the array TAUQ,
* represent the unitary matrix Q as a product of
* elementary reflectors, and the elements above the diagonal,
* with the array TAUP, represent the unitary matrix P as
* a product of elementary reflectors.
* See Further Details.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* D (output) REAL array, dimension (min(M,N))
* The diagonal elements of the bidiagonal matrix B:
* D(i) = A(i,i).
*
* E (output) REAL array, dimension (min(M,N)-1)
* The off-diagonal elements of the bidiagonal matrix B:
* if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
* if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
*
* TAUQ (output) COMPLEX array dimension (min(M,N))
* The scalar factors of the elementary reflectors which
* represent the unitary matrix Q. See Further Details.
*
* TAUP (output) COMPLEX array, dimension (min(M,N))
* The scalar factors of the elementary reflectors which
* represent the unitary matrix P. See Further Details.
*
* WORK (workspace) COMPLEX array, dimension (max(M,N))
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* The matrices Q and P are represented as products of elementary
* reflectors:
*
* If m >= n,
*
* Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
*
* Each H(i) and G(i) has the form:
*
* H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
*
* where tauq and taup are complex scalars, and v and u are complex
* vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
* A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
* A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*
* If m < n,
*
* Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
*
* Each H(i) and G(i) has the form:
*
* H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
*
* where tauq and taup are complex scalars, v and u are complex vectors;
* v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
* u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
* tauq is stored in TAUQ(i) and taup in TAUP(i).
*
* The contents of A on exit are illustrated by the following examples:
*
* m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*
* ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
* ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
* ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
* ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
* ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
* ( v1 v2 v3 v4 v5 )
*
* where d and e denote diagonal and off-diagonal elements of B, vi
* denotes an element of the vector defining H(i), and ui an element of
* the vector defining G(i).
*
* =====================================================================
*
* .. Parameters ..
COMPLEX ZERO, ONE
PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ),
$ ONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
INTEGER I
COMPLEX ALPHA
* ..
* .. External Subroutines ..
EXTERNAL CLACGV, CLARF, CLARFG, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC CONJG, MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.LT.0 ) THEN
CALL XERBLA( 'CGEBD2', -INFO )
RETURN
END IF
*
IF( M.GE.N ) THEN
*
* Reduce to upper bidiagonal form
*
DO 10 I = 1, N
*
* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
*
ALPHA = A( I, I )
CALL CLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1,
$ TAUQ( I ) )
D( I ) = ALPHA
A( I, I ) = ONE
*
* Apply H(i)**H to A(i:m,i+1:n) from the left
*
IF( I.LT.N )
$ CALL CLARF( 'Left', M-I+1, N-I, A( I, I ), 1,
$ CONJG( TAUQ( I ) ), A( I, I+1 ), LDA, WORK )
A( I, I ) = D( I )
*
IF( I.LT.N ) THEN
*
* Generate elementary reflector G(i) to annihilate
* A(i,i+2:n)
*
CALL CLACGV( N-I, A( I, I+1 ), LDA )
ALPHA = A( I, I+1 )
CALL CLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ),
$ LDA, TAUP( I ) )
E( I ) = ALPHA
A( I, I+1 ) = ONE
*
* Apply G(i) to A(i+1:m,i+1:n) from the right
*
CALL CLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
$ TAUP( I ), A( I+1, I+1 ), LDA, WORK )
CALL CLACGV( N-I, A( I, I+1 ), LDA )
A( I, I+1 ) = E( I )
ELSE
TAUP( I ) = ZERO
END IF
10 CONTINUE
ELSE
*
* Reduce to lower bidiagonal form
*
DO 20 I = 1, M
*
* Generate elementary reflector G(i) to annihilate A(i,i+1:n)
*
CALL CLACGV( N-I+1, A( I, I ), LDA )
ALPHA = A( I, I )
CALL CLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA,
$ TAUP( I ) )
D( I ) = ALPHA
A( I, I ) = ONE
*
* Apply G(i) to A(i+1:m,i:n) from the right
*
IF( I.LT.M )
$ CALL CLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
$ TAUP( I ), A( I+1, I ), LDA, WORK )
CALL CLACGV( N-I+1, A( I, I ), LDA )
A( I, I ) = D( I )
*
IF( I.LT.M ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(i+2:m,i)
*
ALPHA = A( I+1, I )
CALL CLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1,
$ TAUQ( I ) )
E( I ) = ALPHA
A( I+1, I ) = ONE
*
* Apply H(i)**H to A(i+1:m,i+1:n) from the left
*
CALL CLARF( 'Left', M-I, N-I, A( I+1, I ), 1,
$ CONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA,
$ WORK )
A( I+1, I ) = E( I )
ELSE
TAUQ( I ) = ZERO
END IF
20 CONTINUE
END IF
RETURN
*
* End of CGEBD2
*
END
|