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
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
|
SUBROUTINE ZLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
$ LDY )
*
* -- LAPACK auxiliary routine (version 3.2) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER LDA, LDX, LDY, M, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ),
$ Y( LDY, * )
* ..
*
* Purpose
* =======
*
* ZLABRD reduces the first NB rows and columns of a complex general
* m by n matrix A to upper or lower real bidiagonal form by a unitary
* transformation Q' * A * P, and returns the matrices X and Y which
* are needed to apply the transformation to the unreduced part of A.
*
* If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
* bidiagonal form.
*
* This is an auxiliary routine called by ZGEBRD
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows in the matrix A.
*
* N (input) INTEGER
* The number of columns in the matrix A.
*
* NB (input) INTEGER
* The number of leading rows and columns of A to be reduced.
*
* A (input/output) COMPLEX*16 array, dimension (LDA,N)
* On entry, the m by n general matrix to be reduced.
* On exit, the first NB rows and columns of the matrix are
* overwritten; the rest of the array is unchanged.
* If m >= n, elements on and below the diagonal in the first NB
* columns, with the array TAUQ, represent the unitary
* matrix Q as a product of elementary reflectors; and
* elements above the diagonal in the first NB rows, with the
* array TAUP, represent the unitary matrix P as a product
* of elementary reflectors.
* If m < n, elements below the diagonal in the first NB
* columns, with the array TAUQ, represent the unitary
* matrix Q as a product of elementary reflectors, and
* elements on and above the diagonal in the first NB rows,
* 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) DOUBLE PRECISION array, dimension (NB)
* The diagonal elements of the first NB rows and columns of
* the reduced matrix. D(i) = A(i,i).
*
* E (output) DOUBLE PRECISION array, dimension (NB)
* The off-diagonal elements of the first NB rows and columns of
* the reduced matrix.
*
* TAUQ (output) COMPLEX*16 array dimension (NB)
* The scalar factors of the elementary reflectors which
* represent the unitary matrix Q. See Further Details.
*
* TAUP (output) COMPLEX*16 array, dimension (NB)
* The scalar factors of the elementary reflectors which
* represent the unitary matrix P. See Further Details.
*
* X (output) COMPLEX*16 array, dimension (LDX,NB)
* The m-by-nb matrix X required to update the unreduced part
* of A.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,M).
*
* Y (output) COMPLEX*16 array, dimension (LDY,NB)
* The n-by-nb matrix Y required to update the unreduced part
* of A.
*
* LDY (input) INTEGER
* The leading dimension of the array Y. LDY >= max(1,N).
*
* Further Details
* ===============
*
* The matrices Q and P are represented as products of elementary
* reflectors:
*
* Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
*
* Each H(i) and G(i) has the form:
*
* H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
*
* where tauq and taup are complex scalars, and v and u are complex
* vectors.
*
* If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
* A(i:m,i); u(1:i) = 0, u(i+1) = 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).
*
* If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
* A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
* A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*
* The elements of the vectors v and u together form the m-by-nb matrix
* V and the nb-by-n matrix U' which are needed, with X and Y, to apply
* the transformation to the unreduced part of the matrix, using a block
* update of the form: A := A - V*Y' - X*U'.
*
* The contents of A on exit are illustrated by the following examples
* with nb = 2:
*
* m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*
* ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
* ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
* ( v1 v2 a a a ) ( v1 1 a a a a )
* ( v1 v2 a a a ) ( v1 v2 a a a a )
* ( v1 v2 a a a ) ( v1 v2 a a a a )
* ( v1 v2 a a a )
*
* where a denotes an element of the original matrix which is unchanged,
* vi denotes an element of the vector defining H(i), and ui an element
* of the vector defining G(i).
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 ZERO, ONE
PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
$ ONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER I
COMPLEX*16 ALPHA
* ..
* .. External Subroutines ..
EXTERNAL ZGEMV, ZLACGV, ZLARFG, ZSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
IF( M.GE.N ) THEN
*
* Reduce to upper bidiagonal form
*
DO 10 I = 1, NB
*
* Update A(i:m,i)
*
CALL ZLACGV( I-1, Y( I, 1 ), LDY )
CALL ZGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
$ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
CALL ZLACGV( I-1, Y( I, 1 ), LDY )
CALL ZGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
$ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
*
* Generate reflection Q(i) to annihilate A(i+1:m,i)
*
ALPHA = A( I, I )
CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1,
$ TAUQ( I ) )
D( I ) = ALPHA
IF( I.LT.N ) THEN
A( I, I ) = ONE
*
* Compute Y(i+1:n,i)
*
CALL ZGEMV( 'Conjugate transpose', M-I+1, N-I, ONE,
$ A( I, I+1 ), LDA, A( I, I ), 1, ZERO,
$ Y( I+1, I ), 1 )
CALL ZGEMV( 'Conjugate transpose', M-I+1, I-1, ONE,
$ A( I, 1 ), LDA, A( I, I ), 1, ZERO,
$ Y( 1, I ), 1 )
CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
$ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL ZGEMV( 'Conjugate transpose', M-I+1, I-1, ONE,
$ X( I, 1 ), LDX, A( I, I ), 1, ZERO,
$ Y( 1, I ), 1 )
CALL ZGEMV( 'Conjugate transpose', I-1, N-I, -ONE,
$ A( 1, I+1 ), LDA, Y( 1, I ), 1, ONE,
$ Y( I+1, I ), 1 )
CALL ZSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
*
* Update A(i,i+1:n)
*
CALL ZLACGV( N-I, A( I, I+1 ), LDA )
CALL ZLACGV( I, A( I, 1 ), LDA )
CALL ZGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
$ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
CALL ZLACGV( I, A( I, 1 ), LDA )
CALL ZLACGV( I-1, X( I, 1 ), LDX )
CALL ZGEMV( 'Conjugate transpose', I-1, N-I, -ONE,
$ A( 1, I+1 ), LDA, X( I, 1 ), LDX, ONE,
$ A( I, I+1 ), LDA )
CALL ZLACGV( I-1, X( I, 1 ), LDX )
*
* Generate reflection P(i) to annihilate A(i,i+2:n)
*
ALPHA = A( I, I+1 )
CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA,
$ TAUP( I ) )
E( I ) = ALPHA
A( I, I+1 ) = ONE
*
* Compute X(i+1:m,i)
*
CALL ZGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
$ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
CALL ZGEMV( 'Conjugate transpose', N-I, I, ONE,
$ Y( I+1, 1 ), LDY, A( I, I+1 ), LDA, ZERO,
$ X( 1, I ), 1 )
CALL ZGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
$ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL ZGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
$ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
$ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL ZSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
CALL ZLACGV( N-I, A( I, I+1 ), LDA )
END IF
10 CONTINUE
ELSE
*
* Reduce to lower bidiagonal form
*
DO 20 I = 1, NB
*
* Update A(i,i:n)
*
CALL ZLACGV( N-I+1, A( I, I ), LDA )
CALL ZLACGV( I-1, A( I, 1 ), LDA )
CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
$ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
CALL ZLACGV( I-1, A( I, 1 ), LDA )
CALL ZLACGV( I-1, X( I, 1 ), LDX )
CALL ZGEMV( 'Conjugate transpose', I-1, N-I+1, -ONE,
$ A( 1, I ), LDA, X( I, 1 ), LDX, ONE, A( I, I ),
$ LDA )
CALL ZLACGV( I-1, X( I, 1 ), LDX )
*
* Generate reflection P(i) to annihilate A(i,i+1:n)
*
ALPHA = A( I, I )
CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA,
$ TAUP( I ) )
D( I ) = ALPHA
IF( I.LT.M ) THEN
A( I, I ) = ONE
*
* Compute X(i+1:m,i)
*
CALL ZGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
$ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
CALL ZGEMV( 'Conjugate transpose', N-I+1, I-1, ONE,
$ Y( I, 1 ), LDY, A( I, I ), LDA, ZERO,
$ X( 1, I ), 1 )
CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
$ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL ZGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
$ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
$ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL ZSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
CALL ZLACGV( N-I+1, A( I, I ), LDA )
*
* Update A(i+1:m,i)
*
CALL ZLACGV( I-1, Y( I, 1 ), LDY )
CALL ZGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
$ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
CALL ZLACGV( I-1, Y( I, 1 ), LDY )
CALL ZGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
$ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
*
* Generate reflection Q(i) to annihilate A(i+2:m,i)
*
ALPHA = A( I+1, I )
CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1,
$ TAUQ( I ) )
E( I ) = ALPHA
A( I+1, I ) = ONE
*
* Compute Y(i+1:n,i)
*
CALL ZGEMV( 'Conjugate transpose', M-I, N-I, ONE,
$ A( I+1, I+1 ), LDA, A( I+1, I ), 1, ZERO,
$ Y( I+1, I ), 1 )
CALL ZGEMV( 'Conjugate transpose', M-I, I-1, ONE,
$ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO,
$ Y( 1, I ), 1 )
CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
$ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL ZGEMV( 'Conjugate transpose', M-I, I, ONE,
$ X( I+1, 1 ), LDX, A( I+1, I ), 1, ZERO,
$ Y( 1, I ), 1 )
CALL ZGEMV( 'Conjugate transpose', I, N-I, -ONE,
$ A( 1, I+1 ), LDA, Y( 1, I ), 1, ONE,
$ Y( I+1, I ), 1 )
CALL ZSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
ELSE
CALL ZLACGV( N-I+1, A( I, I ), LDA )
END IF
20 CONTINUE
END IF
RETURN
*
* End of ZLABRD
*
END
|
