summaryrefslogtreecommitdiff
path: root/SRC/clahr2.f
blob: 50547f2cd849488e5411e058f2d79f2a363b8243 (plain)
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
*> \brief \b CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CLAHR2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/clahr2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/clahr2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/clahr2.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*       .. Scalar Arguments ..
*       INTEGER            K, LDA, LDT, LDY, N, NB
*       ..
*       .. Array Arguments ..
*       COMPLEX            A( LDA, * ), T( LDT, NB ), TAU( NB ),
*      $                   Y( LDY, NB )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
*> matrix A so that elements below the k-th subdiagonal are zero. The
*> reduction is performed by an unitary similarity transformation
*> Q**H * A * Q. The routine returns the matrices V and T which determine
*> Q as a block reflector I - V*T*v**H, and also the matrix Y = A * V * T.
*>
*> This is an auxiliary routine called by CGEHRD.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The offset for the reduction. Elements below the k-th
*>          subdiagonal in the first NB columns are reduced to zero.
*>          K < N.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*>          NB is INTEGER
*>          The number of columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX array, dimension (LDA,N-K+1)
*>          On entry, the n-by-(n-k+1) general matrix A.
*>          On exit, the elements on and above the k-th subdiagonal in
*>          the first NB columns are overwritten with the corresponding
*>          elements of the reduced matrix; the elements below the k-th
*>          subdiagonal, with the array TAU, represent the matrix Q as a
*>          product of elementary reflectors. The other columns of A are
*>          unchanged. See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is COMPLEX array, dimension (NB)
*>          The scalar factors of the elementary reflectors. See Further
*>          Details.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is COMPLEX array, dimension (LDT,NB)
*>          The upper triangular matrix T.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of the array T.  LDT >= NB.
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*>          Y is COMPLEX array, dimension (LDY,NB)
*>          The n-by-nb matrix Y.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*>          LDY is INTEGER
*>          The leading dimension of the array Y. LDY >= N.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complexOTHERauxiliary
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  The matrix Q is represented as a product of nb elementary reflectors
*>
*>     Q = H(1) H(2) . . . H(nb).
*>
*>  Each H(i) has the form
*>
*>     H(i) = I - tau * v * v**H
*>
*>  where tau is a complex scalar, and v is a complex vector with
*>  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
*>  A(i+k+1:n,i), and tau in TAU(i).
*>
*>  The elements of the vectors v together form the (n-k+1)-by-nb matrix
*>  V which is needed, with T and Y, to apply the transformation to the
*>  unreduced part of the matrix, using an update of the form:
*>  A := (I - V*T*V**H) * (A - Y*V**H).
*>
*>  The contents of A on exit are illustrated by the following example
*>  with n = 7, k = 3 and nb = 2:
*>
*>     ( a   a   a   a   a )
*>     ( a   a   a   a   a )
*>     ( a   a   a   a   a )
*>     ( h   h   a   a   a )
*>     ( v1  h   a   a   a )
*>     ( v1  v2  a   a   a )
*>     ( v1  v2  a   a   a )
*>
*>  where a denotes an element of the original matrix A, h denotes a
*>  modified element of the upper Hessenberg matrix H, and vi denotes an
*>  element of the vector defining H(i).
*>
*>  This subroutine is a slight modification of LAPACK-3.0's DLAHRD
*>  incorporating improvements proposed by Quintana-Orti and Van de
*>  Gejin. Note that the entries of A(1:K,2:NB) differ from those
*>  returned by the original LAPACK-3.0's DLAHRD routine. (This
*>  subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
*> \endverbatim
*
*> \par References:
*  ================
*>
*>  Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
*>  performance of reduction to Hessenberg form," ACM Transactions on
*>  Mathematical Software, 32(2):180-194, June 2006.
*>
*  =====================================================================
      SUBROUTINE CLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
*
*  -- LAPACK auxiliary routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      INTEGER            K, LDA, LDT, LDY, N, NB
*     ..
*     .. Array Arguments ..
      COMPLEX            A( LDA, * ), T( LDT, NB ), TAU( NB ),
     $                   Y( LDY, NB )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX            ZERO, ONE
      PARAMETER          ( ZERO = ( 0.0E+0, 0.0E+0 ),
     $                     ONE = ( 1.0E+0, 0.0E+0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER            I
      COMPLEX            EI
*     ..
*     .. External Subroutines ..
      EXTERNAL           CAXPY, CCOPY, CGEMM, CGEMV, CLACPY,
     $                   CLARFG, CSCAL, CTRMM, CTRMV, CLACGV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MIN
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( N.LE.1 )
     $   RETURN
*
      DO 10 I = 1, NB
         IF( I.GT.1 ) THEN
*
*           Update A(K+1:N,I)
*
*           Update I-th column of A - Y * V**H
*
            CALL CLACGV( I-1, A( K+I-1, 1 ), LDA )
            CALL CGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY,
     $                  A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 )
            CALL CLACGV( I-1, A( K+I-1, 1 ), LDA )
*
*           Apply I - V * T**H * V**H to this column (call it b) from the
*           left, using the last column of T as workspace
*
*           Let  V = ( V1 )   and   b = ( b1 )   (first I-1 rows)
*                    ( V2 )             ( b2 )
*
*           where V1 is unit lower triangular
*
*           w := V1**H * b1
*
            CALL CCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 )
            CALL CTRMV( 'Lower', 'Conjugate transpose', 'UNIT',
     $                  I-1, A( K+1, 1 ),
     $                  LDA, T( 1, NB ), 1 )
*
*           w := w + V2**H * b2
*
            CALL CGEMV( 'Conjugate transpose', N-K-I+1, I-1,
     $                  ONE, A( K+I, 1 ),
     $                  LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 )
*
*           w := T**H * w
*
            CALL CTRMV( 'Upper', 'Conjugate transpose', 'NON-UNIT',
     $                  I-1, T, LDT,
     $                  T( 1, NB ), 1 )
*
*           b2 := b2 - V2*w
*
            CALL CGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE,
     $                  A( K+I, 1 ),
     $                  LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 )
*
*           b1 := b1 - V1*w
*
            CALL CTRMV( 'Lower', 'NO TRANSPOSE',
     $                  'UNIT', I-1,
     $                  A( K+1, 1 ), LDA, T( 1, NB ), 1 )
            CALL CAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 )
*
            A( K+I-1, I-1 ) = EI
         END IF
*
*        Generate the elementary reflector H(I) to annihilate
*        A(K+I+1:N,I)
*
         CALL CLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1,
     $                TAU( I ) )
         EI = A( K+I, I )
         A( K+I, I ) = ONE
*
*        Compute  Y(K+1:N,I)
*
         CALL CGEMV( 'NO TRANSPOSE', N-K, N-K-I+1,
     $               ONE, A( K+1, I+1 ),
     $               LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 )
         CALL CGEMV( 'Conjugate transpose', N-K-I+1, I-1,
     $               ONE, A( K+I, 1 ), LDA,
     $               A( K+I, I ), 1, ZERO, T( 1, I ), 1 )
         CALL CGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE,
     $               Y( K+1, 1 ), LDY,
     $               T( 1, I ), 1, ONE, Y( K+1, I ), 1 )
         CALL CSCAL( N-K, TAU( I ), Y( K+1, I ), 1 )
*
*        Compute T(1:I,I)
*
         CALL CSCAL( I-1, -TAU( I ), T( 1, I ), 1 )
         CALL CTRMV( 'Upper', 'No Transpose', 'NON-UNIT',
     $               I-1, T, LDT,
     $               T( 1, I ), 1 )
         T( I, I ) = TAU( I )
*
   10 CONTINUE
      A( K+NB, NB ) = EI
*
*     Compute Y(1:K,1:NB)
*
      CALL CLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY )
      CALL CTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE',
     $            'UNIT', K, NB,
     $            ONE, A( K+1, 1 ), LDA, Y, LDY )
      IF( N.GT.K+NB )
     $   CALL CGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K,
     $               NB, N-K-NB, ONE,
     $               A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y,
     $               LDY )
      CALL CTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE',
     $            'NON-UNIT', K, NB,
     $            ONE, T, LDT, Y, LDY )
*
      RETURN
*
*     End of CLAHR2
*
      END