LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
cunbdb1.f
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1 *> \brief \b CUNBDB1
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CUNBDB1( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
22 * TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
26 * ..
27 * .. Array Arguments ..
28 * REAL PHI(*), THETA(*)
29 * COMPLEX TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
30 * \$ X11(LDX11,*), X21(LDX21,*)
31 * ..
32 *
33 *
34 *> \par Purpose:
35 *> =============
36 *>
37 *>\verbatim
38 *>
39 *> CUNBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny
40 *> matrix X with orthonomal columns:
41 *>
42 *> [ B11 ]
43 *> [ X11 ] [ P1 | ] [ 0 ]
44 *> [-----] = [---------] [-----] Q1**T .
45 *> [ X21 ] [ | P2 ] [ B21 ]
46 *> [ 0 ]
47 *>
48 *> X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,
49 *> M-P, or M-Q. Routines CUNBDB2, CUNBDB3, and CUNBDB4 handle cases in
50 *> which Q is not the minimum dimension.
51 *>
52 *> The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
53 *> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
54 *> Householder vectors.
55 *>
56 *> B11 and B12 are Q-by-Q bidiagonal matrices represented implicitly by
57 *> angles THETA, PHI.
58 *>
59 *>\endverbatim
60 *
61 * Arguments:
62 * ==========
63 *
64 *> \param[in] M
65 *> \verbatim
66 *> M is INTEGER
67 *> The number of rows X11 plus the number of rows in X21.
68 *> \endverbatim
69 *>
70 *> \param[in] P
71 *> \verbatim
72 *> P is INTEGER
73 *> The number of rows in X11. 0 <= P <= M.
74 *> \endverbatim
75 *>
76 *> \param[in] Q
77 *> \verbatim
78 *> Q is INTEGER
79 *> The number of columns in X11 and X21. 0 <= Q <=
80 *> MIN(P,M-P,M-Q).
81 *> \endverbatim
82 *>
83 *> \param[in,out] X11
84 *> \verbatim
85 *> X11 is COMPLEX array, dimension (LDX11,Q)
86 *> On entry, the top block of the matrix X to be reduced. On
87 *> exit, the columns of tril(X11) specify reflectors for P1 and
88 *> the rows of triu(X11,1) specify reflectors for Q1.
89 *> \endverbatim
90 *>
91 *> \param[in] LDX11
92 *> \verbatim
93 *> LDX11 is INTEGER
94 *> The leading dimension of X11. LDX11 >= P.
95 *> \endverbatim
96 *>
97 *> \param[in,out] X21
98 *> \verbatim
99 *> X21 is COMPLEX array, dimension (LDX21,Q)
100 *> On entry, the bottom block of the matrix X to be reduced. On
101 *> exit, the columns of tril(X21) specify reflectors for P2.
102 *> \endverbatim
103 *>
104 *> \param[in] LDX21
105 *> \verbatim
106 *> LDX21 is INTEGER
107 *> The leading dimension of X21. LDX21 >= M-P.
108 *> \endverbatim
109 *>
110 *> \param[out] THETA
111 *> \verbatim
112 *> THETA is REAL array, dimension (Q)
113 *> The entries of the bidiagonal blocks B11, B21 are defined by
114 *> THETA and PHI. See Further Details.
115 *> \endverbatim
116 *>
117 *> \param[out] PHI
118 *> \verbatim
119 *> PHI is REAL array, dimension (Q-1)
120 *> The entries of the bidiagonal blocks B11, B21 are defined by
121 *> THETA and PHI. See Further Details.
122 *> \endverbatim
123 *>
124 *> \param[out] TAUP1
125 *> \verbatim
126 *> TAUP1 is COMPLEX array, dimension (P)
127 *> The scalar factors of the elementary reflectors that define
128 *> P1.
129 *> \endverbatim
130 *>
131 *> \param[out] TAUP2
132 *> \verbatim
133 *> TAUP2 is COMPLEX array, dimension (M-P)
134 *> The scalar factors of the elementary reflectors that define
135 *> P2.
136 *> \endverbatim
137 *>
138 *> \param[out] TAUQ1
139 *> \verbatim
140 *> TAUQ1 is COMPLEX array, dimension (Q)
141 *> The scalar factors of the elementary reflectors that define
142 *> Q1.
143 *> \endverbatim
144 *>
145 *> \param[out] WORK
146 *> \verbatim
147 *> WORK is COMPLEX array, dimension (LWORK)
148 *> \endverbatim
149 *>
150 *> \param[in] LWORK
151 *> \verbatim
152 *> LWORK is INTEGER
153 *> The dimension of the array WORK. LWORK >= M-Q.
154 *>
155 *> If LWORK = -1, then a workspace query is assumed; the routine
156 *> only calculates the optimal size of the WORK array, returns
157 *> this value as the first entry of the WORK array, and no error
158 *> message related to LWORK is issued by XERBLA.
159 *> \endverbatim
160 *>
161 *> \param[out] INFO
162 *> \verbatim
163 *> INFO is INTEGER
164 *> = 0: successful exit.
165 *> < 0: if INFO = -i, the i-th argument had an illegal value.
166 *> \endverbatim
167 *
168 * Authors:
169 * ========
170 *
171 *> \author Univ. of Tennessee
172 *> \author Univ. of California Berkeley
173 *> \author Univ. of Colorado Denver
174 *> \author NAG Ltd.
175 *
176 *> \date July 2012
177 *
178 *> \ingroup complexOTHERcomputational
179 *
180 *> \par Further Details:
181 * =====================
182
183 *> \verbatim
184 *>
185 *> The upper-bidiagonal blocks B11, B21 are represented implicitly by
186 *> angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
187 *> in each bidiagonal band is a product of a sine or cosine of a THETA
188 *> with a sine or cosine of a PHI. See [1] or CUNCSD for details.
189 *>
190 *> P1, P2, and Q1 are represented as products of elementary reflectors.
191 *> See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR
192 *> and CUNGLQ.
193 *> \endverbatim
194 *
195 *> \par References:
196 * ================
197 *>
198 *> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
199 *> Algorithms, 50(1):33-65, 2009.
200 *>
201 * =====================================================================
202  SUBROUTINE cunbdb1( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
203  \$ taup1, taup2, tauq1, work, lwork, info )
204 *
205 * -- LAPACK computational routine (version 3.6.1) --
206 * -- LAPACK is a software package provided by Univ. of Tennessee, --
207 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
208 * July 2012
209 *
210 * .. Scalar Arguments ..
211  INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
212 * ..
213 * .. Array Arguments ..
214  REAL PHI(*), THETA(*)
215  COMPLEX TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
216  \$ x11(ldx11,*), x21(ldx21,*)
217 * ..
218 *
219 * ====================================================================
220 *
221 * .. Parameters ..
222  COMPLEX ONE
223  parameter ( one = (1.0e0,0.0e0) )
224 * ..
225 * .. Local Scalars ..
226  REAL C, S
227  INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
228  \$ lworkmin, lworkopt
229  LOGICAL LQUERY
230 * ..
231 * .. External Subroutines ..
232  EXTERNAL clarf, clarfgp, cunbdb5, csrot, xerbla
233  EXTERNAL clacgv
234 * ..
235 * .. External Functions ..
236  REAL SCNRM2
237  EXTERNAL scnrm2
238 * ..
239 * .. Intrinsic Function ..
240  INTRINSIC atan2, cos, max, sin, sqrt
241 * ..
242 * .. Executable Statements ..
243 *
244 * Test input arguments
245 *
246  info = 0
247  lquery = lwork .EQ. -1
248 *
249  IF( m .LT. 0 ) THEN
250  info = -1
251  ELSE IF( p .LT. q .OR. m-p .LT. q ) THEN
252  info = -2
253  ELSE IF( q .LT. 0 .OR. m-q .LT. q ) THEN
254  info = -3
255  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
256  info = -5
257  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
258  info = -7
259  END IF
260 *
261 * Compute workspace
262 *
263  IF( info .EQ. 0 ) THEN
264  ilarf = 2
265  llarf = max( p-1, m-p-1, q-1 )
266  iorbdb5 = 2
267  lorbdb5 = q-2
268  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
269  lworkmin = lworkopt
270  work(1) = lworkopt
271  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
272  info = -14
273  END IF
274  END IF
275  IF( info .NE. 0 ) THEN
276  CALL xerbla( 'CUNBDB1', -info )
277  RETURN
278  ELSE IF( lquery ) THEN
279  RETURN
280  END IF
281 *
282 * Reduce columns 1, ..., Q of X11 and X21
283 *
284  DO i = 1, q
285 *
286  CALL clarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
287  CALL clarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
288  theta(i) = atan2( REAL( X21(I,I) ), REAL( X11(I,I) ) )
289  c = cos( theta(i) )
290  s = sin( theta(i) )
291  x11(i,i) = one
292  x21(i,i) = one
293  CALL clarf( 'L', p-i+1, q-i, x11(i,i), 1, conjg(taup1(i)),
294  \$ x11(i,i+1), ldx11, work(ilarf) )
295  CALL clarf( 'L', m-p-i+1, q-i, x21(i,i), 1, conjg(taup2(i)),
296  \$ x21(i,i+1), ldx21, work(ilarf) )
297 *
298  IF( i .LT. q ) THEN
299  CALL csrot( q-i, x11(i,i+1), ldx11, x21(i,i+1), ldx21, c,
300  \$ s )
301  CALL clacgv( q-i, x21(i,i+1), ldx21 )
302  CALL clarfgp( q-i, x21(i,i+1), x21(i,i+2), ldx21, tauq1(i) )
303  s = REAL( X21(I,I+1) )
304  x21(i,i+1) = one
305  CALL clarf( 'R', p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
306  \$ x11(i+1,i+1), ldx11, work(ilarf) )
307  CALL clarf( 'R', m-p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
308  \$ x21(i+1,i+1), ldx21, work(ilarf) )
309  CALL clacgv( q-i, x21(i,i+1), ldx21 )
310  c = sqrt( scnrm2( p-i, x11(i+1,i+1), 1 )**2
311  \$ + scnrm2( m-p-i, x21(i+1,i+1), 1 )**2 )
312  phi(i) = atan2( s, c )
313  CALL cunbdb5( p-i, m-p-i, q-i-1, x11(i+1,i+1), 1,
314  \$ x21(i+1,i+1), 1, x11(i+1,i+2), ldx11,
315  \$ x21(i+1,i+2), ldx21, work(iorbdb5), lorbdb5,
316  \$ childinfo )
317  END IF
318 *
319  END DO
320 *
321  RETURN
322 *
323 * End of CUNBDB1
324 *
325  END
326
subroutine cunbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
CUNBDB5
Definition: cunbdb5.f:158
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine cunbdb1(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO)
CUNBDB1
Definition: cunbdb1.f:204
subroutine clarfgp(N, ALPHA, X, INCX, TAU)
CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: clarfgp.f:106
subroutine clarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
CLARF applies an elementary reflector to a general rectangular matrix.
Definition: clarf.f:130
subroutine csrot(N, CX, INCX, CY, INCY, C, S)
CSROT
Definition: csrot.f:100
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:76