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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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*> \ingroup complexOTHERcomputational
177*
178*> \par Further Details:
179*  =====================
180
181*> \verbatim
182*>
183*>  The upper-bidiagonal blocks B11, B21 are represented implicitly by
184*>  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
185*>  in each bidiagonal band is a product of a sine or cosine of a THETA
186*>  with a sine or cosine of a PHI. See [1] or CUNCSD for details.
187*>
188*>  P1, P2, and Q1 are represented as products of elementary reflectors.
189*>  See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR
190*>  and CUNGLQ.
191*> \endverbatim
192*
193*> \par References:
194*  ================
195*>
196*>  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
197*>      Algorithms, 50(1):33-65, 2009.
198*>
199*  =====================================================================
200      SUBROUTINE CUNBDB1( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
201     $                    TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
202*
203*  -- LAPACK computational routine --
204*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
205*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
206*
207*     .. Scalar Arguments ..
208      INTEGER            INFO, LWORK, M, P, Q, LDX11, LDX21
209*     ..
210*     .. Array Arguments ..
211      REAL               PHI(*), THETA(*)
212      COMPLEX            TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
213     $                   X11(LDX11,*), X21(LDX21,*)
214*     ..
215*
216*  ====================================================================
217*
218*     .. Parameters ..
219      COMPLEX            ONE
220      PARAMETER          ( ONE = (1.0E0,0.0E0) )
221*     ..
222*     .. Local Scalars ..
223      REAL               C, S
224      INTEGER            CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
225     $                   LWORKMIN, LWORKOPT
226      LOGICAL            LQUERY
227*     ..
228*     .. External Subroutines ..
229      EXTERNAL           CLARF, CLARFGP, CUNBDB5, CSROT, XERBLA
230      EXTERNAL           CLACGV
231*     ..
232*     .. External Functions ..
233      REAL               SCNRM2
234      EXTERNAL           SCNRM2
235*     ..
236*     .. Intrinsic Function ..
237      INTRINSIC          ATAN2, COS, MAX, SIN, SQRT
238*     ..
239*     .. Executable Statements ..
240*
241*     Test input arguments
242*
243      INFO = 0
244      LQUERY = LWORK .EQ. -1
245*
246      IF( M .LT. 0 ) THEN
247         INFO = -1
248      ELSE IF( P .LT. Q .OR. M-P .LT. Q ) THEN
249         INFO = -2
250      ELSE IF( Q .LT. 0 .OR. M-Q .LT. Q ) THEN
251         INFO = -3
252      ELSE IF( LDX11 .LT. MAX( 1, P ) ) THEN
253         INFO = -5
254      ELSE IF( LDX21 .LT. MAX( 1, M-P ) ) THEN
255         INFO = -7
256      END IF
257*
258*     Compute workspace
259*
260      IF( INFO .EQ. 0 ) THEN
261         ILARF = 2
262         LLARF = MAX( P-1, M-P-1, Q-1 )
263         IORBDB5 = 2
264         LORBDB5 = Q-2
265         LWORKOPT = MAX( ILARF+LLARF-1, IORBDB5+LORBDB5-1 )
266         LWORKMIN = LWORKOPT
267         WORK(1) = LWORKOPT
268         IF( LWORK .LT. LWORKMIN .AND. .NOT.LQUERY ) THEN
269           INFO = -14
270         END IF
271      END IF
272      IF( INFO .NE. 0 ) THEN
273         CALL XERBLA( 'CUNBDB1', -INFO )
274         RETURN
275      ELSE IF( LQUERY ) THEN
276         RETURN
277      END IF
278*
279*     Reduce columns 1, ..., Q of X11 and X21
280*
281      DO I = 1, Q
282*
283         CALL CLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
284         CALL CLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) )
285         THETA(I) = ATAN2( REAL( X21(I,I) ), REAL( X11(I,I) ) )
286         C = COS( THETA(I) )
287         S = SIN( THETA(I) )
288         X11(I,I) = ONE
289         X21(I,I) = ONE
290         CALL CLARF( 'L', P-I+1, Q-I, X11(I,I), 1, CONJG(TAUP1(I)),
291     $               X11(I,I+1), LDX11, WORK(ILARF) )
292         CALL CLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1, CONJG(TAUP2(I)),
293     $               X21(I,I+1), LDX21, WORK(ILARF) )
294*
295         IF( I .LT. Q ) THEN
296            CALL CSROT( Q-I, X11(I,I+1), LDX11, X21(I,I+1), LDX21, C,
297     $                  S )
298            CALL CLACGV( Q-I, X21(I,I+1), LDX21 )
299            CALL CLARFGP( Q-I, X21(I,I+1), X21(I,I+2), LDX21, TAUQ1(I) )
300            S = REAL( X21(I,I+1) )
301            X21(I,I+1) = ONE
302            CALL CLARF( 'R', P-I, Q-I, X21(I,I+1), LDX21, TAUQ1(I),
303     $                  X11(I+1,I+1), LDX11, WORK(ILARF) )
304            CALL CLARF( 'R', M-P-I, Q-I, X21(I,I+1), LDX21, TAUQ1(I),
305     $                  X21(I+1,I+1), LDX21, WORK(ILARF) )
306            CALL CLACGV( Q-I, X21(I,I+1), LDX21 )
307            C = SQRT( SCNRM2( P-I, X11(I+1,I+1), 1 )**2
308     $              + SCNRM2( M-P-I, X21(I+1,I+1), 1 )**2 )
309            PHI(I) = ATAN2( S, C )
310            CALL CUNBDB5( P-I, M-P-I, Q-I-1, X11(I+1,I+1), 1,
311     $                    X21(I+1,I+1), 1, X11(I+1,I+2), LDX11,
312     $                    X21(I+1,I+2), LDX21, WORK(IORBDB5), LORBDB5,
313     $                    CHILDINFO )
314         END IF
315*
316      END DO
317*
318      RETURN
319*
320*     End of CUNBDB1
321*
322      END
323
324