1*> \brief \b SORBDB1
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 SORBDB1( 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*       REAL               TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
30*      $                   X11(LDX11,*), X21(LDX21,*)
31*       ..
32*
33*
34*> \par Purpose:
35*> =============
36*>
37*>\verbatim
38*>
39*> SORBDB1 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 SORBDB2, SORBDB3, and SORBDB4 handle cases in
50*> which Q is not the minimum dimension.
51*>
52*> The orthogonal 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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*
169*  Authors:
170*  ========
171*
172*> \author Univ. of Tennessee
173*> \author Univ. of California Berkeley
174*> \author Univ. of Colorado Denver
175*> \author NAG Ltd.
176*
177*> \date July 2012
178*
179*> \ingroup realOTHERcomputational
180*
181*> \par Further Details:
182*  =====================
183*>
184*> \verbatim
185*>
186*>  The upper-bidiagonal blocks B11, B21 are represented implicitly by
187*>  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
188*>  in each bidiagonal band is a product of a sine or cosine of a THETA
189*>  with a sine or cosine of a PHI. See [1] or SORCSD for details.
190*>
191*>  P1, P2, and Q1 are represented as products of elementary reflectors.
192*>  See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
193*>  and SORGLQ.
194*> \endverbatim
195*
196*> \par References:
197*  ================
198*>
199*>  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
200*>      Algorithms, 50(1):33-65, 2009.
201*>
202*  =====================================================================
203      SUBROUTINE SORBDB1( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
204     $                    TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
205*
206*  -- LAPACK computational routine (version 3.5.0) --
207*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
208*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
209*     July 2012
210*
211*     .. Scalar Arguments ..
212      INTEGER            INFO, LWORK, M, P, Q, LDX11, LDX21
213*     ..
214*     .. Array Arguments ..
215      REAL               PHI(*), THETA(*)
216      REAL               TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
217     $                   X11(LDX11,*), X21(LDX21,*)
218*     ..
219*
220*  ====================================================================
221*
222*     .. Parameters ..
223      REAL               ONE
224      PARAMETER          ( ONE = 1.0E0 )
225*     ..
226*     .. Local Scalars ..
227      REAL               C, S
228      INTEGER            CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
229     $                   LWORKMIN, LWORKOPT
230      LOGICAL            LQUERY
231*     ..
232*     .. External Subroutines ..
233      EXTERNAL           SLARF, SLARFGP, SORBDB5, SROT, XERBLA
234*     ..
235*     .. External Functions ..
236      REAL               SNRM2
237      EXTERNAL           SNRM2
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( 'SORBDB1', -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 SLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
287         CALL SLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) )
288         THETA(I) = ATAN2( X21(I,I), 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 SLARF( 'L', P-I+1, Q-I, X11(I,I), 1, TAUP1(I), X11(I,I+1),
294     $               LDX11, WORK(ILARF) )
295         CALL SLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1, TAUP2(I),
296     $               X21(I,I+1), LDX21, WORK(ILARF) )
297*
298         IF( I .LT. Q ) THEN
299            CALL SROT( Q-I, X11(I,I+1), LDX11, X21(I,I+1), LDX21, C, S )
300            CALL SLARFGP( Q-I, X21(I,I+1), X21(I,I+2), LDX21, TAUQ1(I) )
301            S = X21(I,I+1)
302            X21(I,I+1) = ONE
303            CALL SLARF( 'R', P-I, Q-I, X21(I,I+1), LDX21, TAUQ1(I),
304     $                  X11(I+1,I+1), LDX11, WORK(ILARF) )
305            CALL SLARF( 'R', M-P-I, Q-I, X21(I,I+1), LDX21, TAUQ1(I),
306     $                  X21(I+1,I+1), LDX21, WORK(ILARF) )
307            C = SQRT( SNRM2( P-I, X11(I+1,I+1), 1, X11(I+1,I+1),
308     $          1 )**2 + SNRM2( M-P-I, X21(I+1,I+1), 1, X21(I+1,I+1),
309     $          1 )**2 )
310            PHI(I) = ATAN2( S, C )
311            CALL SORBDB5( P-I, M-P-I, Q-I-1, X11(I+1,I+1), 1,
312     $                    X21(I+1,I+1), 1, X11(I+1,I+2), LDX11,
313     $                    X21(I+1,I+2), LDX21, WORK(IORBDB5), LORBDB5,
314     $                    CHILDINFO )
315         END IF
316*
317      END DO
318*
319      RETURN
320*
321*     End of SORBDB1
322*
323      END
324
325