LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
dorbdb4.f
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1 *> \brief \b DORBDB4
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DORBDB4 + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorbdb4.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DORBDB4( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
22 * TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK,
23 * INFO )
24 *
25 * .. Scalar Arguments ..
26 * INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
27 * ..
28 * .. Array Arguments ..
29 * DOUBLE PRECISION PHI(*), THETA(*)
30 * DOUBLE PRECISION PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*),
31 * $ WORK(*), X11(LDX11,*), X21(LDX21,*)
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *>\verbatim
39 *>
40 *> DORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
41 *> matrix X with orthonomal columns:
42 *>
43 *> [ B11 ]
44 *> [ X11 ] [ P1 | ] [ 0 ]
45 *> [-----] = [---------] [-----] Q1**T .
46 *> [ X21 ] [ | P2 ] [ B21 ]
47 *> [ 0 ]
48 *>
49 *> X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
50 *> M-P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in
51 *> which M-Q is not the minimum dimension.
52 *>
53 *> The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
54 *> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
55 *> Householder vectors.
56 *>
57 *> B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
58 *> implicitly by angles THETA, PHI.
59 *>
60 *>\endverbatim
61 *
62 * Arguments:
63 * ==========
64 *
65 *> \param[in] M
66 *> \verbatim
67 *> M is INTEGER
68 *> The number of rows X11 plus the number of rows in X21.
69 *> \endverbatim
70 *>
71 *> \param[in] P
72 *> \verbatim
73 *> P is INTEGER
74 *> The number of rows in X11. 0 <= P <= M.
75 *> \endverbatim
76 *>
77 *> \param[in] Q
78 *> \verbatim
79 *> Q is INTEGER
80 *> The number of columns in X11 and X21. 0 <= Q <= M and
81 *> M-Q <= min(P,M-P,Q).
82 *> \endverbatim
83 *>
84 *> \param[in,out] X11
85 *> \verbatim
86 *> X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
87 *> On entry, the top block of the matrix X to be reduced. On
88 *> exit, the columns of tril(X11) specify reflectors for P1 and
89 *> the rows of triu(X11,1) specify reflectors for Q1.
90 *> \endverbatim
91 *>
92 *> \param[in] LDX11
93 *> \verbatim
94 *> LDX11 is INTEGER
95 *> The leading dimension of X11. LDX11 >= P.
96 *> \endverbatim
97 *>
98 *> \param[in,out] X21
99 *> \verbatim
100 *> X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
101 *> On entry, the bottom block of the matrix X to be reduced. On
102 *> exit, the columns of tril(X21) specify reflectors for P2.
103 *> \endverbatim
104 *>
105 *> \param[in] LDX21
106 *> \verbatim
107 *> LDX21 is INTEGER
108 *> The leading dimension of X21. LDX21 >= M-P.
109 *> \endverbatim
110 *>
111 *> \param[out] THETA
112 *> \verbatim
113 *> THETA is DOUBLE PRECISION array, dimension (Q)
114 *> The entries of the bidiagonal blocks B11, B21 are defined by
115 *> THETA and PHI. See Further Details.
116 *> \endverbatim
117 *>
118 *> \param[out] PHI
119 *> \verbatim
120 *> PHI is DOUBLE PRECISION array, dimension (Q-1)
121 *> The entries of the bidiagonal blocks B11, B21 are defined by
122 *> THETA and PHI. See Further Details.
123 *> \endverbatim
124 *>
125 *> \param[out] TAUP1
126 *> \verbatim
127 *> TAUP1 is DOUBLE PRECISION array, dimension (P)
128 *> The scalar factors of the elementary reflectors that define
129 *> P1.
130 *> \endverbatim
131 *>
132 *> \param[out] TAUP2
133 *> \verbatim
134 *> TAUP2 is DOUBLE PRECISION array, dimension (M-P)
135 *> The scalar factors of the elementary reflectors that define
136 *> P2.
137 *> \endverbatim
138 *>
139 *> \param[out] TAUQ1
140 *> \verbatim
141 *> TAUQ1 is DOUBLE PRECISION array, dimension (Q)
142 *> The scalar factors of the elementary reflectors that define
143 *> Q1.
144 *> \endverbatim
145 *>
146 *> \param[out] PHANTOM
147 *> \verbatim
148 *> PHANTOM is DOUBLE PRECISION array, dimension (M)
149 *> The routine computes an M-by-1 column vector Y that is
150 *> orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
151 *> PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
152 *> Y(P+1:M), respectively.
153 *> \endverbatim
154 *>
155 *> \param[out] WORK
156 *> \verbatim
157 *> WORK is DOUBLE PRECISION array, dimension (LWORK)
158 *> \endverbatim
159 *>
160 *> \param[in] LWORK
161 *> \verbatim
162 *> LWORK is INTEGER
163 *> The dimension of the array WORK. LWORK >= M-Q.
164 *>
165 *> If LWORK = -1, then a workspace query is assumed; the routine
166 *> only calculates the optimal size of the WORK array, returns
167 *> this value as the first entry of the WORK array, and no error
168 *> message related to LWORK is issued by XERBLA.
169 *> \endverbatim
170 *>
171 *> \param[out] INFO
172 *> \verbatim
173 *> INFO is INTEGER
174 *> = 0: successful exit.
175 *> < 0: if INFO = -i, the i-th argument had an illegal value.
176 *> \endverbatim
177 *
178 * Authors:
179 * ========
180 *
181 *> \author Univ. of Tennessee
182 *> \author Univ. of California Berkeley
183 *> \author Univ. of Colorado Denver
184 *> \author NAG Ltd.
185 *
186 *> \ingroup doubleOTHERcomputational
187 *
188 *> \par Further Details:
189 * =====================
190 *>
191 *> \verbatim
192 *>
193 *> The upper-bidiagonal blocks B11, B21 are represented implicitly by
194 *> angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
195 *> in each bidiagonal band is a product of a sine or cosine of a THETA
196 *> with a sine or cosine of a PHI. See [1] or DORCSD for details.
197 *>
198 *> P1, P2, and Q1 are represented as products of elementary reflectors.
199 *> See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
200 *> and DORGLQ.
201 *> \endverbatim
202 *
203 *> \par References:
204 * ================
205 *>
206 *> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
207 *> Algorithms, 50(1):33-65, 2009.
208 *>
209 * =====================================================================
210  SUBROUTINE dorbdb4( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
211  $ TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK,
212  $ INFO )
213 *
214 * -- LAPACK computational routine --
215 * -- LAPACK is a software package provided by Univ. of Tennessee, --
216 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
217 *
218 * .. Scalar Arguments ..
219  INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
220 * ..
221 * .. Array Arguments ..
222  DOUBLE PRECISION PHI(*), THETA(*)
223  DOUBLE PRECISION PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*),
224  $ work(*), x11(ldx11,*), x21(ldx21,*)
225 * ..
226 *
227 * ====================================================================
228 *
229 * .. Parameters ..
230  DOUBLE PRECISION NEGONE, ONE, ZERO
231  PARAMETER ( NEGONE = -1.0d0, one = 1.0d0, zero = 0.0d0 )
232 * ..
233 * .. Local Scalars ..
234  DOUBLE PRECISION C, S
235  INTEGER CHILDINFO, I, ILARF, IORBDB5, J, LLARF,
236  $ lorbdb5, lworkmin, lworkopt
237  LOGICAL LQUERY
238 * ..
239 * .. External Subroutines ..
240  EXTERNAL dlarf, dlarfgp, dorbdb5, drot, dscal, xerbla
241 * ..
242 * .. External Functions ..
243  DOUBLE PRECISION DNRM2
244  EXTERNAL DNRM2
245 * ..
246 * .. Intrinsic Function ..
247  INTRINSIC atan2, cos, max, sin, sqrt
248 * ..
249 * .. Executable Statements ..
250 *
251 * Test input arguments
252 *
253  info = 0
254  lquery = lwork .EQ. -1
255 *
256  IF( m .LT. 0 ) THEN
257  info = -1
258  ELSE IF( p .LT. m-q .OR. m-p .LT. m-q ) THEN
259  info = -2
260  ELSE IF( q .LT. m-q .OR. q .GT. m ) THEN
261  info = -3
262  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
263  info = -5
264  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
265  info = -7
266  END IF
267 *
268 * Compute workspace
269 *
270  IF( info .EQ. 0 ) THEN
271  ilarf = 2
272  llarf = max( q-1, p-1, m-p-1 )
273  iorbdb5 = 2
274  lorbdb5 = q
275  lworkopt = ilarf + llarf - 1
276  lworkopt = max( lworkopt, iorbdb5 + lorbdb5 - 1 )
277  lworkmin = lworkopt
278  work(1) = lworkopt
279  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
280  info = -14
281  END IF
282  END IF
283  IF( info .NE. 0 ) THEN
284  CALL xerbla( 'DORBDB4', -info )
285  RETURN
286  ELSE IF( lquery ) THEN
287  RETURN
288  END IF
289 *
290 * Reduce columns 1, ..., M-Q of X11 and X21
291 *
292  DO i = 1, m-q
293 *
294  IF( i .EQ. 1 ) THEN
295  DO j = 1, m
296  phantom(j) = zero
297  END DO
298  CALL dorbdb5( p, m-p, q, phantom(1), 1, phantom(p+1), 1,
299  $ x11, ldx11, x21, ldx21, work(iorbdb5),
300  $ lorbdb5, childinfo )
301  CALL dscal( p, negone, phantom(1), 1 )
302  CALL dlarfgp( p, phantom(1), phantom(2), 1, taup1(1) )
303  CALL dlarfgp( m-p, phantom(p+1), phantom(p+2), 1, taup2(1) )
304  theta(i) = atan2( phantom(1), phantom(p+1) )
305  c = cos( theta(i) )
306  s = sin( theta(i) )
307  phantom(1) = one
308  phantom(p+1) = one
309  CALL dlarf( 'L', p, q, phantom(1), 1, taup1(1), x11, ldx11,
310  $ work(ilarf) )
311  CALL dlarf( 'L', m-p, q, phantom(p+1), 1, taup2(1), x21,
312  $ ldx21, work(ilarf) )
313  ELSE
314  CALL dorbdb5( p-i+1, m-p-i+1, q-i+1, x11(i,i-1), 1,
315  $ x21(i,i-1), 1, x11(i,i), ldx11, x21(i,i),
316  $ ldx21, work(iorbdb5), lorbdb5, childinfo )
317  CALL dscal( p-i+1, negone, x11(i,i-1), 1 )
318  CALL dlarfgp( p-i+1, x11(i,i-1), x11(i+1,i-1), 1, taup1(i) )
319  CALL dlarfgp( m-p-i+1, x21(i,i-1), x21(i+1,i-1), 1,
320  $ taup2(i) )
321  theta(i) = atan2( x11(i,i-1), x21(i,i-1) )
322  c = cos( theta(i) )
323  s = sin( theta(i) )
324  x11(i,i-1) = one
325  x21(i,i-1) = one
326  CALL dlarf( 'L', p-i+1, q-i+1, x11(i,i-1), 1, taup1(i),
327  $ x11(i,i), ldx11, work(ilarf) )
328  CALL dlarf( 'L', m-p-i+1, q-i+1, x21(i,i-1), 1, taup2(i),
329  $ x21(i,i), ldx21, work(ilarf) )
330  END IF
331 *
332  CALL drot( q-i+1, x11(i,i), ldx11, x21(i,i), ldx21, s, -c )
333  CALL dlarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
334  c = x21(i,i)
335  x21(i,i) = one
336  CALL dlarf( 'R', p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
337  $ x11(i+1,i), ldx11, work(ilarf) )
338  CALL dlarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
339  $ x21(i+1,i), ldx21, work(ilarf) )
340  IF( i .LT. m-q ) THEN
341  s = sqrt( dnrm2( p-i, x11(i+1,i), 1 )**2
342  $ + dnrm2( m-p-i, x21(i+1,i), 1 )**2 )
343  phi(i) = atan2( s, c )
344  END IF
345 *
346  END DO
347 *
348 * Reduce the bottom-right portion of X11 to [ I 0 ]
349 *
350  DO i = m - q + 1, p
351  CALL dlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
352  x11(i,i) = one
353  CALL dlarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
354  $ x11(i+1,i), ldx11, work(ilarf) )
355  CALL dlarf( 'R', q-p, q-i+1, x11(i,i), ldx11, tauq1(i),
356  $ x21(m-q+1,i), ldx21, work(ilarf) )
357  END DO
358 *
359 * Reduce the bottom-right portion of X21 to [ 0 I ]
360 *
361  DO i = p + 1, q
362  CALL dlarfgp( q-i+1, x21(m-q+i-p,i), x21(m-q+i-p,i+1), ldx21,
363  $ tauq1(i) )
364  x21(m-q+i-p,i) = one
365  CALL dlarf( 'R', q-i, q-i+1, x21(m-q+i-p,i), ldx21, tauq1(i),
366  $ x21(m-q+i-p+1,i), ldx21, work(ilarf) )
367  END DO
368 *
369  RETURN
370 *
371 * End of DORBDB4
372 *
373  END
374 
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:92
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition: dlarf.f:124
subroutine dlarfgp(N, ALPHA, X, INCX, TAU)
DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: dlarfgp.f:104
subroutine dorbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
DORBDB5
Definition: dorbdb5.f:156
subroutine dorbdb4(M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI, TAUP1, TAUP2, TAUQ1, PHANTOM, WORK, LWORK, INFO)
DORBDB4
Definition: dorbdb4.f:213