LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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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
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorbdb4.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorbdb4.f">
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 orthonormal 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 (M-Q)
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-Q)
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 unbdb4
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
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
dlarf
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
dlarfgp
subroutine dlarfgp(n, alpha, x, incx, tau)
DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition dlarfgp.f:104
drot
subroutine drot(n, dx, incx, dy, incy, c, s)
DROT
Definition drot.f:92
dscal
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
dorbdb4
subroutine dorbdb4(m, p, q, x11, ldx11, x21, ldx21, theta, phi, taup1, taup2, tauq1, phantom, work, lwork, info)
DORBDB4
Definition dorbdb4.f:213
dorbdb5
subroutine dorbdb5(m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
DORBDB5
Definition dorbdb5.f:156