LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dorbdb3()

 subroutine dorbdb3 ( integer m, integer p, integer q, double precision, dimension(ldx11,*) x11, integer ldx11, double precision, dimension(ldx21,*) x21, integer ldx21, double precision, dimension(*) theta, double precision, dimension(*) phi, double precision, dimension(*) taup1, double precision, dimension(*) taup2, double precision, dimension(*) tauq1, double precision, dimension(*) work, integer lwork, integer info )

DORBDB3

Download DORBDB3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
``` DORBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonormal columns:

[ B11 ]
[ X11 ]   [ P1 |    ] [  0  ]
[-----] = [---------] [-----] Q1**T .
[ X21 ]   [    | P2 ] [ B21 ]
[  0  ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P,
Q, or M-Q. Routines DORBDB1, DORBDB2, and DORBDB4 handle cases in
which M-P is not the minimum dimension.

The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
Householder vectors.

B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented
implicitly by angles THETA, PHI.```
Parameters
 [in] M ``` M is INTEGER The number of rows X11 plus the number of rows in X21.``` [in] P ``` P is INTEGER The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).``` [in] Q ``` Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= M.``` [in,out] X11 ``` X11 is DOUBLE PRECISION array, dimension (LDX11,Q) On entry, the top block of the matrix X to be reduced. On exit, the columns of tril(X11) specify reflectors for P1 and the rows of triu(X11,1) specify reflectors for Q1.``` [in] LDX11 ``` LDX11 is INTEGER The leading dimension of X11. LDX11 >= P.``` [in,out] X21 ``` X21 is DOUBLE PRECISION array, dimension (LDX21,Q) On entry, the bottom block of the matrix X to be reduced. On exit, the columns of tril(X21) specify reflectors for P2.``` [in] LDX21 ``` LDX21 is INTEGER The leading dimension of X21. LDX21 >= M-P.``` [out] THETA ``` THETA is DOUBLE PRECISION array, dimension (Q) The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.``` [out] PHI ``` PHI is DOUBLE PRECISION array, dimension (Q-1) The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.``` [out] TAUP1 ``` TAUP1 is DOUBLE PRECISION array, dimension (P) The scalar factors of the elementary reflectors that define P1.``` [out] TAUP2 ``` TAUP2 is DOUBLE PRECISION array, dimension (M-P) The scalar factors of the elementary reflectors that define P2.``` [out] TAUQ1 ``` TAUQ1 is DOUBLE PRECISION array, dimension (Q) The scalar factors of the elementary reflectors that define Q1.``` [out] WORK ` WORK is DOUBLE PRECISION array, dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK >= M-Q. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.```
Further Details:
```  The upper-bidiagonal blocks B11, B21 are represented implicitly by
angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
in each bidiagonal band is a product of a sine or cosine of a THETA
with a sine or cosine of a PHI. See [1] or DORCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors.
See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
and DORGLQ.```
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 199 of file dorbdb3.f.

201*
202* -- LAPACK computational routine --
203* -- LAPACK is a software package provided by Univ. of Tennessee, --
204* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
205*
206* .. Scalar Arguments ..
207 INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
208* ..
209* .. Array Arguments ..
210 DOUBLE PRECISION PHI(*), THETA(*)
211 DOUBLE PRECISION TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
212 \$ X11(LDX11,*), X21(LDX21,*)
213* ..
214*
215* ====================================================================
216*
217* .. Parameters ..
218 DOUBLE PRECISION ONE
219 parameter( one = 1.0d0 )
220* ..
221* .. Local Scalars ..
222 DOUBLE PRECISION C, S
223 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
224 \$ LWORKMIN, LWORKOPT
225 LOGICAL LQUERY
226* ..
227* .. External Subroutines ..
228 EXTERNAL dlarf, dlarfgp, dorbdb5, drot, xerbla
229* ..
230* .. External Functions ..
231 DOUBLE PRECISION DNRM2
232 EXTERNAL dnrm2
233* ..
234* .. Intrinsic Function ..
235 INTRINSIC atan2, cos, max, sin, sqrt
236* ..
237* .. Executable Statements ..
238*
239* Test input arguments
240*
241 info = 0
242 lquery = lwork .EQ. -1
243*
244 IF( m .LT. 0 ) THEN
245 info = -1
246 ELSE IF( 2*p .LT. m .OR. p .GT. m ) THEN
247 info = -2
248 ELSE IF( q .LT. m-p .OR. m-q .LT. m-p ) THEN
249 info = -3
250 ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
251 info = -5
252 ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
253 info = -7
254 END IF
255*
256* Compute workspace
257*
258 IF( info .EQ. 0 ) THEN
259 ilarf = 2
260 llarf = max( p, m-p-1, q-1 )
261 iorbdb5 = 2
262 lorbdb5 = q-1
263 lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
264 lworkmin = lworkopt
265 work(1) = lworkopt
266 IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
267 info = -14
268 END IF
269 END IF
270 IF( info .NE. 0 ) THEN
271 CALL xerbla( 'DORBDB3', -info )
272 RETURN
273 ELSE IF( lquery ) THEN
274 RETURN
275 END IF
276*
277* Reduce rows 1, ..., M-P of X11 and X21
278*
279 DO i = 1, m-p
280*
281 IF( i .GT. 1 ) THEN
282 CALL drot( q-i+1, x11(i-1,i), ldx11, x21(i,i), ldx11, c, s )
283 END IF
284*
285 CALL dlarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
286 s = x21(i,i)
287 x21(i,i) = one
288 CALL dlarf( 'R', p-i+1, q-i+1, x21(i,i), ldx21, tauq1(i),
289 \$ x11(i,i), ldx11, work(ilarf) )
290 CALL dlarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
291 \$ x21(i+1,i), ldx21, work(ilarf) )
292 c = sqrt( dnrm2( p-i+1, x11(i,i), 1 )**2
293 \$ + dnrm2( m-p-i, x21(i+1,i), 1 )**2 )
294 theta(i) = atan2( s, c )
295*
296 CALL dorbdb5( p-i+1, m-p-i, q-i, x11(i,i), 1, x21(i+1,i), 1,
297 \$ x11(i,i+1), ldx11, x21(i+1,i+1), ldx21,
298 \$ work(iorbdb5), lorbdb5, childinfo )
299 CALL dlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
300 IF( i .LT. m-p ) THEN
301 CALL dlarfgp( m-p-i, x21(i+1,i), x21(i+2,i), 1, taup2(i) )
302 phi(i) = atan2( x21(i+1,i), x11(i,i) )
303 c = cos( phi(i) )
304 s = sin( phi(i) )
305 x21(i+1,i) = one
306 CALL dlarf( 'L', m-p-i, q-i, x21(i+1,i), 1, taup2(i),
307 \$ x21(i+1,i+1), ldx21, work(ilarf) )
308 END IF
309 x11(i,i) = one
310 CALL dlarf( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,i+1),
311 \$ ldx11, work(ilarf) )
312*
313 END DO
314*
315* Reduce the bottom-right portion of X11 to the identity matrix
316*
317 DO i = m-p + 1, q
318 CALL dlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
319 x11(i,i) = one
320 CALL dlarf( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,i+1),
321 \$ ldx11, work(ilarf) )
322 END DO
323*
324 RETURN
325*
326* End of DORBDB3
327*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
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
real(wp) function dnrm2(n, x, incx)
DNRM2
Definition dnrm2.f90:89
subroutine drot(n, dx, incx, dy, incy, c, s)
DROT
Definition drot.f:92
subroutine dorbdb5(m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
DORBDB5
Definition dorbdb5.f:156
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