LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dorbdb1()

 subroutine dorbdb1 ( 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 )

DORBDB1

Purpose:
``` DORBDB1 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. Q must be no larger than P,
M-P, or M-Q. Routines DORBDB2, DORBDB3, and DORBDB4 handle cases in
which Q 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 Q-by-Q 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.``` [in] Q ``` Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).``` [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 201 of file dorbdb1.f.

203*
204* -- LAPACK computational routine --
205* -- LAPACK is a software package provided by Univ. of Tennessee, --
206* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
207*
208* .. Scalar Arguments ..
209 INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
210* ..
211* .. Array Arguments ..
212 DOUBLE PRECISION PHI(*), THETA(*)
213 DOUBLE PRECISION TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
214 \$ X11(LDX11,*), X21(LDX21,*)
215* ..
216*
217* ====================================================================
218*
219* .. Parameters ..
220 DOUBLE PRECISION ONE
221 parameter( one = 1.0d0 )
222* ..
223* .. Local Scalars ..
224 DOUBLE PRECISION C, S
225 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
226 \$ LWORKMIN, LWORKOPT
227 LOGICAL LQUERY
228* ..
229* .. External Subroutines ..
230 EXTERNAL dlarf, dlarfgp, dorbdb5, drot, xerbla
231* ..
232* .. External Functions ..
233 DOUBLE PRECISION DNRM2
234 EXTERNAL dnrm2
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( 'DORBDB1', -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 dlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
284 CALL dlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
285 theta(i) = atan2( x21(i,i), 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 dlarf( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,i+1),
291 \$ ldx11, work(ilarf) )
292 CALL dlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
293 \$ x21(i,i+1), ldx21, work(ilarf) )
294*
295 IF( i .LT. q ) THEN
296 CALL drot( q-i, x11(i,i+1), ldx11, x21(i,i+1), ldx21, c, s )
297 CALL dlarfgp( q-i, x21(i,i+1), x21(i,i+2), ldx21, tauq1(i) )
298 s = x21(i,i+1)
299 x21(i,i+1) = one
300 CALL dlarf( 'R', p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
301 \$ x11(i+1,i+1), ldx11, work(ilarf) )
302 CALL dlarf( 'R', m-p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
303 \$ x21(i+1,i+1), ldx21, work(ilarf) )
304 c = sqrt( dnrm2( p-i, x11(i+1,i+1), 1 )**2
305 \$ + dnrm2( m-p-i, x21(i+1,i+1), 1 )**2 )
306 phi(i) = atan2( s, c )
307 CALL dorbdb5( p-i, m-p-i, q-i-1, x11(i+1,i+1), 1,
308 \$ x21(i+1,i+1), 1, x11(i+1,i+2), ldx11,
309 \$ x21(i+1,i+2), ldx21, work(iorbdb5), lorbdb5,
310 \$ childinfo )
311 END IF
312*
313 END DO
314*
315 RETURN
316*
317* End of DORBDB1
318*
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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