LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dorbdb2()

 subroutine dorbdb2 ( 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 )

DORBDB2

Purpose:
``` DORBDB2 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. P must be no larger than M-P,
Q, or M-Q. Routines DORBDB1, DORBDB3, and DORBDB4 handle cases in
which 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 P-by-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 <= min(M-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-1) The scalar factors of the elementary reflectors that define P1.``` [out] TAUP2 ``` TAUP2 is DOUBLE PRECISION array, dimension (Q) 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 200 of file dorbdb2.f.

202*
203* -- LAPACK computational routine --
204* -- LAPACK is a software package provided by Univ. of Tennessee, --
205* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
206*
207* .. Scalar Arguments ..
208 INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
209* ..
210* .. Array Arguments ..
211 DOUBLE PRECISION PHI(*), THETA(*)
212 DOUBLE PRECISION TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
213 \$ X11(LDX11,*), X21(LDX21,*)
214* ..
215*
216* ====================================================================
217*
218* .. Parameters ..
219 DOUBLE PRECISION NEGONE, ONE
220 parameter( negone = -1.0d0, one = 1.0d0 )
221* ..
222* .. Local Scalars ..
223 DOUBLE PRECISION C, S
224 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
225 \$ LWORKMIN, LWORKOPT
226 LOGICAL LQUERY
227* ..
228* .. External Subroutines ..
229 EXTERNAL dlarf, dlarfgp, dorbdb5, drot, dscal, xerbla
230* ..
231* .. External Functions ..
232 DOUBLE PRECISION DNRM2
233 EXTERNAL dnrm2
234* ..
235* .. Intrinsic Function ..
236 INTRINSIC atan2, cos, max, sin, sqrt
237* ..
238* .. Executable Statements ..
239*
240* Test input arguments
241*
242 info = 0
243 lquery = lwork .EQ. -1
244*
245 IF( m .LT. 0 ) THEN
246 info = -1
247 ELSE IF( p .LT. 0 .OR. p .GT. m-p ) THEN
248 info = -2
249 ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) THEN
250 info = -3
251 ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
252 info = -5
253 ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
254 info = -7
255 END IF
256*
257* Compute workspace
258*
259 IF( info .EQ. 0 ) THEN
260 ilarf = 2
261 llarf = max( p-1, m-p, q-1 )
262 iorbdb5 = 2
263 lorbdb5 = q-1
264 lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
265 lworkmin = lworkopt
266 work(1) = lworkopt
267 IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
268 info = -14
269 END IF
270 END IF
271 IF( info .NE. 0 ) THEN
272 CALL xerbla( 'DORBDB2', -info )
273 RETURN
274 ELSE IF( lquery ) THEN
275 RETURN
276 END IF
277*
278* Reduce rows 1, ..., P of X11 and X21
279*
280 DO i = 1, p
281*
282 IF( i .GT. 1 ) THEN
283 CALL drot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c, s )
284 END IF
285 CALL dlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
286 c = x11(i,i)
287 x11(i,i) = one
288 CALL dlarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
289 \$ x11(i+1,i), ldx11, work(ilarf) )
290 CALL dlarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
291 \$ x21(i,i), ldx21, work(ilarf) )
292 s = sqrt( dnrm2( p-i, x11(i+1,i), 1 )**2
293 \$ + dnrm2( m-p-i+1, x21(i,i), 1 )**2 )
294 theta(i) = atan2( s, c )
295*
296 CALL dorbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
297 \$ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
298 \$ work(iorbdb5), lorbdb5, childinfo )
299 CALL dscal( p-i, negone, x11(i+1,i), 1 )
300 CALL dlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
301 IF( i .LT. p ) THEN
302 CALL dlarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
303 phi(i) = atan2( x11(i+1,i), x21(i,i) )
304 c = cos( phi(i) )
305 s = sin( phi(i) )
306 x11(i+1,i) = one
307 CALL dlarf( 'L', p-i, q-i, x11(i+1,i), 1, taup1(i),
308 \$ x11(i+1,i+1), ldx11, work(ilarf) )
309 END IF
310 x21(i,i) = one
311 CALL dlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
312 \$ x21(i,i+1), ldx21, work(ilarf) )
313*
314 END DO
315*
316* Reduce the bottom-right portion of X21 to the identity matrix
317*
318 DO i = p + 1, q
319 CALL dlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
320 x21(i,i) = one
321 CALL dlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
322 \$ x21(i,i+1), ldx21, work(ilarf) )
323 END DO
324*
325 RETURN
326*
327* End of DORBDB2
328*
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 dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
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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