LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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

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Purpose:
``` DORBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonomal 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)
XERBLA
Definition: xerbla.f:60
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:92
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
real(wp) function dnrm2(n, x, incx)
DNRM2
Definition: dnrm2.f90:89
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