LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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 orthonomal 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)
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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