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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
dlarf
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
dlarfgp
subroutine dlarfgp(n, alpha, x, incx, tau)
DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition dlarfgp.f:104
dnrm2
real(wp) function dnrm2(n, x, incx)
DNRM2
Definition dnrm2.f90:89
drot
subroutine drot(n, dx, incx, dy, incy, c, s)
DROT
Definition drot.f:92
dorbdb5
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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