LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ dorbdb4()

subroutine dorbdb4 ( 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(*)  PHANTOM,
double precision, dimension(*)  WORK,
integer  LWORK,
integer  INFO 
)

DORBDB4

Download DORBDB4 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DORBDB4 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-Q must be no larger than P,
 M-P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in
 which M-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 (M-Q)-by-(M-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 <= M and
           M-Q <= min(P,M-P,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]PHANTOM
          PHANTOM is DOUBLE PRECISION array, dimension (M)
           The routine computes an M-by-1 column vector Y that is
           orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
           PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
           Y(P+1:M), respectively.
[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.
Date
July 2012
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 215 of file dorbdb4.f.

215 *
216 * -- LAPACK computational routine (version 3.7.1) --
217 * -- LAPACK is a software package provided by Univ. of Tennessee, --
218 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
219 * July 2012
220 *
221 * .. Scalar Arguments ..
222  INTEGER info, lwork, m, p, q, ldx11, ldx21
223 * ..
224 * .. Array Arguments ..
225  DOUBLE PRECISION phi(*), theta(*)
226  DOUBLE PRECISION phantom(*), taup1(*), taup2(*), tauq1(*),
227  $ work(*), x11(ldx11,*), x21(ldx21,*)
228 * ..
229 *
230 * ====================================================================
231 *
232 * .. Parameters ..
233  DOUBLE PRECISION negone, one, zero
234  parameter( negone = -1.0d0, one = 1.0d0, zero = 0.0d0 )
235 * ..
236 * .. Local Scalars ..
237  DOUBLE PRECISION c, s
238  INTEGER childinfo, i, ilarf, iorbdb5, j, llarf,
239  $ lorbdb5, lworkmin, lworkopt
240  LOGICAL lquery
241 * ..
242 * .. External Subroutines ..
243  EXTERNAL dlarf, dlarfgp, dorbdb5, drot, dscal, xerbla
244 * ..
245 * .. External Functions ..
246  DOUBLE PRECISION dnrm2
247  EXTERNAL dnrm2
248 * ..
249 * .. Intrinsic Function ..
250  INTRINSIC atan2, cos, max, sin, sqrt
251 * ..
252 * .. Executable Statements ..
253 *
254 * Test input arguments
255 *
256  info = 0
257  lquery = lwork .EQ. -1
258 *
259  IF( m .LT. 0 ) THEN
260  info = -1
261  ELSE IF( p .LT. m-q .OR. m-p .LT. m-q ) THEN
262  info = -2
263  ELSE IF( q .LT. m-q .OR. q .GT. m ) THEN
264  info = -3
265  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
266  info = -5
267  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
268  info = -7
269  END IF
270 *
271 * Compute workspace
272 *
273  IF( info .EQ. 0 ) THEN
274  ilarf = 2
275  llarf = max( q-1, p-1, m-p-1 )
276  iorbdb5 = 2
277  lorbdb5 = q
278  lworkopt = ilarf + llarf - 1
279  lworkopt = max( lworkopt, iorbdb5 + lorbdb5 - 1 )
280  lworkmin = lworkopt
281  work(1) = lworkopt
282  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
283  info = -14
284  END IF
285  END IF
286  IF( info .NE. 0 ) THEN
287  CALL xerbla( 'DORBDB4', -info )
288  RETURN
289  ELSE IF( lquery ) THEN
290  RETURN
291  END IF
292 *
293 * Reduce columns 1, ..., M-Q of X11 and X21
294 *
295  DO i = 1, m-q
296 *
297  IF( i .EQ. 1 ) THEN
298  DO j = 1, m
299  phantom(j) = zero
300  END DO
301  CALL dorbdb5( p, m-p, q, phantom(1), 1, phantom(p+1), 1,
302  $ x11, ldx11, x21, ldx21, work(iorbdb5),
303  $ lorbdb5, childinfo )
304  CALL dscal( p, negone, phantom(1), 1 )
305  CALL dlarfgp( p, phantom(1), phantom(2), 1, taup1(1) )
306  CALL dlarfgp( m-p, phantom(p+1), phantom(p+2), 1, taup2(1) )
307  theta(i) = atan2( phantom(1), phantom(p+1) )
308  c = cos( theta(i) )
309  s = sin( theta(i) )
310  phantom(1) = one
311  phantom(p+1) = one
312  CALL dlarf( 'L', p, q, phantom(1), 1, taup1(1), x11, ldx11,
313  $ work(ilarf) )
314  CALL dlarf( 'L', m-p, q, phantom(p+1), 1, taup2(1), x21,
315  $ ldx21, work(ilarf) )
316  ELSE
317  CALL dorbdb5( p-i+1, m-p-i+1, q-i+1, x11(i,i-1), 1,
318  $ x21(i,i-1), 1, x11(i,i), ldx11, x21(i,i),
319  $ ldx21, work(iorbdb5), lorbdb5, childinfo )
320  CALL dscal( p-i+1, negone, x11(i,i-1), 1 )
321  CALL dlarfgp( p-i+1, x11(i,i-1), x11(i+1,i-1), 1, taup1(i) )
322  CALL dlarfgp( m-p-i+1, x21(i,i-1), x21(i+1,i-1), 1,
323  $ taup2(i) )
324  theta(i) = atan2( x11(i,i-1), x21(i,i-1) )
325  c = cos( theta(i) )
326  s = sin( theta(i) )
327  x11(i,i-1) = one
328  x21(i,i-1) = one
329  CALL dlarf( 'L', p-i+1, q-i+1, x11(i,i-1), 1, taup1(i),
330  $ x11(i,i), ldx11, work(ilarf) )
331  CALL dlarf( 'L', m-p-i+1, q-i+1, x21(i,i-1), 1, taup2(i),
332  $ x21(i,i), ldx21, work(ilarf) )
333  END IF
334 *
335  CALL drot( q-i+1, x11(i,i), ldx11, x21(i,i), ldx21, s, -c )
336  CALL dlarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
337  c = x21(i,i)
338  x21(i,i) = one
339  CALL dlarf( 'R', p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
340  $ x11(i+1,i), ldx11, work(ilarf) )
341  CALL dlarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
342  $ x21(i+1,i), ldx21, work(ilarf) )
343  IF( i .LT. m-q ) THEN
344  s = sqrt( dnrm2( p-i, x11(i+1,i), 1 )**2
345  $ + dnrm2( m-p-i, x21(i+1,i), 1 )**2 )
346  phi(i) = atan2( s, c )
347  END IF
348 *
349  END DO
350 *
351 * Reduce the bottom-right portion of X11 to [ I 0 ]
352 *
353  DO i = m - q + 1, p
354  CALL dlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
355  x11(i,i) = one
356  CALL dlarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
357  $ x11(i+1,i), ldx11, work(ilarf) )
358  CALL dlarf( 'R', q-p, q-i+1, x11(i,i), ldx11, tauq1(i),
359  $ x21(m-q+1,i), ldx21, work(ilarf) )
360  END DO
361 *
362 * Reduce the bottom-right portion of X21 to [ 0 I ]
363 *
364  DO i = p + 1, q
365  CALL dlarfgp( q-i+1, x21(m-q+i-p,i), x21(m-q+i-p,i+1), ldx21,
366  $ tauq1(i) )
367  x21(m-q+i-p,i) = one
368  CALL dlarf( 'R', q-i, q-i+1, x21(m-q+i-p,i), ldx21, tauq1(i),
369  $ x21(m-q+i-p+1,i), ldx21, work(ilarf) )
370  END DO
371 *
372  RETURN
373 *
374 * End of DORBDB4
375 *
subroutine dlarfgp(N, ALPHA, X, INCX, TAU)
DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: dlarfgp.f:106
subroutine dlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition: dlarf.f:126
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:94
subroutine dorbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
DORBDB5
Definition: dorbdb5.f:158
double precision function dnrm2(N, X, INCX)
DNRM2
Definition: dnrm2.f:76
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:81
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