LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ cunbdb2()

subroutine cunbdb2 ( integer  M,
integer  P,
integer  Q,
complex, dimension(ldx11,*)  X11,
integer  LDX11,
complex, dimension(ldx21,*)  X21,
integer  LDX21,
real, dimension(*)  THETA,
real, dimension(*)  PHI,
complex, dimension(*)  TAUP1,
complex, dimension(*)  TAUP2,
complex, dimension(*)  TAUQ1,
complex, dimension(*)  WORK,
integer  LWORK,
integer  INFO 
)

CUNBDB2

Download CUNBDB2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CUNBDB2 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. P must be no larger than M-P,
 Q, or M-Q. Routines CUNBDB1, CUNBDB3, and CUNBDB4 handle cases in
 which P is not the minimum dimension.

 The unitary 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 COMPLEX 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 COMPLEX 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 REAL array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.
[out]PHI
          PHI is REAL 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 COMPLEX array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.
[out]TAUP2
          TAUP2 is COMPLEX array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.
[out]TAUQ1
          TAUQ1 is COMPLEX array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.
[out]WORK
          WORK is COMPLEX 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 CUNCSD for details.

  P1, P2, and Q1 are represented as products of elementary reflectors.
  See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR
  and CUNGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 204 of file cunbdb2.f.

204 *
205 * -- LAPACK computational routine (version 3.8.0) --
206 * -- LAPACK is a software package provided by Univ. of Tennessee, --
207 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
208 * July 2012
209 *
210 * .. Scalar Arguments ..
211  INTEGER info, lwork, m, p, q, ldx11, ldx21
212 * ..
213 * .. Array Arguments ..
214  REAL phi(*), theta(*)
215  COMPLEX taup1(*), taup2(*), tauq1(*), work(*),
216  $ x11(ldx11,*), x21(ldx21,*)
217 * ..
218 *
219 * ====================================================================
220 *
221 * .. Parameters ..
222  COMPLEX negone, one
223  parameter( negone = (-1.0e0,0.0e0),
224  $ one = (1.0e0,0.0e0) )
225 * ..
226 * .. Local Scalars ..
227  REAL c, s
228  INTEGER childinfo, i, ilarf, iorbdb5, llarf, lorbdb5,
229  $ lworkmin, lworkopt
230  LOGICAL lquery
231 * ..
232 * .. External Subroutines ..
233  EXTERNAL clarf, clarfgp, cunbdb5, csrot, cscal, clacgv,
234  $ xerbla
235 * ..
236 * .. External Functions ..
237  REAL scnrm2
238  EXTERNAL scnrm2
239 * ..
240 * .. Intrinsic Function ..
241  INTRINSIC atan2, cos, max, sin, sqrt
242 * ..
243 * .. Executable Statements ..
244 *
245 * Test input arguments
246 *
247  info = 0
248  lquery = lwork .EQ. -1
249 *
250  IF( m .LT. 0 ) THEN
251  info = -1
252  ELSE IF( p .LT. 0 .OR. p .GT. m-p ) THEN
253  info = -2
254  ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) THEN
255  info = -3
256  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
257  info = -5
258  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
259  info = -7
260  END IF
261 *
262 * Compute workspace
263 *
264  IF( info .EQ. 0 ) THEN
265  ilarf = 2
266  llarf = max( p-1, m-p, q-1 )
267  iorbdb5 = 2
268  lorbdb5 = q-1
269  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
270  lworkmin = lworkopt
271  work(1) = lworkopt
272  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
273  info = -14
274  END IF
275  END IF
276  IF( info .NE. 0 ) THEN
277  CALL xerbla( 'CUNBDB2', -info )
278  RETURN
279  ELSE IF( lquery ) THEN
280  RETURN
281  END IF
282 *
283 * Reduce rows 1, ..., P of X11 and X21
284 *
285  DO i = 1, p
286 *
287  IF( i .GT. 1 ) THEN
288  CALL csrot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c,
289  $ s )
290  END IF
291  CALL clacgv( q-i+1, x11(i,i), ldx11 )
292  CALL clarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
293  c = REAL( X11(I,I) )
294  x11(i,i) = one
295  CALL clarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
296  $ x11(i+1,i), ldx11, work(ilarf) )
297  CALL clarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
298  $ x21(i,i), ldx21, work(ilarf) )
299  CALL clacgv( q-i+1, x11(i,i), ldx11 )
300  s = sqrt( scnrm2( p-i, x11(i+1,i), 1 )**2
301  $ + scnrm2( m-p-i+1, x21(i,i), 1 )**2 )
302  theta(i) = atan2( s, c )
303 *
304  CALL cunbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
305  $ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
306  $ work(iorbdb5), lorbdb5, childinfo )
307  CALL cscal( p-i, negone, x11(i+1,i), 1 )
308  CALL clarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
309  IF( i .LT. p ) THEN
310  CALL clarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
311  phi(i) = atan2( REAL( X11(I+1,I) ), REAL( X21(I,I) ) )
312  c = cos( phi(i) )
313  s = sin( phi(i) )
314  x11(i+1,i) = one
315  CALL clarf( 'L', p-i, q-i, x11(i+1,i), 1, conjg(taup1(i)),
316  $ x11(i+1,i+1), ldx11, work(ilarf) )
317  END IF
318  x21(i,i) = one
319  CALL clarf( 'L', m-p-i+1, q-i, x21(i,i), 1, conjg(taup2(i)),
320  $ x21(i,i+1), ldx21, work(ilarf) )
321 *
322  END DO
323 *
324 * Reduce the bottom-right portion of X21 to the identity matrix
325 *
326  DO i = p + 1, q
327  CALL clarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
328  x21(i,i) = one
329  CALL clarf( 'L', m-p-i+1, q-i, x21(i,i), 1, conjg(taup2(i)),
330  $ x21(i,i+1), ldx21, work(ilarf) )
331  END DO
332 *
333  RETURN
334 *
335 * End of CUNBDB2
336 *
real function scnrm2(N, X, INCX)
SCNRM2
Definition: scnrm2.f:77
subroutine cunbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
CUNBDB5
Definition: cunbdb5.f:158
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:80
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:76
subroutine csrot(N, CX, INCX, CY, INCY, C, S)
CSROT
Definition: csrot.f:100
subroutine clarfgp(N, ALPHA, X, INCX, TAU)
CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: clarfgp.f:106
subroutine clarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
CLARF applies an elementary reflector to a general rectangular matrix.
Definition: clarf.f:130
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