LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ cunbdb3()

subroutine cunbdb3 ( 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 
)

CUNBDB3

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Purpose:
 CUNBDB3 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 CUNBDB1, CUNBDB2, and CUNBDB4 handle cases in
 which M-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 (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 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 cunbdb3.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 one
223  parameter( one = (1.0e0,0.0e0) )
224 * ..
225 * .. Local Scalars ..
226  REAL c, s
227  INTEGER childinfo, i, ilarf, iorbdb5, llarf, lorbdb5,
228  $ lworkmin, lworkopt
229  LOGICAL lquery
230 * ..
231 * .. External Subroutines ..
232  EXTERNAL clarf, clarfgp, cunbdb5, csrot, clacgv, xerbla
233 * ..
234 * .. External Functions ..
235  REAL scnrm2
236  EXTERNAL scnrm2
237 * ..
238 * .. Intrinsic Function ..
239  INTRINSIC atan2, cos, max, sin, sqrt
240 * ..
241 * .. Executable Statements ..
242 *
243 * Test input arguments
244 *
245  info = 0
246  lquery = lwork .EQ. -1
247 *
248  IF( m .LT. 0 ) THEN
249  info = -1
250  ELSE IF( 2*p .LT. m .OR. p .GT. m ) THEN
251  info = -2
252  ELSE IF( q .LT. m-p .OR. m-q .LT. m-p ) THEN
253  info = -3
254  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
255  info = -5
256  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
257  info = -7
258  END IF
259 *
260 * Compute workspace
261 *
262  IF( info .EQ. 0 ) THEN
263  ilarf = 2
264  llarf = max( p, m-p-1, q-1 )
265  iorbdb5 = 2
266  lorbdb5 = q-1
267  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
268  lworkmin = lworkopt
269  work(1) = lworkopt
270  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
271  info = -14
272  END IF
273  END IF
274  IF( info .NE. 0 ) THEN
275  CALL xerbla( 'CUNBDB3', -info )
276  RETURN
277  ELSE IF( lquery ) THEN
278  RETURN
279  END IF
280 *
281 * Reduce rows 1, ..., M-P of X11 and X21
282 *
283  DO i = 1, m-p
284 *
285  IF( i .GT. 1 ) THEN
286  CALL csrot( q-i+1, x11(i-1,i), ldx11, x21(i,i), ldx11, c,
287  $ s )
288  END IF
289 *
290  CALL clacgv( q-i+1, x21(i,i), ldx21 )
291  CALL clarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
292  s = REAL( X21(I,I) )
293  x21(i,i) = one
294  CALL clarf( 'R', p-i+1, q-i+1, x21(i,i), ldx21, tauq1(i),
295  $ x11(i,i), ldx11, work(ilarf) )
296  CALL clarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
297  $ x21(i+1,i), ldx21, work(ilarf) )
298  CALL clacgv( q-i+1, x21(i,i), ldx21 )
299  c = sqrt( scnrm2( p-i+1, x11(i,i), 1 )**2
300  $ + scnrm2( m-p-i, x21(i+1,i), 1 )**2 )
301  theta(i) = atan2( s, c )
302 *
303  CALL cunbdb5( p-i+1, m-p-i, q-i, x11(i,i), 1, x21(i+1,i), 1,
304  $ x11(i,i+1), ldx11, x21(i+1,i+1), ldx21,
305  $ work(iorbdb5), lorbdb5, childinfo )
306  CALL clarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
307  IF( i .LT. m-p ) THEN
308  CALL clarfgp( m-p-i, x21(i+1,i), x21(i+2,i), 1, taup2(i) )
309  phi(i) = atan2( REAL( X21(I+1,I) ), REAL( X11(I,I) ) )
310  c = cos( phi(i) )
311  s = sin( phi(i) )
312  x21(i+1,i) = one
313  CALL clarf( 'L', m-p-i, q-i, x21(i+1,i), 1, conjg(taup2(i)),
314  $ x21(i+1,i+1), ldx21, work(ilarf) )
315  END IF
316  x11(i,i) = one
317  CALL clarf( 'L', p-i+1, q-i, x11(i,i), 1, conjg(taup1(i)),
318  $ x11(i,i+1), ldx11, work(ilarf) )
319 *
320  END DO
321 *
322 * Reduce the bottom-right portion of X11 to the identity matrix
323 *
324  DO i = m-p + 1, q
325  CALL clarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
326  x11(i,i) = one
327  CALL clarf( 'L', p-i+1, q-i, x11(i,i), 1, conjg(taup1(i)),
328  $ x11(i,i+1), ldx11, work(ilarf) )
329  END DO
330 *
331  RETURN
332 *
333 * End of CUNBDB3
334 *
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 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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