LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ cunbdb4()

subroutine cunbdb4 ( 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(*)  PHANTOM,
complex, dimension(*)  WORK,
integer  LWORK,
integer  INFO 
)

CUNBDB4

Download CUNBDB4 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CUNBDB4 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 CUNBDB1, CUNBDB2, and CUNBDB3 handle cases in
 which M-Q 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-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 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]PHANTOM
          PHANTOM is COMPLEX 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 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 215 of file cunbdb4.f.

215 *
216 * -- LAPACK computational routine (version 3.8.0) --
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  REAL phi(*), theta(*)
226  COMPLEX phantom(*), taup1(*), taup2(*), tauq1(*),
227  $ work(*), x11(ldx11,*), x21(ldx21,*)
228 * ..
229 *
230 * ====================================================================
231 *
232 * .. Parameters ..
233  COMPLEX negone, one, zero
234  parameter( negone = (-1.0e0,0.0e0), one = (1.0e0,0.0e0),
235  $ zero = (0.0e0,0.0e0) )
236 * ..
237 * .. Local Scalars ..
238  REAL c, s
239  INTEGER childinfo, i, ilarf, iorbdb5, j, llarf,
240  $ lorbdb5, lworkmin, lworkopt
241  LOGICAL lquery
242 * ..
243 * .. External Subroutines ..
244  EXTERNAL clarf, clarfgp, cunbdb5, csrot, cscal, clacgv,
245  $ xerbla
246 * ..
247 * .. External Functions ..
248  REAL scnrm2
249  EXTERNAL scnrm2
250 * ..
251 * .. Intrinsic Function ..
252  INTRINSIC atan2, cos, max, sin, sqrt
253 * ..
254 * .. Executable Statements ..
255 *
256 * Test input arguments
257 *
258  info = 0
259  lquery = lwork .EQ. -1
260 *
261  IF( m .LT. 0 ) THEN
262  info = -1
263  ELSE IF( p .LT. m-q .OR. m-p .LT. m-q ) THEN
264  info = -2
265  ELSE IF( q .LT. m-q .OR. q .GT. m ) THEN
266  info = -3
267  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
268  info = -5
269  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
270  info = -7
271  END IF
272 *
273 * Compute workspace
274 *
275  IF( info .EQ. 0 ) THEN
276  ilarf = 2
277  llarf = max( q-1, p-1, m-p-1 )
278  iorbdb5 = 2
279  lorbdb5 = q
280  lworkopt = ilarf + llarf - 1
281  lworkopt = max( lworkopt, iorbdb5 + lorbdb5 - 1 )
282  lworkmin = lworkopt
283  work(1) = lworkopt
284  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
285  info = -14
286  END IF
287  END IF
288  IF( info .NE. 0 ) THEN
289  CALL xerbla( 'CUNBDB4', -info )
290  RETURN
291  ELSE IF( lquery ) THEN
292  RETURN
293  END IF
294 *
295 * Reduce columns 1, ..., M-Q of X11 and X21
296 *
297  DO i = 1, m-q
298 *
299  IF( i .EQ. 1 ) THEN
300  DO j = 1, m
301  phantom(j) = zero
302  END DO
303  CALL cunbdb5( p, m-p, q, phantom(1), 1, phantom(p+1), 1,
304  $ x11, ldx11, x21, ldx21, work(iorbdb5),
305  $ lorbdb5, childinfo )
306  CALL cscal( p, negone, phantom(1), 1 )
307  CALL clarfgp( p, phantom(1), phantom(2), 1, taup1(1) )
308  CALL clarfgp( m-p, phantom(p+1), phantom(p+2), 1, taup2(1) )
309  theta(i) = atan2( REAL( PHANTOM(1) ), REAL( PHANTOM(P+1) ) )
310  c = cos( theta(i) )
311  s = sin( theta(i) )
312  phantom(1) = one
313  phantom(p+1) = one
314  CALL clarf( 'L', p, q, phantom(1), 1, conjg(taup1(1)), x11,
315  $ ldx11, work(ilarf) )
316  CALL clarf( 'L', m-p, q, phantom(p+1), 1, conjg(taup2(1)),
317  $ x21, ldx21, work(ilarf) )
318  ELSE
319  CALL cunbdb5( p-i+1, m-p-i+1, q-i+1, x11(i,i-1), 1,
320  $ x21(i,i-1), 1, x11(i,i), ldx11, x21(i,i),
321  $ ldx21, work(iorbdb5), lorbdb5, childinfo )
322  CALL cscal( p-i+1, negone, x11(i,i-1), 1 )
323  CALL clarfgp( p-i+1, x11(i,i-1), x11(i+1,i-1), 1, taup1(i) )
324  CALL clarfgp( m-p-i+1, x21(i,i-1), x21(i+1,i-1), 1,
325  $ taup2(i) )
326  theta(i) = atan2( REAL( X11(I,I-1) ), REAL( X21(I,I-1) ) )
327  c = cos( theta(i) )
328  s = sin( theta(i) )
329  x11(i,i-1) = one
330  x21(i,i-1) = one
331  CALL clarf( 'L', p-i+1, q-i+1, x11(i,i-1), 1,
332  $ conjg(taup1(i)), x11(i,i), ldx11, work(ilarf) )
333  CALL clarf( 'L', m-p-i+1, q-i+1, x21(i,i-1), 1,
334  $ conjg(taup2(i)), x21(i,i), ldx21, work(ilarf) )
335  END IF
336 *
337  CALL csrot( q-i+1, x11(i,i), ldx11, x21(i,i), ldx21, s, -c )
338  CALL clacgv( q-i+1, x21(i,i), ldx21 )
339  CALL clarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
340  c = REAL( X21(I,I) )
341  x21(i,i) = one
342  CALL clarf( 'R', p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
343  $ x11(i+1,i), ldx11, work(ilarf) )
344  CALL clarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
345  $ x21(i+1,i), ldx21, work(ilarf) )
346  CALL clacgv( q-i+1, x21(i,i), ldx21 )
347  IF( i .LT. m-q ) THEN
348  s = sqrt( scnrm2( p-i, x11(i+1,i), 1 )**2
349  $ + scnrm2( m-p-i, x21(i+1,i), 1 )**2 )
350  phi(i) = atan2( s, c )
351  END IF
352 *
353  END DO
354 *
355 * Reduce the bottom-right portion of X11 to [ I 0 ]
356 *
357  DO i = m - q + 1, p
358  CALL clacgv( q-i+1, x11(i,i), ldx11 )
359  CALL clarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
360  x11(i,i) = one
361  CALL clarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
362  $ x11(i+1,i), ldx11, work(ilarf) )
363  CALL clarf( 'R', q-p, q-i+1, x11(i,i), ldx11, tauq1(i),
364  $ x21(m-q+1,i), ldx21, work(ilarf) )
365  CALL clacgv( q-i+1, x11(i,i), ldx11 )
366  END DO
367 *
368 * Reduce the bottom-right portion of X21 to [ 0 I ]
369 *
370  DO i = p + 1, q
371  CALL clacgv( q-i+1, x21(m-q+i-p,i), ldx21 )
372  CALL clarfgp( q-i+1, x21(m-q+i-p,i), x21(m-q+i-p,i+1), ldx21,
373  $ tauq1(i) )
374  x21(m-q+i-p,i) = one
375  CALL clarf( 'R', q-i, q-i+1, x21(m-q+i-p,i), ldx21, tauq1(i),
376  $ x21(m-q+i-p+1,i), ldx21, work(ilarf) )
377  CALL clacgv( q-i+1, x21(m-q+i-p,i), ldx21 )
378  END DO
379 *
380  RETURN
381 *
382 * End of CUNBDB4
383 *
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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