LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ 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-1)
           The scalar factors of the elementary reflectors that define
           P1.
[out]TAUP2
          TAUP2 is COMPLEX array, dimension (Q)
           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.
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 200 of file cunbdb2.f.

202*
203* -- LAPACK computational routine --
204* -- LAPACK is a software package provided by Univ. of Tennessee, --
205* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
206*
207* .. Scalar Arguments ..
208 INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
209* ..
210* .. Array Arguments ..
211 REAL PHI(*), THETA(*)
212 COMPLEX TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
213 $ X11(LDX11,*), X21(LDX21,*)
214* ..
215*
216* ====================================================================
217*
218* .. Parameters ..
219 COMPLEX NEGONE, ONE
220 parameter( negone = (-1.0e0,0.0e0),
221 $ one = (1.0e0,0.0e0) )
222* ..
223* .. Local Scalars ..
224 REAL C, S
225 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
226 $ LWORKMIN, LWORKOPT
227 LOGICAL LQUERY
228* ..
229* .. External Subroutines ..
230 EXTERNAL clarf, clarfgp, cunbdb5, csrot, cscal, clacgv,
231 $ xerbla
232* ..
233* .. External Functions ..
234 REAL SCNRM2
235 EXTERNAL scnrm2
236* ..
237* .. Intrinsic Function ..
238 INTRINSIC atan2, cos, max, sin, sqrt
239* ..
240* .. Executable Statements ..
241*
242* Test input arguments
243*
244 info = 0
245 lquery = lwork .EQ. -1
246*
247 IF( m .LT. 0 ) THEN
248 info = -1
249 ELSE IF( p .LT. 0 .OR. p .GT. m-p ) THEN
250 info = -2
251 ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) THEN
252 info = -3
253 ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
254 info = -5
255 ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
256 info = -7
257 END IF
258*
259* Compute workspace
260*
261 IF( info .EQ. 0 ) THEN
262 ilarf = 2
263 llarf = max( p-1, m-p, q-1 )
264 iorbdb5 = 2
265 lorbdb5 = q-1
266 lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
267 lworkmin = lworkopt
268 work(1) = lworkopt
269 IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
270 info = -14
271 END IF
272 END IF
273 IF( info .NE. 0 ) THEN
274 CALL xerbla( 'CUNBDB2', -info )
275 RETURN
276 ELSE IF( lquery ) THEN
277 RETURN
278 END IF
279*
280* Reduce rows 1, ..., P of X11 and X21
281*
282 DO i = 1, p
283*
284 IF( i .GT. 1 ) THEN
285 CALL csrot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c,
286 $ s )
287 END IF
288 CALL clacgv( q-i+1, x11(i,i), ldx11 )
289 CALL clarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
290 c = real( x11(i,i) )
291 x11(i,i) = one
292 CALL clarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
293 $ x11(i+1,i), ldx11, work(ilarf) )
294 CALL clarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
295 $ x21(i,i), ldx21, work(ilarf) )
296 CALL clacgv( q-i+1, x11(i,i), ldx11 )
297 s = sqrt( scnrm2( p-i, x11(i+1,i), 1 )**2
298 $ + scnrm2( m-p-i+1, x21(i,i), 1 )**2 )
299 theta(i) = atan2( s, c )
300*
301 CALL cunbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
302 $ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
303 $ work(iorbdb5), lorbdb5, childinfo )
304 CALL cscal( p-i, negone, x11(i+1,i), 1 )
305 CALL clarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
306 IF( i .LT. p ) THEN
307 CALL clarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
308 phi(i) = atan2( real( x11(i+1,i) ), real( x21(i,i) ) )
309 c = cos( phi(i) )
310 s = sin( phi(i) )
311 x11(i+1,i) = one
312 CALL clarf( 'L', p-i, q-i, x11(i+1,i), 1, conjg(taup1(i)),
313 $ x11(i+1,i+1), ldx11, work(ilarf) )
314 END IF
315 x21(i,i) = one
316 CALL clarf( 'L', m-p-i+1, q-i, x21(i,i), 1, conjg(taup2(i)),
317 $ x21(i,i+1), ldx21, work(ilarf) )
318*
319 END DO
320*
321* Reduce the bottom-right portion of X21 to the identity matrix
322*
323 DO i = p + 1, q
324 CALL clarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
325 x21(i,i) = one
326 CALL clarf( 'L', m-p-i+1, q-i, x21(i,i), 1, conjg(taup2(i)),
327 $ x21(i,i+1), ldx21, work(ilarf) )
328 END DO
329*
330 RETURN
331*
332* End of CUNBDB2
333*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
subroutine csrot(N, CX, INCX, CY, INCY, C, S)
CSROT
Definition: csrot.f:98
subroutine clarfgp(N, ALPHA, X, INCX, TAU)
CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: clarfgp.f:104
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:74
subroutine clarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
CLARF applies an elementary reflector to a general rectangular matrix.
Definition: clarf.f:128
subroutine cunbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
CUNBDB5
Definition: cunbdb5.f:156
real(wp) function scnrm2(n, x, incx)
SCNRM2
Definition: scnrm2.f90:90
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