LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ cunbdb1()

subroutine cunbdb1 ( 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 
)

CUNBDB1

Download CUNBDB1 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CUNBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonormal columns:

                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]

 X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,
 M-P, or M-Q. Routines CUNBDB2, CUNBDB3, and CUNBDB4 handle cases in
 which 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 Q-by-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 <=
           MIN(P,M-P,M-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]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 cunbdb1.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 ONE
220 parameter( one = (1.0e0,0.0e0) )
221* ..
222* .. Local Scalars ..
223 REAL C, S
224 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
225 $ LWORKMIN, LWORKOPT
226 LOGICAL LQUERY
227* ..
228* .. External Subroutines ..
229 EXTERNAL clarf, clarfgp, cunbdb5, csrot, xerbla
230 EXTERNAL clacgv
231* ..
232* .. External Functions ..
233 REAL SCNRM2, SROUNDUP_LWORK
234 EXTERNAL scnrm2, sroundup_lwork
235* ..
236* .. Intrinsic Function ..
237 INTRINSIC atan2, cos, max, sin, sqrt
238* ..
239* .. Executable Statements ..
240*
241* Test input arguments
242*
243 info = 0
244 lquery = lwork .EQ. -1
245*
246 IF( m .LT. 0 ) THEN
247 info = -1
248 ELSE IF( p .LT. q .OR. m-p .LT. q ) THEN
249 info = -2
250 ELSE IF( q .LT. 0 .OR. m-q .LT. q ) THEN
251 info = -3
252 ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
253 info = -5
254 ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
255 info = -7
256 END IF
257*
258* Compute workspace
259*
260 IF( info .EQ. 0 ) THEN
261 ilarf = 2
262 llarf = max( p-1, m-p-1, q-1 )
263 iorbdb5 = 2
264 lorbdb5 = q-2
265 lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
266 lworkmin = lworkopt
267 work(1) = sroundup_lwork(lworkopt)
268 IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
269 info = -14
270 END IF
271 END IF
272 IF( info .NE. 0 ) THEN
273 CALL xerbla( 'CUNBDB1', -info )
274 RETURN
275 ELSE IF( lquery ) THEN
276 RETURN
277 END IF
278*
279* Reduce columns 1, ..., Q of X11 and X21
280*
281 DO i = 1, q
282*
283 CALL clarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
284 CALL clarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
285 theta(i) = atan2( real( x21(i,i) ), real( x11(i,i) ) )
286 c = cos( theta(i) )
287 s = sin( theta(i) )
288 x11(i,i) = one
289 x21(i,i) = one
290 CALL clarf( 'L', p-i+1, q-i, x11(i,i), 1, conjg(taup1(i)),
291 $ x11(i,i+1), ldx11, work(ilarf) )
292 CALL clarf( 'L', m-p-i+1, q-i, x21(i,i), 1, conjg(taup2(i)),
293 $ x21(i,i+1), ldx21, work(ilarf) )
294*
295 IF( i .LT. q ) THEN
296 CALL csrot( q-i, x11(i,i+1), ldx11, x21(i,i+1), ldx21, c,
297 $ s )
298 CALL clacgv( q-i, x21(i,i+1), ldx21 )
299 CALL clarfgp( q-i, x21(i,i+1), x21(i,i+2), ldx21, tauq1(i) )
300 s = real( x21(i,i+1) )
301 x21(i,i+1) = one
302 CALL clarf( 'R', p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
303 $ x11(i+1,i+1), ldx11, work(ilarf) )
304 CALL clarf( 'R', m-p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
305 $ x21(i+1,i+1), ldx21, work(ilarf) )
306 CALL clacgv( q-i, x21(i,i+1), ldx21 )
307 c = sqrt( scnrm2( p-i, x11(i+1,i+1), 1 )**2
308 $ + scnrm2( m-p-i, x21(i+1,i+1), 1 )**2 )
309 phi(i) = atan2( s, c )
310 CALL cunbdb5( p-i, m-p-i, q-i-1, x11(i+1,i+1), 1,
311 $ x21(i+1,i+1), 1, x11(i+1,i+2), ldx11,
312 $ x21(i+1,i+2), ldx21, work(iorbdb5), lorbdb5,
313 $ childinfo )
314 END IF
315*
316 END DO
317*
318 RETURN
319*
320* End of CUNBDB1
321*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
clacgv
subroutine clacgv(n, x, incx)
CLACGV conjugates a complex vector.
Definition clacgv.f:74
clarf
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
clarfgp
subroutine clarfgp(n, alpha, x, incx, tau)
CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition clarfgp.f:104
scnrm2
real(wp) function scnrm2(n, x, incx)
SCNRM2
Definition scnrm2.f90:90
csrot
subroutine csrot(n, cx, incx, cy, incy, c, s)
CSROT
Definition csrot.f:98
sroundup_lwork
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
Definition sroundup_lwork.f:59
cunbdb5
subroutine cunbdb5(m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
CUNBDB5
Definition cunbdb5.f:156
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