LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
Loading...
Searching...
No Matches

◆ zunbdb2()

subroutine zunbdb2 ( integer  m,
integer  p,
integer  q,
complex*16, dimension(ldx11,*)  x11,
integer  ldx11,
complex*16, dimension(ldx21,*)  x21,
integer  ldx21,
double precision, dimension(*)  theta,
double precision, dimension(*)  phi,
complex*16, dimension(*)  taup1,
complex*16, dimension(*)  taup2,
complex*16, dimension(*)  tauq1,
complex*16, dimension(*)  work,
integer  lwork,
integer  info 
)

ZUNBDB2

Download ZUNBDB2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 ZUNBDB2 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. P must be no larger than M-P,
 Q, or M-Q. Routines ZUNBDB1, ZUNBDB3, and ZUNBDB4 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*16 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*16 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 DOUBLE PRECISION array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.
[out]PHI
          PHI is DOUBLE PRECISION 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*16 array, dimension (P-1)
           The scalar factors of the elementary reflectors that define
           P1.
[out]TAUP2
          TAUP2 is COMPLEX*16 array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           P2.
[out]TAUQ1
          TAUQ1 is COMPLEX*16 array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.
[out]WORK
          WORK is COMPLEX*16 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 ZUNCSD for details.

  P1, P2, and Q1 are represented as products of elementary reflectors.
  See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR
  and ZUNGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 199 of file zunbdb2.f.

201*
202* -- LAPACK computational routine --
203* -- LAPACK is a software package provided by Univ. of Tennessee, --
204* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
205*
206* .. Scalar Arguments ..
207 INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
208* ..
209* .. Array Arguments ..
210 DOUBLE PRECISION PHI(*), THETA(*)
211 COMPLEX*16 TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
212 $ X11(LDX11,*), X21(LDX21,*)
213* ..
214*
215* ====================================================================
216*
217* .. Parameters ..
218 COMPLEX*16 NEGONE, ONE
219 parameter( negone = (-1.0d0,0.0d0),
220 $ one = (1.0d0,0.0d0) )
221* ..
222* .. Local Scalars ..
223 DOUBLE PRECISION C, S
224 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
225 $ LWORKMIN, LWORKOPT
226 LOGICAL LQUERY
227* ..
228* .. External Subroutines ..
229 EXTERNAL zlarf, zlarfgp, zunbdb5, zdrot, zscal, zlacgv,
230 $ xerbla
231* ..
232* .. External Functions ..
233 DOUBLE PRECISION DZNRM2
234 EXTERNAL dznrm2
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. 0 .OR. p .GT. m-p ) THEN
249 info = -2
250 ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) 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, q-1 )
263 iorbdb5 = 2
264 lorbdb5 = q-1
265 lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
266 lworkmin = lworkopt
267 work(1) = 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( 'ZUNBDB2', -info )
274 RETURN
275 ELSE IF( lquery ) THEN
276 RETURN
277 END IF
278*
279* Reduce rows 1, ..., P of X11 and X21
280*
281 DO i = 1, p
282*
283 IF( i .GT. 1 ) THEN
284 CALL zdrot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c,
285 $ s )
286 END IF
287 CALL zlacgv( q-i+1, x11(i,i), ldx11 )
288 CALL zlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
289 c = dble( x11(i,i) )
290 x11(i,i) = one
291 CALL zlarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
292 $ x11(i+1,i), ldx11, work(ilarf) )
293 CALL zlarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
294 $ x21(i,i), ldx21, work(ilarf) )
295 CALL zlacgv( q-i+1, x11(i,i), ldx11 )
296 s = sqrt( dznrm2( p-i, x11(i+1,i), 1 )**2
297 $ + dznrm2( m-p-i+1, x21(i,i), 1 )**2 )
298 theta(i) = atan2( s, c )
299*
300 CALL zunbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
301 $ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
302 $ work(iorbdb5), lorbdb5, childinfo )
303 CALL zscal( p-i, negone, x11(i+1,i), 1 )
304 CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
305 IF( i .LT. p ) THEN
306 CALL zlarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
307 phi(i) = atan2( dble( x11(i+1,i) ), dble( x21(i,i) ) )
308 c = cos( phi(i) )
309 s = sin( phi(i) )
310 x11(i+1,i) = one
311 CALL zlarf( 'L', p-i, q-i, x11(i+1,i), 1, dconjg(taup1(i)),
312 $ x11(i+1,i+1), ldx11, work(ilarf) )
313 END IF
314 x21(i,i) = one
315 CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
316 $ x21(i,i+1), ldx21, work(ilarf) )
317*
318 END DO
319*
320* Reduce the bottom-right portion of X21 to the identity matrix
321*
322 DO i = p + 1, q
323 CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
324 x21(i,i) = one
325 CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
326 $ x21(i,i+1), ldx21, work(ilarf) )
327 END DO
328*
329 RETURN
330*
331* End of ZUNBDB2
332*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
zlacgv
subroutine zlacgv(n, x, incx)
ZLACGV conjugates a complex vector.
Definition zlacgv.f:74
zlarf
subroutine zlarf(side, m, n, v, incv, tau, c, ldc, work)
ZLARF applies an elementary reflector to a general rectangular matrix.
Definition zlarf.f:128
zlarfgp
subroutine zlarfgp(n, alpha, x, incx, tau)
ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition zlarfgp.f:104
dznrm2
real(wp) function dznrm2(n, x, incx)
DZNRM2
Definition dznrm2.f90:90
zdrot
subroutine zdrot(n, zx, incx, zy, incy, c, s)
ZDROT
Definition zdrot.f:98
zscal
subroutine zscal(n, za, zx, incx)
ZSCAL
Definition zscal.f:78
zunbdb5
subroutine zunbdb5(m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
ZUNBDB5
Definition zunbdb5.f:156
Here is the call graph for this function:
Here is the caller graph for this function: