LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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◆ zunbdb1()

subroutine zunbdb1 ( 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 )

ZUNBDB1

Download ZUNBDB1 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> ZUNBDB1 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 ZUNBDB2, ZUNBDB3, and ZUNBDB4 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*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)
!>           The scalar factors of the elementary reflectors that define
!>           P1.
!> 
[out]TAUP2
!>          TAUP2 is COMPLEX*16 array, dimension (M-P)
!>           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 zunbdb1.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 DOUBLE PRECISION PHI(*), THETA(*)
212 COMPLEX*16 TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
213 $ X11(LDX11,*), X21(LDX21,*)
214* ..
215*
216* ====================================================================
217*
218* .. Parameters ..
219 COMPLEX*16 ONE
220 parameter( 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 zlarf1f, zlarfgp, zunbdb5, zdrot,
230 $ xerbla
231 EXTERNAL zlacgv
232* ..
233* .. External Functions ..
234 DOUBLE PRECISION DZNRM2
235 EXTERNAL dznrm2
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. q .OR. m-p .LT. q ) THEN
250 info = -2
251 ELSE IF( q .LT. 0 .OR. m-q .LT. q ) 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-1, q-1 )
264 iorbdb5 = 2
265 lorbdb5 = q-2
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( 'ZUNBDB1', -info )
275 RETURN
276 ELSE IF( lquery ) THEN
277 RETURN
278 END IF
279*
280* Reduce columns 1, ..., Q of X11 and X21
281*
282 DO i = 1, q
283*
284 CALL zlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
285 CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
286 theta(i) = atan2( dble( x21(i,i) ), dble( x11(i,i) ) )
287 c = cos( theta(i) )
288 s = sin( theta(i) )
289 c = cos( theta(i) )
290 s = sin( theta(i) )
291 CALL zlarf1f( 'L', p-i+1, q-i, x11(i,i), 1, conjg(taup1(i)),
292 $ x11(i,i+1), ldx11, work(ilarf) )
293 CALL zlarf1f( 'L', m-p-i+1, q-i, x21(i,i), 1,
294 $ conjg(taup2(i)), x21(i,i+1), ldx21,
295 $ work(ilarf) )
296*
297 IF( i .LT. q ) THEN
298 CALL zdrot( q-i, x11(i,i+1), ldx11, x21(i,i+1), ldx21, c,
299 $ s )
300 CALL zlacgv( q-i, x21(i,i+1), ldx21 )
301 CALL zlarfgp( q-i, x21(i,i+1), x21(i,i+2), ldx21,
302 $ tauq1(i) )
303 s = dble( x21(i,i+1) )
304 CALL zlarf1f( 'R', p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
305 $ x11(i+1,i+1), ldx11, work(ilarf) )
306 CALL zlarf1f( 'R', m-p-i, q-i, x21(i,i+1), ldx21,
307 $ tauq1(i), x21(i+1,i+1), ldx21,
308 $ work(ilarf) )
309 CALL zlacgv( q-i, x21(i,i+1), ldx21 )
310 c = sqrt( dznrm2( p-i, x11(i+1,i+1), 1 )**2
311 $ + dznrm2( m-p-i, x21(i+1,i+1), 1 )**2 )
312 phi(i) = atan2( s, c )
313 CALL zunbdb5( p-i, m-p-i, q-i-1, x11(i+1,i+1), 1,
314 $ x21(i+1,i+1), 1, x11(i+1,i+2), ldx11,
315 $ x21(i+1,i+2), ldx21, work(iorbdb5), lorbdb5,
316 $ childinfo )
317 END IF
318*
319 END DO
320*
321 RETURN
322*
323* End of ZUNBDB1
324*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zlacgv(n, x, incx)
ZLACGV conjugates a complex vector.
Definition zlacgv.f:72
subroutine zlarf1f(side, m, n, v, incv, tau, c, ldc, work)
ZLARF1F applies an elementary reflector to a general rectangular
Definition zlarf1f.f:157
subroutine zlarfgp(n, alpha, x, incx, tau)
ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition zlarfgp.f:102
real(wp) function dznrm2(n, x, incx)
DZNRM2
Definition dznrm2.f90:90
subroutine zdrot(n, zx, incx, zy, incy, c, s)
ZDROT
Definition zdrot.f:98
subroutine zunbdb5(m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
ZUNBDB5
Definition zunbdb5.f:155
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