LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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◆ dorbdb4()

subroutine dorbdb4 ( integer m,
integer p,
integer q,
double precision, dimension(ldx11,*) x11,
integer ldx11,
double precision, dimension(ldx21,*) x21,
integer ldx21,
double precision, dimension(*) theta,
double precision, dimension(*) phi,
double precision, dimension(*) taup1,
double precision, dimension(*) taup2,
double precision, dimension(*) tauq1,
double precision, dimension(*) phantom,
double precision, dimension(*) work,
integer lwork,
integer info )

DORBDB4

Download DORBDB4 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> DORBDB4 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. M-Q must be no larger than P,
!> M-P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in
!> which M-Q is not the minimum dimension.
!>
!> The orthogonal 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (M-Q)
!>           The scalar factors of the elementary reflectors that define
!>           P1.
!>
[out]TAUP2
!>          TAUP2 is DOUBLE PRECISION array, dimension (M-Q)
!>           The scalar factors of the elementary reflectors that define
!>           P2.
!>
[out]TAUQ1
!>          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
!>           The scalar factors of the elementary reflectors that define
!>           Q1.
!>
[out]PHANTOM
!>          PHANTOM is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DORCSD for details.
!>
!>  P1, P2, and Q1 are represented as products of elementary reflectors.
!>  See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
!>  and DORGLQ.
!>
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 208 of file dorbdb4.f.

212*
213* -- LAPACK computational routine --
214* -- LAPACK is a software package provided by Univ. of Tennessee, --
215* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
216*
217* .. Scalar Arguments ..
218 INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
219* ..
220* .. Array Arguments ..
221 DOUBLE PRECISION PHI(*), THETA(*)
222 DOUBLE PRECISION PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*),
223 $ WORK(*), X11(LDX11,*), X21(LDX21,*)
224* ..
225*
226* ====================================================================
227*
228* .. Parameters ..
229 DOUBLE PRECISION NEGONE, ONE, ZERO
230 parameter( negone = -1.0d0, one = 1.0d0, zero = 0.0d0 )
231* ..
232* .. Local Scalars ..
233 DOUBLE PRECISION C, S
234 INTEGER CHILDINFO, I, ILARF, IORBDB5, J, LLARF,
235 $ LORBDB5, LWORKMIN, LWORKOPT
236 LOGICAL LQUERY
237* ..
238* .. External Subroutines ..
239 EXTERNAL dlarf, dlarfgp, dorbdb5, drot, dscal,
240 $ xerbla
241* ..
242* .. External Functions ..
243 DOUBLE PRECISION DNRM2
244 EXTERNAL dnrm2
245* ..
246* .. Intrinsic Function ..
247 INTRINSIC atan2, cos, max, sin, sqrt
248* ..
249* .. Executable Statements ..
250*
251* Test input arguments
252*
253 info = 0
254 lquery = lwork .EQ. -1
255*
256 IF( m .LT. 0 ) THEN
257 info = -1
258 ELSE IF( p .LT. m-q .OR. m-p .LT. m-q ) THEN
259 info = -2
260 ELSE IF( q .LT. m-q .OR. q .GT. m ) THEN
261 info = -3
262 ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
263 info = -5
264 ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
265 info = -7
266 END IF
267*
268* Compute workspace
269*
270 IF( info .EQ. 0 ) THEN
271 ilarf = 2
272 llarf = max( q-1, p-1, m-p-1 )
273 iorbdb5 = 2
274 lorbdb5 = q
275 lworkopt = ilarf + llarf - 1
276 lworkopt = max( lworkopt, iorbdb5 + lorbdb5 - 1 )
277 lworkmin = lworkopt
278 work(1) = lworkopt
279 IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
280 info = -14
281 END IF
282 END IF
283 IF( info .NE. 0 ) THEN
284 CALL xerbla( 'DORBDB4', -info )
285 RETURN
286 ELSE IF( lquery ) THEN
287 RETURN
288 END IF
289*
290* Reduce columns 1, ..., M-Q of X11 and X21
291*
292 DO i = 1, m-q
293*
294 IF( i .EQ. 1 ) THEN
295 DO j = 1, m
296 phantom(j) = zero
297 END DO
298 CALL dorbdb5( p, m-p, q, phantom(1), 1, phantom(p+1), 1,
299 $ x11, ldx11, x21, ldx21, work(iorbdb5),
300 $ lorbdb5, childinfo )
301 CALL dscal( p, negone, phantom(1), 1 )
302 CALL dlarfgp( p, phantom(1), phantom(2), 1, taup1(1) )
303 CALL dlarfgp( m-p, phantom(p+1), phantom(p+2), 1,
304 $ taup2(1) )
305 theta(i) = atan2( phantom(1), phantom(p+1) )
306 c = cos( theta(i) )
307 s = sin( theta(i) )
308 CALL dlarf1f( 'L', p, q, phantom(1), 1, taup1(1), x11,
309 $ ldx11, work(ilarf) )
310 CALL dlarf1f( 'L', m-p, q, phantom(p+1), 1, taup2(1),
311 $ x21, ldx21, work(ilarf) )
312 ELSE
313 CALL dorbdb5( p-i+1, m-p-i+1, q-i+1, x11(i,i-1), 1,
314 $ x21(i,i-1), 1, x11(i,i), ldx11, x21(i,i),
315 $ ldx21, work(iorbdb5), lorbdb5, childinfo )
316 CALL dscal( p-i+1, negone, x11(i,i-1), 1 )
317 CALL dlarfgp( p-i+1, x11(i,i-1), x11(i+1,i-1), 1,
318 $ taup1(i) )
319 CALL dlarfgp( m-p-i+1, x21(i,i-1), x21(i+1,i-1), 1,
320 $ taup2(i) )
321 theta(i) = atan2( x11(i,i-1), x21(i,i-1) )
322 c = cos( theta(i) )
323 s = sin( theta(i) )
324 CALL dlarf1f( 'L', p-i+1, q-i+1, x11(i,i-1), 1, taup1(i),
325 $ x11(i,i), ldx11, work(ilarf) )
326 CALL dlarf1f( 'L', m-p-i+1, q-i+1, x21(i,i-1), 1,
327 $ taup2(i), x21(i,i), ldx21, work(ilarf) )
328 END IF
329*
330 CALL drot( q-i+1, x11(i,i), ldx11, x21(i,i), ldx21, s, -c )
331 CALL dlarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
332 c = x21(i,i)
333 CALL dlarf1f( 'R', p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
334 $ x11(i+1,i), ldx11, work(ilarf) )
335 CALL dlarf1f( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
336 $ x21(i+1,i), ldx21, work(ilarf) )
337 IF( i .LT. m-q ) THEN
338 s = sqrt( dnrm2( p-i, x11(i+1,i), 1 )**2
339 $ + dnrm2( m-p-i, x21(i+1,i), 1 )**2 )
340 phi(i) = atan2( s, c )
341 END IF
342*
343 END DO
344*
345* Reduce the bottom-right portion of X11 to [ I 0 ]
346*
347 DO i = m - q + 1, p
348 CALL dlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
349 CALL dlarf1f( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
350 $ x11(i+1,i), ldx11, work(ilarf) )
351 CALL dlarf1f( 'R', q-p, q-i+1, x11(i,i), ldx11, tauq1(i),
352 $ x21(m-q+1,i), ldx21, work(ilarf) )
353 END DO
354*
355* Reduce the bottom-right portion of X21 to [ 0 I ]
356*
357 DO i = p + 1, q
358 CALL dlarfgp( q-i+1, x21(m-q+i-p,i), x21(m-q+i-p,i+1),
359 $ ldx21,
360 $ tauq1(i) )
361 CALL dlarf1f( 'R', q-i, q-i+1, x21(m-q+i-p,i), ldx21,
362 $ tauq1(i),
363 $ x21(m-q+i-p+1,i), ldx21, work(ilarf) )
364 END DO
365*
366 RETURN
367*
368* End of DORBDB4
369*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
dlarf1f
subroutine dlarf1f(side, m, n, v, incv, tau, c, ldc, work)
DLARF1F applies an elementary reflector to a general rectangular
Definition dlarf1f.f:157
dlarf
subroutine dlarf(side, m, n, v, incv, tau, c, ldc, work)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition dlarf.f:122
dlarfgp
subroutine dlarfgp(n, alpha, x, incx, tau)
DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition dlarfgp.f:102
dnrm2
real(wp) function dnrm2(n, x, incx)
DNRM2
Definition dnrm2.f90:89
drot
subroutine drot(n, dx, incx, dy, incy, c, s)
DROT
Definition drot.f:92
dscal
subroutine dscal(n, da, dx, incx)
DSCAL
Definition dscal.f:79
dorbdb5
subroutine dorbdb5(m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
DORBDB5
Definition dorbdb5.f:155
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