LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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◆ sorbdb2()

subroutine sorbdb2 ( integer m,
integer p,
integer q,
real, dimension(ldx11,*) x11,
integer ldx11,
real, dimension(ldx21,*) x21,
integer ldx21,
real, dimension(*) theta,
real, dimension(*) phi,
real, dimension(*) taup1,
real, dimension(*) taup2,
real, dimension(*) tauq1,
real, dimension(*) work,
integer lwork,
integer info )

SORBDB2

Download SORBDB2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
!>
!> SORBDB2 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 SORBDB1, SORBDB3, and SORBDB4 handle cases in
!> which P 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 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 REAL 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 REAL 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 REAL array, dimension (P-1)
!>           The scalar factors of the elementary reflectors that define
!>           P1.
!>
[out]TAUP2
!>          TAUP2 is REAL array, dimension (Q)
!>           The scalar factors of the elementary reflectors that define
!>           P2.
!>
[out]TAUQ1
!>          TAUQ1 is REAL array, dimension (Q)
!>           The scalar factors of the elementary reflectors that define
!>           Q1.
!>
[out]WORK
!>          WORK is REAL 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 SORCSD for details.
!>
!>  P1, P2, and Q1 are represented as products of elementary reflectors.
!>  See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
!>  and SORGLQ.
!>
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 197 of file sorbdb2.f.

200*
201* -- LAPACK computational routine --
202* -- LAPACK is a software package provided by Univ. of Tennessee, --
203* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
204*
205* .. Scalar Arguments ..
206 INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
207* ..
208* .. Array Arguments ..
209 REAL PHI(*), THETA(*)
210 REAL TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
211 $ X11(LDX11,*), X21(LDX21,*)
212* ..
213*
214* ====================================================================
215*
216* .. Parameters ..
217 REAL NEGONE
218 parameter( negone = -1.0e0 )
219* ..
220* .. Local Scalars ..
221 REAL C, S
222 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
223 $ LWORKMIN, LWORKOPT
224 LOGICAL LQUERY
225* ..
226* .. External Subroutines ..
227 EXTERNAL slarf1f, slarfgp, sorbdb5, srot, sscal,
228 $ xerbla
229* ..
230* .. External Functions ..
231 REAL SNRM2
232 EXTERNAL snrm2
233* ..
234* .. Intrinsic Function ..
235 INTRINSIC atan2, cos, max, sin, sqrt
236* ..
237* .. Executable Statements ..
238*
239* Test input arguments
240*
241 info = 0
242 lquery = lwork .EQ. -1
243*
244 IF( m .LT. 0 ) THEN
245 info = -1
246 ELSE IF( p .LT. 0 .OR. p .GT. m-p ) THEN
247 info = -2
248 ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) THEN
249 info = -3
250 ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
251 info = -5
252 ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
253 info = -7
254 END IF
255*
256* Compute workspace
257*
258 IF( info .EQ. 0 ) THEN
259 ilarf = 2
260 llarf = max( p-1, m-p, q-1 )
261 iorbdb5 = 2
262 lorbdb5 = q-1
263 lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
264 lworkmin = lworkopt
265 work(1) = real( lworkopt )
266 IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
267 info = -14
268 END IF
269 END IF
270 IF( info .NE. 0 ) THEN
271 CALL xerbla( 'SORBDB2', -info )
272 RETURN
273 ELSE IF( lquery ) THEN
274 RETURN
275 END IF
276*
277* Reduce rows 1, ..., P of X11 and X21
278*
279 DO i = 1, p
280*
281 IF( i .GT. 1 ) THEN
282 CALL srot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c,
283 $ s )
284 END IF
285 CALL slarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
286 c = x11(i,i)
287 CALL slarf1f( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
288 $ x11(i+1,i), ldx11, work(ilarf) )
289 CALL slarf1f( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11,
290 $ tauq1(i), x21(i,i), ldx21, work(ilarf) )
291 s = sqrt( snrm2( p-i, x11(i+1,i), 1 )**2
292 $ + snrm2( m-p-i+1, x21(i,i), 1 )**2 )
293 theta(i) = atan2( s, c )
294*
295 CALL sorbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
296 $ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
297 $ work(iorbdb5), lorbdb5, childinfo )
298 CALL sscal( p-i, negone, x11(i+1,i), 1 )
299 CALL slarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
300 IF( i .LT. p ) THEN
301 CALL slarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
302 phi(i) = atan2( x11(i+1,i), x21(i,i) )
303 c = cos( phi(i) )
304 s = sin( phi(i) )
305 CALL slarf1f( 'L', p-i, q-i, x11(i+1,i), 1, taup1(i),
306 $ x11(i+1,i+1), ldx11, work(ilarf) )
307 END IF
308 CALL slarf1f( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
309 $ x21(i,i+1), ldx21, work(ilarf) )
310*
311 END DO
312*
313* Reduce the bottom-right portion of X21 to the identity matrix
314*
315 DO i = p + 1, q
316 CALL slarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
317 CALL slarf1f( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
318 $ x21(i,i+1), ldx21, work(ilarf) )
319 END DO
320*
321 RETURN
322*
323* End of SORBDB2
324*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
slarfgp
subroutine slarfgp(n, alpha, x, incx, tau)
SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition slarfgp.f:102
snrm2
real(wp) function snrm2(n, x, incx)
SNRM2
Definition snrm2.f90:89
srot
subroutine srot(n, sx, incx, sy, incy, c, s)
SROT
Definition srot.f:92
sscal
subroutine sscal(n, sa, sx, incx)
SSCAL
Definition sscal.f:79
sorbdb5
subroutine sorbdb5(m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
SORBDB5
Definition sorbdb5.f:155
slarf1f
subroutine slarf1f(side, m, n, v, incv, tau, c, ldc, work)
SLARF1F applies an elementary reflector to a general rectangular
Definition slarf1f.f:123
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