LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ 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 orthonomal 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)
           The scalar factors of the elementary reflectors that define
           P1.
[out]TAUP2
          TAUP2 is REAL array, dimension (M-P)
           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 199 of file sorbdb2.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  REAL PHI(*), THETA(*)
211  REAL TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
212  $ X11(LDX11,*), X21(LDX21,*)
213 * ..
214 *
215 * ====================================================================
216 *
217 * .. Parameters ..
218  REAL NEGONE, ONE
219  parameter( negone = -1.0e0, one = 1.0e0 )
220 * ..
221 * .. Local Scalars ..
222  REAL C, S
223  INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
224  $ LWORKMIN, LWORKOPT
225  LOGICAL LQUERY
226 * ..
227 * .. External Subroutines ..
228  EXTERNAL slarf, slarfgp, sorbdb5, srot, sscal, 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) = 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, s )
283  END IF
284  CALL slarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
285  c = x11(i,i)
286  x11(i,i) = one
287  CALL slarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
288  $ x11(i+1,i), ldx11, work(ilarf) )
289  CALL slarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
290  $ 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  x11(i+1,i) = one
306  CALL slarf( 'L', p-i, q-i, x11(i+1,i), 1, taup1(i),
307  $ x11(i+1,i+1), ldx11, work(ilarf) )
308  END IF
309  x21(i,i) = one
310  CALL slarf( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
311  $ x21(i,i+1), ldx21, work(ilarf) )
312 *
313  END DO
314 *
315 * Reduce the bottom-right portion of X21 to the identity matrix
316 *
317  DO i = p + 1, q
318  CALL slarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
319  x21(i,i) = one
320  CALL slarf( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
321  $ x21(i,i+1), ldx21, work(ilarf) )
322  END DO
323 *
324  RETURN
325 *
326 * End of SORBDB2
327 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slarfgp(N, ALPHA, X, INCX, TAU)
SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: slarfgp.f:104
subroutine slarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
SLARF applies an elementary reflector to a general rectangular matrix.
Definition: slarf.f:124
subroutine sorbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
SORBDB5
Definition: sorbdb5.f:156
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:92
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real(wp) function snrm2(n, x, incx)
SNRM2
Definition: snrm2.f90:89
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