 LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ sorbdb3()

 subroutine sorbdb3 ( 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 )

SORBDB3

Purpose:
``` SORBDB3 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. M-P must be no larger than P,
Q, or M-Q. Routines SORBDB1, SORBDB2, and SORBDB4 handle cases in
which M-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 (M-P)-by-(M-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 <= M. M-P <= min(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.```
Date
July 2012
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  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:
 Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 204 of file sorbdb3.f.

204 *
205 * -- LAPACK computational routine (version 3.7.1) --
206 * -- LAPACK is a software package provided by Univ. of Tennessee, --
207 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
208 * July 2012
209 *
210 * .. Scalar Arguments ..
211  INTEGER info, lwork, m, p, q, ldx11, ldx21
212 * ..
213 * .. Array Arguments ..
214  REAL phi(*), theta(*)
215  REAL taup1(*), taup2(*), tauq1(*), work(*),
216  \$ x11(ldx11,*), x21(ldx21,*)
217 * ..
218 *
219 * ====================================================================
220 *
221 * .. Parameters ..
222  REAL one
223  parameter( one = 1.0e0 )
224 * ..
225 * .. Local Scalars ..
226  REAL c, s
227  INTEGER childinfo, i, ilarf, iorbdb5, llarf, lorbdb5,
228  \$ lworkmin, lworkopt
229  LOGICAL lquery
230 * ..
231 * .. External Subroutines ..
232  EXTERNAL slarf, slarfgp, sorbdb5, srot, xerbla
233 * ..
234 * .. External Functions ..
235  REAL snrm2
236  EXTERNAL snrm2
237 * ..
238 * .. Intrinsic Function ..
239  INTRINSIC atan2, cos, max, sin, sqrt
240 * ..
241 * .. Executable Statements ..
242 *
243 * Test input arguments
244 *
245  info = 0
246  lquery = lwork .EQ. -1
247 *
248  IF( m .LT. 0 ) THEN
249  info = -1
250  ELSE IF( 2*p .LT. m .OR. p .GT. m ) THEN
251  info = -2
252  ELSE IF( q .LT. m-p .OR. m-q .LT. m-p ) THEN
253  info = -3
254  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
255  info = -5
256  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
257  info = -7
258  END IF
259 *
260 * Compute workspace
261 *
262  IF( info .EQ. 0 ) THEN
263  ilarf = 2
264  llarf = max( p, m-p-1, q-1 )
265  iorbdb5 = 2
266  lorbdb5 = q-1
267  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
268  lworkmin = lworkopt
269  work(1) = lworkopt
270  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
271  info = -14
272  END IF
273  END IF
274  IF( info .NE. 0 ) THEN
275  CALL xerbla( 'SORBDB3', -info )
276  RETURN
277  ELSE IF( lquery ) THEN
278  RETURN
279  END IF
280 *
281 * Reduce rows 1, ..., M-P of X11 and X21
282 *
283  DO i = 1, m-p
284 *
285  IF( i .GT. 1 ) THEN
286  CALL srot( q-i+1, x11(i-1,i), ldx11, x21(i,i), ldx11, c, s )
287  END IF
288 *
289  CALL slarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
290  s = x21(i,i)
291  x21(i,i) = one
292  CALL slarf( 'R', p-i+1, q-i+1, x21(i,i), ldx21, tauq1(i),
293  \$ x11(i,i), ldx11, work(ilarf) )
294  CALL slarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
295  \$ x21(i+1,i), ldx21, work(ilarf) )
296  c = sqrt( snrm2( p-i+1, x11(i,i), 1 )**2
297  \$ + snrm2( m-p-i, x21(i+1,i), 1 )**2 )
298  theta(i) = atan2( s, c )
299 *
300  CALL sorbdb5( p-i+1, m-p-i, q-i, x11(i,i), 1, x21(i+1,i), 1,
301  \$ x11(i,i+1), ldx11, x21(i+1,i+1), ldx21,
302  \$ work(iorbdb5), lorbdb5, childinfo )
303  CALL slarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
304  IF( i .LT. m-p ) THEN
305  CALL slarfgp( m-p-i, x21(i+1,i), x21(i+2,i), 1, taup2(i) )
306  phi(i) = atan2( x21(i+1,i), x11(i,i) )
307  c = cos( phi(i) )
308  s = sin( phi(i) )
309  x21(i+1,i) = one
310  CALL slarf( 'L', m-p-i, q-i, x21(i+1,i), 1, taup2(i),
311  \$ x21(i+1,i+1), ldx21, work(ilarf) )
312  END IF
313  x11(i,i) = one
314  CALL slarf( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,i+1),
315  \$ ldx11, work(ilarf) )
316 *
317  END DO
318 *
319 * Reduce the bottom-right portion of X11 to the identity matrix
320 *
321  DO i = m-p + 1, q
322  CALL slarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
323  x11(i,i) = one
324  CALL slarf( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,i+1),
325  \$ ldx11, work(ilarf) )
326  END DO
327 *
328  RETURN
329 *
330 * End of SORBDB3
331 *
real function snrm2(N, X, INCX)
SNRM2
Definition: snrm2.f:76
subroutine sorbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
SORBDB5
Definition: sorbdb5.f:158
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
SLARF applies an elementary reflector to a general rectangular matrix.
Definition: slarf.f:126
subroutine slarfgp(N, ALPHA, X, INCX, TAU)
SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: slarfgp.f:106
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:94
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