LAPACK  3.8.0 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

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.```
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 [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 203 of file sorbdb2.f.

203 *
204 * -- LAPACK computational routine (version 3.7.1) --
205 * -- LAPACK is a software package provided by Univ. of Tennessee, --
206 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
207 * July 2012
208 *
209 * .. Scalar Arguments ..
210  INTEGER info, lwork, m, p, q, ldx11, ldx21
211 * ..
212 * .. Array Arguments ..
213  REAL phi(*), theta(*)
214  REAL taup1(*), taup2(*), tauq1(*), work(*),
215  \$ x11(ldx11,*), x21(ldx21,*)
216 * ..
217 *
218 * ====================================================================
219 *
220 * .. Parameters ..
221  REAL negone, one
222  parameter( negone = -1.0e0, one = 1.0e0 )
223 * ..
224 * .. Local Scalars ..
225  REAL c, s
226  INTEGER childinfo, i, ilarf, iorbdb5, llarf, lorbdb5,
227  \$ lworkmin, lworkopt
228  LOGICAL lquery
229 * ..
230 * .. External Subroutines ..
231  EXTERNAL slarf, slarfgp, sorbdb5, srot, sscal, xerbla
232 * ..
233 * .. External Functions ..
234  REAL snrm2
235  EXTERNAL snrm2
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. 0 .OR. p .GT. m-p ) THEN
250  info = -2
251  ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) 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, q-1 )
264  iorbdb5 = 2
265  lorbdb5 = q-1
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( 'SORBDB2', -info )
275  RETURN
276  ELSE IF( lquery ) THEN
277  RETURN
278  END IF
279 *
280 * Reduce rows 1, ..., P of X11 and X21
281 *
282  DO i = 1, p
283 *
284  IF( i .GT. 1 ) THEN
285  CALL srot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c, s )
286  END IF
287  CALL slarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
288  c = x11(i,i)
289  x11(i,i) = one
290  CALL slarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
291  \$ x11(i+1,i), ldx11, work(ilarf) )
292  CALL slarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
293  \$ x21(i,i), ldx21, work(ilarf) )
294  s = sqrt( snrm2( p-i, x11(i+1,i), 1 )**2
295  \$ + snrm2( m-p-i+1, x21(i,i), 1 )**2 )
296  theta(i) = atan2( s, c )
297 *
298  CALL sorbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
299  \$ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
300  \$ work(iorbdb5), lorbdb5, childinfo )
301  CALL sscal( p-i, negone, x11(i+1,i), 1 )
302  CALL slarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
303  IF( i .LT. p ) THEN
304  CALL slarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
305  phi(i) = atan2( x11(i+1,i), x21(i,i) )
306  c = cos( phi(i) )
307  s = sin( phi(i) )
308  x11(i+1,i) = one
309  CALL slarf( 'L', p-i, q-i, x11(i+1,i), 1, taup1(i),
310  \$ x11(i+1,i+1), ldx11, work(ilarf) )
311  END IF
312  x21(i,i) = one
313  CALL slarf( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
314  \$ x21(i,i+1), ldx21, work(ilarf) )
315 *
316  END DO
317 *
318 * Reduce the bottom-right portion of X21 to the identity matrix
319 *
320  DO i = p + 1, q
321  CALL slarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
322  x21(i,i) = one
323  CALL slarf( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
324  \$ x21(i,i+1), ldx21, work(ilarf) )
325  END DO
326 *
327  RETURN
328 *
329 * End of SORBDB2
330 *
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 sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:81
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:94
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