LAPACK  3.10.1 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.```
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 200 of file sorbdb3.f.

202 *
203 * -- LAPACK computational routine --
204 * -- LAPACK is a software package provided by Univ. of Tennessee, --
205 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
206 *
207 * .. Scalar Arguments ..
208  INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
209 * ..
210 * .. Array Arguments ..
211  REAL PHI(*), THETA(*)
212  REAL TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
213  \$ X11(LDX11,*), X21(LDX21,*)
214 * ..
215 *
216 * ====================================================================
217 *
218 * .. Parameters ..
219  REAL ONE
220  parameter( one = 1.0e0 )
221 * ..
222 * .. Local Scalars ..
223  REAL C, S
224  INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
225  \$ LWORKMIN, LWORKOPT
226  LOGICAL LQUERY
227 * ..
228 * .. External Subroutines ..
229  EXTERNAL slarf, slarfgp, sorbdb5, srot, xerbla
230 * ..
231 * .. External Functions ..
232  REAL SNRM2
233  EXTERNAL snrm2
234 * ..
235 * .. Intrinsic Function ..
236  INTRINSIC atan2, cos, max, sin, sqrt
237 * ..
238 * .. Executable Statements ..
239 *
240 * Test input arguments
241 *
242  info = 0
243  lquery = lwork .EQ. -1
244 *
245  IF( m .LT. 0 ) THEN
246  info = -1
247  ELSE IF( 2*p .LT. m .OR. p .GT. m ) THEN
248  info = -2
249  ELSE IF( q .LT. m-p .OR. m-q .LT. m-p ) THEN
250  info = -3
251  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
252  info = -5
253  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
254  info = -7
255  END IF
256 *
257 * Compute workspace
258 *
259  IF( info .EQ. 0 ) THEN
260  ilarf = 2
261  llarf = max( p, m-p-1, q-1 )
262  iorbdb5 = 2
263  lorbdb5 = q-1
264  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
265  lworkmin = lworkopt
266  work(1) = lworkopt
267  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
268  info = -14
269  END IF
270  END IF
271  IF( info .NE. 0 ) THEN
272  CALL xerbla( 'SORBDB3', -info )
273  RETURN
274  ELSE IF( lquery ) THEN
275  RETURN
276  END IF
277 *
278 * Reduce rows 1, ..., M-P of X11 and X21
279 *
280  DO i = 1, m-p
281 *
282  IF( i .GT. 1 ) THEN
283  CALL srot( q-i+1, x11(i-1,i), ldx11, x21(i,i), ldx11, c, s )
284  END IF
285 *
286  CALL slarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
287  s = x21(i,i)
288  x21(i,i) = one
289  CALL slarf( 'R', p-i+1, q-i+1, x21(i,i), ldx21, tauq1(i),
290  \$ x11(i,i), ldx11, work(ilarf) )
291  CALL slarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
292  \$ x21(i+1,i), ldx21, work(ilarf) )
293  c = sqrt( snrm2( p-i+1, x11(i,i), 1 )**2
294  \$ + snrm2( m-p-i, x21(i+1,i), 1 )**2 )
295  theta(i) = atan2( s, c )
296 *
297  CALL sorbdb5( p-i+1, m-p-i, q-i, x11(i,i), 1, x21(i+1,i), 1,
298  \$ x11(i,i+1), ldx11, x21(i+1,i+1), ldx21,
299  \$ work(iorbdb5), lorbdb5, childinfo )
300  CALL slarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
301  IF( i .LT. m-p ) THEN
302  CALL slarfgp( m-p-i, x21(i+1,i), x21(i+2,i), 1, taup2(i) )
303  phi(i) = atan2( x21(i+1,i), x11(i,i) )
304  c = cos( phi(i) )
305  s = sin( phi(i) )
306  x21(i+1,i) = one
307  CALL slarf( 'L', m-p-i, q-i, x21(i+1,i), 1, taup2(i),
308  \$ x21(i+1,i+1), ldx21, work(ilarf) )
309  END IF
310  x11(i,i) = one
311  CALL slarf( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,i+1),
312  \$ ldx11, work(ilarf) )
313 *
314  END DO
315 *
316 * Reduce the bottom-right portion of X11 to the identity matrix
317 *
318  DO i = m-p + 1, q
319  CALL slarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
320  x11(i,i) = one
321  CALL slarf( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,i+1),
322  \$ ldx11, work(ilarf) )
323  END DO
324 *
325  RETURN
326 *
327 * End of SORBDB3
328 *
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
real(wp) function snrm2(n, x, incx)
SNRM2
Definition: snrm2.f90:89
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