 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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  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 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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