LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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-1) The scalar factors of the elementary reflectors that define P1.``` [out] TAUP2 ``` TAUP2 is REAL array, dimension (Q) 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 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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