LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ sorbdb1()

 subroutine sorbdb1 ( 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 )

SORBDB1

Purpose:
``` SORBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny
matrix X with orthonormal columns:

[ B11 ]
[ X11 ]   [ P1 |    ] [  0  ]
[-----] = [---------] [-----] Q1**T .
[ X21 ]   [    | P2 ] [ B21 ]
[  0  ]

X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,
M-P, or M-Q. Routines SORBDB2, SORBDB3, and SORBDB4 handle cases in
which Q 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 Q-by-Q 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.``` [in] Q ``` Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).``` [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 201 of file sorbdb1.f.

203*
204* -- LAPACK computational routine --
205* -- LAPACK is a software package provided by Univ. of Tennessee, --
206* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
207*
208* .. Scalar Arguments ..
209 INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
210* ..
211* .. Array Arguments ..
212 REAL PHI(*), THETA(*)
213 REAL TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
214 \$ X11(LDX11,*), X21(LDX21,*)
215* ..
216*
217* ====================================================================
218*
219* .. Parameters ..
220 REAL ONE
221 parameter( one = 1.0e0 )
222* ..
223* .. Local Scalars ..
224 REAL C, S
225 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
226 \$ LWORKMIN, LWORKOPT
227 LOGICAL LQUERY
228* ..
229* .. External Subroutines ..
230 EXTERNAL slarf, slarfgp, sorbdb5, srot, xerbla
231* ..
232* .. External Functions ..
233 REAL SNRM2
234 EXTERNAL snrm2
235* ..
236* .. Intrinsic Function ..
237 INTRINSIC atan2, cos, max, sin, sqrt
238* ..
239* .. Executable Statements ..
240*
241* Test input arguments
242*
243 info = 0
244 lquery = lwork .EQ. -1
245*
246 IF( m .LT. 0 ) THEN
247 info = -1
248 ELSE IF( p .LT. q .OR. m-p .LT. q ) THEN
249 info = -2
250 ELSE IF( q .LT. 0 .OR. m-q .LT. q ) THEN
251 info = -3
252 ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
253 info = -5
254 ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
255 info = -7
256 END IF
257*
258* Compute workspace
259*
260 IF( info .EQ. 0 ) THEN
261 ilarf = 2
262 llarf = max( p-1, m-p-1, q-1 )
263 iorbdb5 = 2
264 lorbdb5 = q-2
265 lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
266 lworkmin = lworkopt
267 work(1) = lworkopt
268 IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
269 info = -14
270 END IF
271 END IF
272 IF( info .NE. 0 ) THEN
273 CALL xerbla( 'SORBDB1', -info )
274 RETURN
275 ELSE IF( lquery ) THEN
276 RETURN
277 END IF
278*
279* Reduce columns 1, ..., Q of X11 and X21
280*
281 DO i = 1, q
282*
283 CALL slarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
284 CALL slarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
285 theta(i) = atan2( x21(i,i), x11(i,i) )
286 c = cos( theta(i) )
287 s = sin( theta(i) )
288 x11(i,i) = one
289 x21(i,i) = one
290 CALL slarf( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,i+1),
291 \$ ldx11, work(ilarf) )
292 CALL slarf( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
293 \$ x21(i,i+1), ldx21, work(ilarf) )
294*
295 IF( i .LT. q ) THEN
296 CALL srot( q-i, x11(i,i+1), ldx11, x21(i,i+1), ldx21, c, s )
297 CALL slarfgp( q-i, x21(i,i+1), x21(i,i+2), ldx21, tauq1(i) )
298 s = x21(i,i+1)
299 x21(i,i+1) = one
300 CALL slarf( 'R', p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
301 \$ x11(i+1,i+1), ldx11, work(ilarf) )
302 CALL slarf( 'R', m-p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
303 \$ x21(i+1,i+1), ldx21, work(ilarf) )
304 c = sqrt( snrm2( p-i, x11(i+1,i+1), 1 )**2
305 \$ + snrm2( m-p-i, x21(i+1,i+1), 1 )**2 )
306 phi(i) = atan2( s, c )
307 CALL sorbdb5( p-i, m-p-i, q-i-1, x11(i+1,i+1), 1,
308 \$ x21(i+1,i+1), 1, x11(i+1,i+2), ldx11,
309 \$ x21(i+1,i+2), ldx21, work(iorbdb5), lorbdb5,
310 \$ childinfo )
311 END IF
312*
313 END DO
314*
315 RETURN
316*
317* End of SORBDB1
318*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
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 slarfgp(n, alpha, x, incx, tau)
SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition slarfgp.f:104
real(wp) function snrm2(n, x, incx)
SNRM2
Definition snrm2.f90:89
subroutine srot(n, sx, incx, sy, incy, c, s)
SROT
Definition srot.f:92
subroutine sorbdb5(m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
SORBDB5
Definition sorbdb5.f:156
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