LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
Searching...
No Matches

## ◆ cunbdb3()

 subroutine cunbdb3 ( integer m, integer p, integer q, complex, dimension(ldx11,*) x11, integer ldx11, complex, dimension(ldx21,*) x21, integer ldx21, real, dimension(*) theta, real, dimension(*) phi, complex, dimension(*) taup1, complex, dimension(*) taup2, complex, dimension(*) tauq1, complex, dimension(*) work, integer lwork, integer info )

CUNBDB3

Purpose:
``` CUNBDB3 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. M-P must be no larger than P,
Q, or M-Q. Routines CUNBDB1, CUNBDB2, and CUNBDB4 handle cases in
which M-P is not the minimum dimension.

The unitary 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 COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (P) The scalar factors of the elementary reflectors that define P1.``` [out] TAUP2 ``` TAUP2 is COMPLEX array, dimension (M-P) The scalar factors of the elementary reflectors that define P2.``` [out] TAUQ1 ``` TAUQ1 is COMPLEX array, dimension (Q) The scalar factors of the elementary reflectors that define Q1.``` [out] WORK ` WORK is COMPLEX 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 CUNCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors.
See CUNCSD2BY1 for details on generating P1, P2, and Q1 using CUNGQR
and CUNGLQ.```
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 200 of file cunbdb3.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 COMPLEX TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
213 \$ X11(LDX11,*), X21(LDX21,*)
214* ..
215*
216* ====================================================================
217*
218* .. Parameters ..
219 COMPLEX ONE
220 parameter( one = (1.0e0,0.0e0) )
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 clarf, clarfgp, cunbdb5, csrot, clacgv, xerbla
230* ..
231* .. External Functions ..
232 REAL SCNRM2, SROUNDUP_LWORK
233 EXTERNAL scnrm2, sroundup_lwork
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) = sroundup_lwork(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( 'CUNBDB3', -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 csrot( q-i+1, x11(i-1,i), ldx11, x21(i,i), ldx11, c,
284 \$ s )
285 END IF
286*
287 CALL clacgv( q-i+1, x21(i,i), ldx21 )
288 CALL clarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
289 s = real( x21(i,i) )
290 x21(i,i) = one
291 CALL clarf( 'R', p-i+1, q-i+1, x21(i,i), ldx21, tauq1(i),
292 \$ x11(i,i), ldx11, work(ilarf) )
293 CALL clarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
294 \$ x21(i+1,i), ldx21, work(ilarf) )
295 CALL clacgv( q-i+1, x21(i,i), ldx21 )
296 c = sqrt( scnrm2( p-i+1, x11(i,i), 1 )**2
297 \$ + scnrm2( m-p-i, x21(i+1,i), 1 )**2 )
298 theta(i) = atan2( s, c )
299*
300 CALL cunbdb5( p-i+1, m-p-i, q-i, x11(i,i), 1, x21(i+1,i), 1,
301 \$ x11(i,i+1), ldx11, x21(i+1,i+1), ldx21,
302 \$ work(iorbdb5), lorbdb5, childinfo )
303 CALL clarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
304 IF( i .LT. m-p ) THEN
305 CALL clarfgp( m-p-i, x21(i+1,i), x21(i+2,i), 1, taup2(i) )
306 phi(i) = atan2( real( x21(i+1,i) ), real( x11(i,i) ) )
307 c = cos( phi(i) )
308 s = sin( phi(i) )
309 x21(i+1,i) = one
310 CALL clarf( 'L', m-p-i, q-i, x21(i+1,i), 1, conjg(taup2(i)),
311 \$ x21(i+1,i+1), ldx21, work(ilarf) )
312 END IF
313 x11(i,i) = one
314 CALL clarf( 'L', p-i+1, q-i, x11(i,i), 1, conjg(taup1(i)),
315 \$ x11(i,i+1), ldx11, work(ilarf) )
316*
317 END DO
318*
319* Reduce the bottom-right portion of X11 to the identity matrix
320*
321 DO i = m-p + 1, q
322 CALL clarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
323 x11(i,i) = one
324 CALL clarf( 'L', p-i+1, q-i, x11(i,i), 1, conjg(taup1(i)),
325 \$ x11(i,i+1), ldx11, work(ilarf) )
326 END DO
327*
328 RETURN
329*
330* End of CUNBDB3
331*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine clacgv(n, x, incx)
CLACGV conjugates a complex vector.
Definition clacgv.f:74
subroutine clarf(side, m, n, v, incv, tau, c, ldc, work)
CLARF applies an elementary reflector to a general rectangular matrix.
Definition clarf.f:128
subroutine clarfgp(n, alpha, x, incx, tau)
CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition clarfgp.f:104
real(wp) function scnrm2(n, x, incx)
SCNRM2
Definition scnrm2.f90:90
subroutine csrot(n, cx, incx, cy, incy, c, s)
CSROT
Definition csrot.f:98
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
subroutine cunbdb5(m1, m2, n, x1, incx1, x2, incx2, q1, ldq1, q2, ldq2, work, lwork, info)
CUNBDB5
Definition cunbdb5.f:156
Here is the call graph for this function:
Here is the caller graph for this function: