LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zunbdb2()

 subroutine zunbdb2 ( integer M, integer P, integer Q, complex*16, dimension(ldx11,*) X11, integer LDX11, complex*16, dimension(ldx21,*) X21, integer LDX21, double precision, dimension(*) THETA, double precision, dimension(*) PHI, complex*16, dimension(*) TAUP1, complex*16, dimension(*) TAUP2, complex*16, dimension(*) TAUQ1, complex*16, dimension(*) WORK, integer LWORK, integer INFO )

ZUNBDB2

Purpose:
``` ZUNBDB2 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 ZUNBDB1, ZUNBDB3, and ZUNBDB4 handle cases in
which 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 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 COMPLEX*16 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*16 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 DOUBLE PRECISION array, dimension (Q) The entries of the bidiagonal blocks B11, B21 are defined by THETA and PHI. See Further Details.``` [out] PHI ``` PHI is DOUBLE PRECISION 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*16 array, dimension (P-1) The scalar factors of the elementary reflectors that define P1.``` [out] TAUP2 ``` TAUP2 is COMPLEX*16 array, dimension (Q) The scalar factors of the elementary reflectors that define P2.``` [out] TAUQ1 ``` TAUQ1 is COMPLEX*16 array, dimension (Q) The scalar factors of the elementary reflectors that define Q1.``` [out] WORK ` WORK is COMPLEX*16 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 ZUNCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors.
See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR
and ZUNGLQ.```
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 199 of file zunbdb2.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 DOUBLE PRECISION PHI(*), THETA(*)
211 COMPLEX*16 TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
212 \$ X11(LDX11,*), X21(LDX21,*)
213* ..
214*
215* ====================================================================
216*
217* .. Parameters ..
218 COMPLEX*16 NEGONE, ONE
219 parameter( negone = (-1.0d0,0.0d0),
220 \$ one = (1.0d0,0.0d0) )
221* ..
222* .. Local Scalars ..
223 DOUBLE PRECISION C, S
224 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
225 \$ LWORKMIN, LWORKOPT
226 LOGICAL LQUERY
227* ..
228* .. External Subroutines ..
229 EXTERNAL zlarf, zlarfgp, zunbdb5, zdrot, zscal, zlacgv,
230 \$ xerbla
231* ..
232* .. External Functions ..
233 DOUBLE PRECISION DZNRM2
234 EXTERNAL dznrm2
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. 0 .OR. p .GT. m-p ) THEN
249 info = -2
250 ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) 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, q-1 )
263 iorbdb5 = 2
264 lorbdb5 = q-1
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( 'ZUNBDB2', -info )
274 RETURN
275 ELSE IF( lquery ) THEN
276 RETURN
277 END IF
278*
279* Reduce rows 1, ..., P of X11 and X21
280*
281 DO i = 1, p
282*
283 IF( i .GT. 1 ) THEN
284 CALL zdrot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c,
285 \$ s )
286 END IF
287 CALL zlacgv( q-i+1, x11(i,i), ldx11 )
288 CALL zlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
289 c = dble( x11(i,i) )
290 x11(i,i) = one
291 CALL zlarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
292 \$ x11(i+1,i), ldx11, work(ilarf) )
293 CALL zlarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
294 \$ x21(i,i), ldx21, work(ilarf) )
295 CALL zlacgv( q-i+1, x11(i,i), ldx11 )
296 s = sqrt( dznrm2( p-i, x11(i+1,i), 1 )**2
297 \$ + dznrm2( m-p-i+1, x21(i,i), 1 )**2 )
298 theta(i) = atan2( s, c )
299*
300 CALL zunbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
301 \$ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
302 \$ work(iorbdb5), lorbdb5, childinfo )
303 CALL zscal( p-i, negone, x11(i+1,i), 1 )
304 CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
305 IF( i .LT. p ) THEN
306 CALL zlarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
307 phi(i) = atan2( dble( x11(i+1,i) ), dble( x21(i,i) ) )
308 c = cos( phi(i) )
309 s = sin( phi(i) )
310 x11(i+1,i) = one
311 CALL zlarf( 'L', p-i, q-i, x11(i+1,i), 1, dconjg(taup1(i)),
312 \$ x11(i+1,i+1), ldx11, work(ilarf) )
313 END IF
314 x21(i,i) = one
315 CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
316 \$ x21(i,i+1), ldx21, work(ilarf) )
317*
318 END DO
319*
320* Reduce the bottom-right portion of X21 to the identity matrix
321*
322 DO i = p + 1, q
323 CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
324 x21(i,i) = one
325 CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
326 \$ x21(i,i+1), ldx21, work(ilarf) )
327 END DO
328*
329 RETURN
330*
331* End of ZUNBDB2
332*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zdrot(N, ZX, INCX, ZY, INCY, C, S)
ZDROT
Definition: zdrot.f:98
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine zlarfgp(N, ALPHA, X, INCX, TAU)
ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: zlarfgp.f:104
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74
subroutine zlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
ZLARF applies an elementary reflector to a general rectangular matrix.
Definition: zlarf.f:128
subroutine zunbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
ZUNBDB5
Definition: zunbdb5.f:156
real(wp) function dznrm2(n, x, incx)
DZNRM2
Definition: dznrm2.f90:90
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