LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zunbdb3()

 subroutine zunbdb3 ( 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 )

ZUNBDB3

Purpose:
``` ZUNBDB3 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 ZUNBDB1, ZUNBDB2, and ZUNBDB4 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*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) The scalar factors of the elementary reflectors that define P1.``` [out] TAUP2 ``` TAUP2 is COMPLEX*16 array, dimension (M-P) 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 zunbdb3.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 ONE
219 parameter( one = (1.0d0,0.0d0) )
220* ..
221* .. Local Scalars ..
222 DOUBLE PRECISION C, S
223 INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
224 \$ LWORKMIN, LWORKOPT
225 LOGICAL LQUERY
226* ..
227* .. External Subroutines ..
228 EXTERNAL zlarf, zlarfgp, zunbdb5, zdrot, zlacgv, xerbla
229* ..
230* .. External Functions ..
231 DOUBLE PRECISION DZNRM2
232 EXTERNAL dznrm2
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( 2*p .LT. m .OR. p .GT. m ) THEN
247 info = -2
248 ELSE IF( q .LT. m-p .OR. m-q .LT. m-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, m-p-1, 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( 'ZUNBDB3', -info )
272 RETURN
273 ELSE IF( lquery ) THEN
274 RETURN
275 END IF
276*
277* Reduce rows 1, ..., M-P of X11 and X21
278*
279 DO i = 1, m-p
280*
281 IF( i .GT. 1 ) THEN
282 CALL zdrot( q-i+1, x11(i-1,i), ldx11, x21(i,i), ldx11, c,
283 \$ s )
284 END IF
285*
286 CALL zlacgv( q-i+1, x21(i,i), ldx21 )
287 CALL zlarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
288 s = dble( x21(i,i) )
289 x21(i,i) = one
290 CALL zlarf( 'R', p-i+1, q-i+1, x21(i,i), ldx21, tauq1(i),
291 \$ x11(i,i), ldx11, work(ilarf) )
292 CALL zlarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
293 \$ x21(i+1,i), ldx21, work(ilarf) )
294 CALL zlacgv( q-i+1, x21(i,i), ldx21 )
295 c = sqrt( dznrm2( p-i+1, x11(i,i), 1 )**2
296 \$ + dznrm2( m-p-i, x21(i+1,i), 1 )**2 )
297 theta(i) = atan2( s, c )
298*
299 CALL zunbdb5( p-i+1, m-p-i, q-i, x11(i,i), 1, x21(i+1,i), 1,
300 \$ x11(i,i+1), ldx11, x21(i+1,i+1), ldx21,
301 \$ work(iorbdb5), lorbdb5, childinfo )
302 CALL zlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
303 IF( i .LT. m-p ) THEN
304 CALL zlarfgp( m-p-i, x21(i+1,i), x21(i+2,i), 1, taup2(i) )
305 phi(i) = atan2( dble( x21(i+1,i) ), dble( x11(i,i) ) )
306 c = cos( phi(i) )
307 s = sin( phi(i) )
308 x21(i+1,i) = one
309 CALL zlarf( 'L', m-p-i, q-i, x21(i+1,i), 1,
310 \$ dconjg(taup2(i)), x21(i+1,i+1), ldx21,
311 \$ work(ilarf) )
312 END IF
313 x11(i,i) = one
314 CALL zlarf( 'L', p-i+1, q-i, x11(i,i), 1, dconjg(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 zlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
323 x11(i,i) = one
324 CALL zlarf( 'L', p-i+1, q-i, x11(i,i), 1, dconjg(taup1(i)),
325 \$ x11(i,i+1), ldx11, work(ilarf) )
326 END DO
327*
328 RETURN
329*
330* End of ZUNBDB3
331*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zdrot(N, ZX, INCX, ZY, INCY, C, S)
ZDROT
Definition: zdrot.f:98
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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