 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ 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 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 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  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:
 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
233  EXTERNAL scnrm2
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) = 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)
XERBLA
Definition: xerbla.f:60
subroutine csrot(N, CX, INCX, CY, INCY, C, S)
CSROT
Definition: csrot.f:98
subroutine clarfgp(N, ALPHA, X, INCX, TAU)
CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: clarfgp.f:104
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 cunbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
CUNBDB5
Definition: cunbdb5.f:156
real(wp) function scnrm2(n, x, incx)
SCNRM2
Definition: scnrm2.f90:90
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