LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ cunbdb2()

 subroutine cunbdb2 ( 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 )

CUNBDB2

Purpose:
``` CUNBDB2 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 CUNBDB1, CUNBDB3, and CUNBDB4 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 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 cunbdb2.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 NEGONE, ONE
220  parameter( negone = (-1.0e0,0.0e0),
221  \$ one = (1.0e0,0.0e0) )
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 clarf, clarfgp, cunbdb5, csrot, cscal, clacgv,
231  \$ xerbla
232 * ..
233 * .. External Functions ..
234  REAL SCNRM2
235  EXTERNAL scnrm2
236 * ..
237 * .. Intrinsic Function ..
238  INTRINSIC atan2, cos, max, sin, sqrt
239 * ..
240 * .. Executable Statements ..
241 *
242 * Test input arguments
243 *
244  info = 0
245  lquery = lwork .EQ. -1
246 *
247  IF( m .LT. 0 ) THEN
248  info = -1
249  ELSE IF( p .LT. 0 .OR. p .GT. m-p ) THEN
250  info = -2
251  ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) THEN
252  info = -3
253  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
254  info = -5
255  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
256  info = -7
257  END IF
258 *
259 * Compute workspace
260 *
261  IF( info .EQ. 0 ) THEN
262  ilarf = 2
263  llarf = max( p-1, m-p, q-1 )
264  iorbdb5 = 2
265  lorbdb5 = q-1
266  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
267  lworkmin = lworkopt
268  work(1) = lworkopt
269  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
270  info = -14
271  END IF
272  END IF
273  IF( info .NE. 0 ) THEN
274  CALL xerbla( 'CUNBDB2', -info )
275  RETURN
276  ELSE IF( lquery ) THEN
277  RETURN
278  END IF
279 *
280 * Reduce rows 1, ..., P of X11 and X21
281 *
282  DO i = 1, p
283 *
284  IF( i .GT. 1 ) THEN
285  CALL csrot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c,
286  \$ s )
287  END IF
288  CALL clacgv( q-i+1, x11(i,i), ldx11 )
289  CALL clarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
290  c = real( x11(i,i) )
291  x11(i,i) = one
292  CALL clarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
293  \$ x11(i+1,i), ldx11, work(ilarf) )
294  CALL clarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
295  \$ x21(i,i), ldx21, work(ilarf) )
296  CALL clacgv( q-i+1, x11(i,i), ldx11 )
297  s = sqrt( scnrm2( p-i, x11(i+1,i), 1 )**2
298  \$ + scnrm2( m-p-i+1, x21(i,i), 1 )**2 )
299  theta(i) = atan2( s, c )
300 *
301  CALL cunbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
302  \$ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
303  \$ work(iorbdb5), lorbdb5, childinfo )
304  CALL cscal( p-i, negone, x11(i+1,i), 1 )
305  CALL clarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
306  IF( i .LT. p ) THEN
307  CALL clarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
308  phi(i) = atan2( real( x11(i+1,i) ), real( x21(i,i) ) )
309  c = cos( phi(i) )
310  s = sin( phi(i) )
311  x11(i+1,i) = one
312  CALL clarf( 'L', p-i, q-i, x11(i+1,i), 1, conjg(taup1(i)),
313  \$ x11(i+1,i+1), ldx11, work(ilarf) )
314  END IF
315  x21(i,i) = one
316  CALL clarf( 'L', m-p-i+1, q-i, x21(i,i), 1, conjg(taup2(i)),
317  \$ x21(i,i+1), ldx21, work(ilarf) )
318 *
319  END DO
320 *
321 * Reduce the bottom-right portion of X21 to the identity matrix
322 *
323  DO i = p + 1, q
324  CALL clarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
325  x21(i,i) = one
326  CALL clarf( 'L', m-p-i+1, q-i, x21(i,i), 1, conjg(taup2(i)),
327  \$ x21(i,i+1), ldx21, work(ilarf) )
328  END DO
329 *
330  RETURN
331 *
332 * End of CUNBDB2
333 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
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
Here is the call graph for this function:
Here is the caller graph for this function: