 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ cunbdb1()

 subroutine cunbdb1 ( 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 )

CUNBDB1

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Purpose:
``` CUNBDB1 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. Q must be no larger than P,
M-P, or M-Q. Routines CUNBDB2, CUNBDB3, and CUNBDB4 handle cases in
which Q 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 Q-by-Q 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.``` [in] Q ``` Q is INTEGER The number of columns in X11 and X21. 0 <= Q <= MIN(P,M-P,M-Q).``` [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 cunbdb1.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, xerbla
230  EXTERNAL clacgv
231 * ..
232 * .. External Functions ..
233  REAL SCNRM2
234  EXTERNAL scnrm2
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. q .OR. m-p .LT. q ) THEN
249  info = -2
250  ELSE IF( q .LT. 0 .OR. m-q .LT. q ) 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-1, q-1 )
263  iorbdb5 = 2
264  lorbdb5 = q-2
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( 'CUNBDB1', -info )
274  RETURN
275  ELSE IF( lquery ) THEN
276  RETURN
277  END IF
278 *
279 * Reduce columns 1, ..., Q of X11 and X21
280 *
281  DO i = 1, q
282 *
283  CALL clarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
284  CALL clarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
285  theta(i) = atan2( real( x21(i,i) ), real( x11(i,i) ) )
286  c = cos( theta(i) )
287  s = sin( theta(i) )
288  x11(i,i) = one
289  x21(i,i) = one
290  CALL clarf( 'L', p-i+1, q-i, x11(i,i), 1, conjg(taup1(i)),
291  \$ x11(i,i+1), ldx11, work(ilarf) )
292  CALL clarf( 'L', m-p-i+1, q-i, x21(i,i), 1, conjg(taup2(i)),
293  \$ x21(i,i+1), ldx21, work(ilarf) )
294 *
295  IF( i .LT. q ) THEN
296  CALL csrot( q-i, x11(i,i+1), ldx11, x21(i,i+1), ldx21, c,
297  \$ s )
298  CALL clacgv( q-i, x21(i,i+1), ldx21 )
299  CALL clarfgp( q-i, x21(i,i+1), x21(i,i+2), ldx21, tauq1(i) )
300  s = real( x21(i,i+1) )
301  x21(i,i+1) = one
302  CALL clarf( 'R', p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
303  \$ x11(i+1,i+1), ldx11, work(ilarf) )
304  CALL clarf( 'R', m-p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
305  \$ x21(i+1,i+1), ldx21, work(ilarf) )
306  CALL clacgv( q-i, x21(i,i+1), ldx21 )
307  c = sqrt( scnrm2( p-i, x11(i+1,i+1), 1 )**2
308  \$ + scnrm2( m-p-i, x21(i+1,i+1), 1 )**2 )
309  phi(i) = atan2( s, c )
310  CALL cunbdb5( p-i, m-p-i, q-i-1, x11(i+1,i+1), 1,
311  \$ x21(i+1,i+1), 1, x11(i+1,i+2), ldx11,
312  \$ x21(i+1,i+2), ldx21, work(iorbdb5), lorbdb5,
313  \$ childinfo )
314  END IF
315 *
316  END DO
317 *
318  RETURN
319 *
320 * End of CUNBDB1
321 *
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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