LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ cunbdb4()

 subroutine cunbdb4 ( 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(*) PHANTOM, complex, dimension(*) WORK, integer LWORK, integer INFO )

CUNBDB4

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Purpose:
``` CUNBDB4 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-Q must be no larger than P,
M-P, or Q. Routines CUNBDB1, CUNBDB2, and CUNBDB3 handle cases in
which M-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 (M-Q)-by-(M-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 <= M and M-Q <= min(P,M-P,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] PHANTOM ``` PHANTOM is COMPLEX array, dimension (M) The routine computes an M-by-1 column vector Y that is orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and Y(P+1:M), respectively.``` [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 210 of file cunbdb4.f.

213 *
214 * -- LAPACK computational routine --
215 * -- LAPACK is a software package provided by Univ. of Tennessee, --
216 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
217 *
218 * .. Scalar Arguments ..
219  INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
220 * ..
221 * .. Array Arguments ..
222  REAL PHI(*), THETA(*)
223  COMPLEX PHANTOM(*), TAUP1(*), TAUP2(*), TAUQ1(*),
224  \$ WORK(*), X11(LDX11,*), X21(LDX21,*)
225 * ..
226 *
227 * ====================================================================
228 *
229 * .. Parameters ..
230  COMPLEX NEGONE, ONE, ZERO
231  parameter( negone = (-1.0e0,0.0e0), one = (1.0e0,0.0e0),
232  \$ zero = (0.0e0,0.0e0) )
233 * ..
234 * .. Local Scalars ..
235  REAL C, S
236  INTEGER CHILDINFO, I, ILARF, IORBDB5, J, LLARF,
237  \$ LORBDB5, LWORKMIN, LWORKOPT
238  LOGICAL LQUERY
239 * ..
240 * .. External Subroutines ..
241  EXTERNAL clarf, clarfgp, cunbdb5, csrot, cscal, clacgv,
242  \$ xerbla
243 * ..
244 * .. External Functions ..
245  REAL SCNRM2
246  EXTERNAL scnrm2
247 * ..
248 * .. Intrinsic Function ..
249  INTRINSIC atan2, cos, max, sin, sqrt
250 * ..
251 * .. Executable Statements ..
252 *
253 * Test input arguments
254 *
255  info = 0
256  lquery = lwork .EQ. -1
257 *
258  IF( m .LT. 0 ) THEN
259  info = -1
260  ELSE IF( p .LT. m-q .OR. m-p .LT. m-q ) THEN
261  info = -2
262  ELSE IF( q .LT. m-q .OR. q .GT. m ) THEN
263  info = -3
264  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
265  info = -5
266  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
267  info = -7
268  END IF
269 *
270 * Compute workspace
271 *
272  IF( info .EQ. 0 ) THEN
273  ilarf = 2
274  llarf = max( q-1, p-1, m-p-1 )
275  iorbdb5 = 2
276  lorbdb5 = q
277  lworkopt = ilarf + llarf - 1
278  lworkopt = max( lworkopt, iorbdb5 + lorbdb5 - 1 )
279  lworkmin = lworkopt
280  work(1) = lworkopt
281  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
282  info = -14
283  END IF
284  END IF
285  IF( info .NE. 0 ) THEN
286  CALL xerbla( 'CUNBDB4', -info )
287  RETURN
288  ELSE IF( lquery ) THEN
289  RETURN
290  END IF
291 *
292 * Reduce columns 1, ..., M-Q of X11 and X21
293 *
294  DO i = 1, m-q
295 *
296  IF( i .EQ. 1 ) THEN
297  DO j = 1, m
298  phantom(j) = zero
299  END DO
300  CALL cunbdb5( p, m-p, q, phantom(1), 1, phantom(p+1), 1,
301  \$ x11, ldx11, x21, ldx21, work(iorbdb5),
302  \$ lorbdb5, childinfo )
303  CALL cscal( p, negone, phantom(1), 1 )
304  CALL clarfgp( p, phantom(1), phantom(2), 1, taup1(1) )
305  CALL clarfgp( m-p, phantom(p+1), phantom(p+2), 1, taup2(1) )
306  theta(i) = atan2( real( phantom(1) ), real( phantom(p+1) ) )
307  c = cos( theta(i) )
308  s = sin( theta(i) )
309  phantom(1) = one
310  phantom(p+1) = one
311  CALL clarf( 'L', p, q, phantom(1), 1, conjg(taup1(1)), x11,
312  \$ ldx11, work(ilarf) )
313  CALL clarf( 'L', m-p, q, phantom(p+1), 1, conjg(taup2(1)),
314  \$ x21, ldx21, work(ilarf) )
315  ELSE
316  CALL cunbdb5( p-i+1, m-p-i+1, q-i+1, x11(i,i-1), 1,
317  \$ x21(i,i-1), 1, x11(i,i), ldx11, x21(i,i),
318  \$ ldx21, work(iorbdb5), lorbdb5, childinfo )
319  CALL cscal( p-i+1, negone, x11(i,i-1), 1 )
320  CALL clarfgp( p-i+1, x11(i,i-1), x11(i+1,i-1), 1, taup1(i) )
321  CALL clarfgp( m-p-i+1, x21(i,i-1), x21(i+1,i-1), 1,
322  \$ taup2(i) )
323  theta(i) = atan2( real( x11(i,i-1) ), real( x21(i,i-1) ) )
324  c = cos( theta(i) )
325  s = sin( theta(i) )
326  x11(i,i-1) = one
327  x21(i,i-1) = one
328  CALL clarf( 'L', p-i+1, q-i+1, x11(i,i-1), 1,
329  \$ conjg(taup1(i)), x11(i,i), ldx11, work(ilarf) )
330  CALL clarf( 'L', m-p-i+1, q-i+1, x21(i,i-1), 1,
331  \$ conjg(taup2(i)), x21(i,i), ldx21, work(ilarf) )
332  END IF
333 *
334  CALL csrot( q-i+1, x11(i,i), ldx11, x21(i,i), ldx21, s, -c )
335  CALL clacgv( q-i+1, x21(i,i), ldx21 )
336  CALL clarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
337  c = real( x21(i,i) )
338  x21(i,i) = one
339  CALL clarf( 'R', p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
340  \$ x11(i+1,i), ldx11, work(ilarf) )
341  CALL clarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
342  \$ x21(i+1,i), ldx21, work(ilarf) )
343  CALL clacgv( q-i+1, x21(i,i), ldx21 )
344  IF( i .LT. m-q ) THEN
345  s = sqrt( scnrm2( p-i, x11(i+1,i), 1 )**2
346  \$ + scnrm2( m-p-i, x21(i+1,i), 1 )**2 )
347  phi(i) = atan2( s, c )
348  END IF
349 *
350  END DO
351 *
352 * Reduce the bottom-right portion of X11 to [ I 0 ]
353 *
354  DO i = m - q + 1, p
355  CALL clacgv( q-i+1, x11(i,i), ldx11 )
356  CALL clarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
357  x11(i,i) = one
358  CALL clarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
359  \$ x11(i+1,i), ldx11, work(ilarf) )
360  CALL clarf( 'R', q-p, q-i+1, x11(i,i), ldx11, tauq1(i),
361  \$ x21(m-q+1,i), ldx21, work(ilarf) )
362  CALL clacgv( q-i+1, x11(i,i), ldx11 )
363  END DO
364 *
365 * Reduce the bottom-right portion of X21 to [ 0 I ]
366 *
367  DO i = p + 1, q
368  CALL clacgv( q-i+1, x21(m-q+i-p,i), ldx21 )
369  CALL clarfgp( q-i+1, x21(m-q+i-p,i), x21(m-q+i-p,i+1), ldx21,
370  \$ tauq1(i) )
371  x21(m-q+i-p,i) = one
372  CALL clarf( 'R', q-i, q-i+1, x21(m-q+i-p,i), ldx21, tauq1(i),
373  \$ x21(m-q+i-p+1,i), ldx21, work(ilarf) )
374  CALL clacgv( q-i+1, x21(m-q+i-p,i), ldx21 )
375  END DO
376 *
377  RETURN
378 *
379 * End of CUNBDB4
380 *
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
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