LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine zunbdb2 ( 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 )

ZUNBDB2

# Purpose:

``` ZUNBDB2 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 ZUNBDB1, ZUNBDB3, and ZUNBDB4 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*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.```
Date
July 2012
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 203 of file zunbdb2.f.

203 *
204 * -- LAPACK computational routine (version 3.6.1) --
205 * -- LAPACK is a software package provided by Univ. of Tennessee, --
206 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
207 * July 2012
208 *
209 * .. Scalar Arguments ..
210  INTEGER info, lwork, m, p, q, ldx11, ldx21
211 * ..
212 * .. Array Arguments ..
213  DOUBLE PRECISION phi(*), theta(*)
214  COMPLEX*16 taup1(*), taup2(*), tauq1(*), work(*),
215  \$ x11(ldx11,*), x21(ldx21,*)
216 * ..
217 *
218 * ====================================================================
219 *
220 * .. Parameters ..
221  COMPLEX*16 negone, one
222  parameter ( negone = (-1.0d0,0.0d0),
223  \$ one = (1.0d0,0.0d0) )
224 * ..
225 * .. Local Scalars ..
226  DOUBLE PRECISION c, s
227  INTEGER childinfo, i, ilarf, iorbdb5, llarf, lorbdb5,
228  \$ lworkmin, lworkopt
229  LOGICAL lquery
230 * ..
231 * .. External Subroutines ..
232  EXTERNAL zlarf, zlarfgp, zunbdb5, zdrot, zscal, xerbla
233 * ..
234 * .. External Functions ..
235  DOUBLE PRECISION dznrm2
236  EXTERNAL dznrm2
237 * ..
238 * .. Intrinsic Function ..
239  INTRINSIC atan2, cos, max, sin, sqrt
240 * ..
241 * .. Executable Statements ..
242 *
243 * Test input arguments
244 *
245  info = 0
246  lquery = lwork .EQ. -1
247 *
248  IF( m .LT. 0 ) THEN
249  info = -1
250  ELSE IF( p .LT. 0 .OR. p .GT. m-p ) THEN
251  info = -2
252  ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) THEN
253  info = -3
254  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
255  info = -5
256  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
257  info = -7
258  END IF
259 *
260 * Compute workspace
261 *
262  IF( info .EQ. 0 ) THEN
263  ilarf = 2
264  llarf = max( p-1, m-p, q-1 )
265  iorbdb5 = 2
266  lorbdb5 = q-1
267  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
268  lworkmin = lworkopt
269  work(1) = lworkopt
270  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
271  info = -14
272  END IF
273  END IF
274  IF( info .NE. 0 ) THEN
275  CALL xerbla( 'ZUNBDB2', -info )
276  RETURN
277  ELSE IF( lquery ) THEN
278  RETURN
279  END IF
280 *
281 * Reduce rows 1, ..., P of X11 and X21
282 *
283  DO i = 1, p
284 *
285  IF( i .GT. 1 ) THEN
286  CALL zdrot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c,
287  \$ s )
288  END IF
289  CALL zlacgv( q-i+1, x11(i,i), ldx11 )
290  CALL zlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
291  c = dble( x11(i,i) )
292  x11(i,i) = one
293  CALL zlarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
294  \$ x11(i+1,i), ldx11, work(ilarf) )
295  CALL zlarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
296  \$ x21(i,i), ldx21, work(ilarf) )
297  CALL zlacgv( q-i+1, x11(i,i), ldx11 )
298  s = sqrt( dznrm2( p-i, x11(i+1,i), 1 )**2
299  \$ + dznrm2( m-p-i+1, x21(i,i), 1 )**2 )
300  theta(i) = atan2( s, c )
301 *
302  CALL zunbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
303  \$ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
304  \$ work(iorbdb5), lorbdb5, childinfo )
305  CALL zscal( p-i, negone, x11(i+1,i), 1 )
306  CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
307  IF( i .LT. p ) THEN
308  CALL zlarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
309  phi(i) = atan2( dble( x11(i+1,i) ), dble( x21(i,i) ) )
310  c = cos( phi(i) )
311  s = sin( phi(i) )
312  x11(i+1,i) = one
313  CALL zlarf( 'L', p-i, q-i, x11(i+1,i), 1, dconjg(taup1(i)),
314  \$ x11(i+1,i+1), ldx11, work(ilarf) )
315  END IF
316  x21(i,i) = one
317  CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
318  \$ x21(i,i+1), ldx21, work(ilarf) )
319 *
320  END DO
321 *
322 * Reduce the bottom-right portion of X21 to the identity matrix
323 *
324  DO i = p + 1, q
325  CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
326  x21(i,i) = one
327  CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
328  \$ x21(i,i+1), ldx21, work(ilarf) )
329  END DO
330 *
331  RETURN
332 *
333 * End of ZUNBDB2
334 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zdrot(N, CX, INCX, CY, INCY, C, S)
ZDROT
Definition: zdrot.f:100
double precision function dznrm2(N, X, INCX)
DZNRM2
Definition: dznrm2.f:56
subroutine zunbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
ZUNBDB5
Definition: zunbdb5.f:158
subroutine zlarfgp(N, ALPHA, X, INCX, TAU)
ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: zlarfgp.f:106
subroutine zlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
ZLARF applies an elementary reflector to a general rectangular matrix.
Definition: zlarf.f:130
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:54
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76

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