LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zunbdb2()

 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 3.8.0) --
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, zlacgv,
233  \$ xerbla
234 * ..
235 * .. External Functions ..
236  DOUBLE PRECISION dznrm2
237  EXTERNAL dznrm2
238 * ..
239 * .. Intrinsic Function ..
240  INTRINSIC atan2, cos, max, sin, sqrt
241 * ..
242 * .. Executable Statements ..
243 *
244 * Test input arguments
245 *
246  info = 0
247  lquery = lwork .EQ. -1
248 *
249  IF( m .LT. 0 ) THEN
250  info = -1
251  ELSE IF( p .LT. 0 .OR. p .GT. m-p ) THEN
252  info = -2
253  ELSE IF( q .LT. 0 .OR. q .LT. p .OR. m-q .LT. p ) THEN
254  info = -3
255  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
256  info = -5
257  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
258  info = -7
259  END IF
260 *
261 * Compute workspace
262 *
263  IF( info .EQ. 0 ) THEN
264  ilarf = 2
265  llarf = max( p-1, m-p, q-1 )
266  iorbdb5 = 2
267  lorbdb5 = q-1
268  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
269  lworkmin = lworkopt
270  work(1) = lworkopt
271  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
272  info = -14
273  END IF
274  END IF
275  IF( info .NE. 0 ) THEN
276  CALL xerbla( 'ZUNBDB2', -info )
277  RETURN
278  ELSE IF( lquery ) THEN
279  RETURN
280  END IF
281 *
282 * Reduce rows 1, ..., P of X11 and X21
283 *
284  DO i = 1, p
285 *
286  IF( i .GT. 1 ) THEN
287  CALL zdrot( q-i+1, x11(i,i), ldx11, x21(i-1,i), ldx21, c,
288  \$ s )
289  END IF
290  CALL zlacgv( q-i+1, x11(i,i), ldx11 )
291  CALL zlarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
292  c = dble( x11(i,i) )
293  x11(i,i) = one
294  CALL zlarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
295  \$ x11(i+1,i), ldx11, work(ilarf) )
296  CALL zlarf( 'R', m-p-i+1, q-i+1, x11(i,i), ldx11, tauq1(i),
297  \$ x21(i,i), ldx21, work(ilarf) )
298  CALL zlacgv( q-i+1, x11(i,i), ldx11 )
299  s = sqrt( dznrm2( p-i, x11(i+1,i), 1 )**2
300  \$ + dznrm2( m-p-i+1, x21(i,i), 1 )**2 )
301  theta(i) = atan2( s, c )
302 *
303  CALL zunbdb5( p-i, m-p-i+1, q-i, x11(i+1,i), 1, x21(i,i), 1,
304  \$ x11(i+1,i+1), ldx11, x21(i,i+1), ldx21,
305  \$ work(iorbdb5), lorbdb5, childinfo )
306  CALL zscal( p-i, negone, x11(i+1,i), 1 )
307  CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
308  IF( i .LT. p ) THEN
309  CALL zlarfgp( p-i, x11(i+1,i), x11(i+2,i), 1, taup1(i) )
310  phi(i) = atan2( dble( x11(i+1,i) ), dble( x21(i,i) ) )
311  c = cos( phi(i) )
312  s = sin( phi(i) )
313  x11(i+1,i) = one
314  CALL zlarf( 'L', p-i, q-i, x11(i+1,i), 1, dconjg(taup1(i)),
315  \$ x11(i+1,i+1), ldx11, work(ilarf) )
316  END IF
317  x21(i,i) = one
318  CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
319  \$ x21(i,i+1), ldx21, work(ilarf) )
320 *
321  END DO
322 *
323 * Reduce the bottom-right portion of X21 to the identity matrix
324 *
325  DO i = p + 1, q
326  CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
327  x21(i,i) = one
328  CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
329  \$ x21(i,i+1), ldx21, work(ilarf) )
330  END DO
331 *
332  RETURN
333 *
334 * End of ZUNBDB2
335 *
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76
subroutine zdrot(N, CX, INCX, CY, INCY, C, S)
ZDROT
Definition: zdrot.f:100
subroutine zlarfgp(N, ALPHA, X, INCX, TAU)
ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: zlarfgp.f:106
double precision function dznrm2(N, X, INCX)
DZNRM2
Definition: dznrm2.f:77
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:80
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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 zunbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
ZUNBDB5
Definition: zunbdb5.f:158
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