LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ zunbdb1()

 subroutine zunbdb1 ( 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 )

ZUNBDB1

Purpose:
``` ZUNBDB1 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 ZUNBDB2, ZUNBDB3, and ZUNBDB4 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*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 205 of file zunbdb1.f.

205 *
206 * -- LAPACK computational routine (version 3.7.1) --
207 * -- LAPACK is a software package provided by Univ. of Tennessee, --
208 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
209 * July 2012
210 *
211 * .. Scalar Arguments ..
212  INTEGER info, lwork, m, p, q, ldx11, ldx21
213 * ..
214 * .. Array Arguments ..
215  DOUBLE PRECISION phi(*), theta(*)
216  COMPLEX*16 taup1(*), taup2(*), tauq1(*), work(*),
217  \$ x11(ldx11,*), x21(ldx21,*)
218 * ..
219 *
220 * ====================================================================
221 *
222 * .. Parameters ..
223  COMPLEX*16 one
224  parameter( one = (1.0d0,0.0d0) )
225 * ..
226 * .. Local Scalars ..
227  DOUBLE PRECISION c, s
228  INTEGER childinfo, i, ilarf, iorbdb5, llarf, lorbdb5,
229  \$ lworkmin, lworkopt
230  LOGICAL lquery
231 * ..
232 * .. External Subroutines ..
233  EXTERNAL zlarf, zlarfgp, zunbdb5, zdrot, xerbla
234  EXTERNAL zlacgv
235 * ..
236 * .. External Functions ..
237  DOUBLE PRECISION dznrm2
238  EXTERNAL dznrm2
239 * ..
240 * .. Intrinsic Function ..
241  INTRINSIC atan2, cos, max, sin, sqrt
242 * ..
243 * .. Executable Statements ..
244 *
245 * Test input arguments
246 *
247  info = 0
248  lquery = lwork .EQ. -1
249 *
250  IF( m .LT. 0 ) THEN
251  info = -1
252  ELSE IF( p .LT. q .OR. m-p .LT. q ) THEN
253  info = -2
254  ELSE IF( q .LT. 0 .OR. m-q .LT. q ) THEN
255  info = -3
256  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
257  info = -5
258  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
259  info = -7
260  END IF
261 *
262 * Compute workspace
263 *
264  IF( info .EQ. 0 ) THEN
265  ilarf = 2
266  llarf = max( p-1, m-p-1, q-1 )
267  iorbdb5 = 2
268  lorbdb5 = q-2
269  lworkopt = max( ilarf+llarf-1, iorbdb5+lorbdb5-1 )
270  lworkmin = lworkopt
271  work(1) = lworkopt
272  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
273  info = -14
274  END IF
275  END IF
276  IF( info .NE. 0 ) THEN
277  CALL xerbla( 'ZUNBDB1', -info )
278  RETURN
279  ELSE IF( lquery ) THEN
280  RETURN
281  END IF
282 *
283 * Reduce columns 1, ..., Q of X11 and X21
284 *
285  DO i = 1, q
286 *
287  CALL zlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
288  CALL zlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
289  theta(i) = atan2( dble( x21(i,i) ), dble( x11(i,i) ) )
290  c = cos( theta(i) )
291  s = sin( theta(i) )
292  x11(i,i) = one
293  x21(i,i) = one
294  CALL zlarf( 'L', p-i+1, q-i, x11(i,i), 1, dconjg(taup1(i)),
295  \$ x11(i,i+1), ldx11, work(ilarf) )
296  CALL zlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, dconjg(taup2(i)),
297  \$ x21(i,i+1), ldx21, work(ilarf) )
298 *
299  IF( i .LT. q ) THEN
300  CALL zdrot( q-i, x11(i,i+1), ldx11, x21(i,i+1), ldx21, c,
301  \$ s )
302  CALL zlacgv( q-i, x21(i,i+1), ldx21 )
303  CALL zlarfgp( q-i, x21(i,i+1), x21(i,i+2), ldx21, tauq1(i) )
304  s = dble( x21(i,i+1) )
305  x21(i,i+1) = one
306  CALL zlarf( 'R', p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
307  \$ x11(i+1,i+1), ldx11, work(ilarf) )
308  CALL zlarf( 'R', m-p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
309  \$ x21(i+1,i+1), ldx21, work(ilarf) )
310  CALL zlacgv( q-i, x21(i,i+1), ldx21 )
311  c = sqrt( dznrm2( p-i, x11(i+1,i+1), 1 )**2
312  \$ + dznrm2( m-p-i, x21(i+1,i+1), 1 )**2 )
313  phi(i) = atan2( s, c )
314  CALL zunbdb5( p-i, m-p-i, q-i-1, x11(i+1,i+1), 1,
315  \$ x21(i+1,i+1), 1, x11(i+1,i+2), ldx11,
316  \$ x21(i+1,i+2), ldx21, work(iorbdb5), lorbdb5,
317  \$ childinfo )
318  END IF
319 *
320  END DO
321 *
322  RETURN
323 *
324 * End of ZUNBDB1
325 *
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 zdrot(N, CX, INCX, CY, INCY, C, S)
ZDROT
Definition: zdrot.f:100
double precision function dznrm2(N, X, INCX)
DZNRM2
Definition: dznrm2.f:77
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76
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
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