LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine dorbdb1 ( integer M, integer P, integer Q, double precision, dimension(ldx11,*) X11, integer LDX11, double precision, dimension(ldx21,*) X21, integer LDX21, double precision, dimension(*) THETA, double precision, dimension(*) PHI, double precision, dimension(*) TAUP1, double precision, dimension(*) TAUP2, double precision, dimension(*) TAUQ1, double precision, dimension(*) WORK, integer LWORK, integer INFO )

DORBDB1

Purpose:

DORBDB1 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 DORBDB2, DORBDB3, and DORBDB4 handle cases in
which Q is not the minimum dimension.

The orthogonal 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (P) The scalar factors of the elementary reflectors that define P1. [out] TAUP2 TAUP2 is DOUBLE PRECISION array, dimension (M-P) The scalar factors of the elementary reflectors that define P2. [out] TAUQ1 TAUQ1 is DOUBLE PRECISION array, dimension (Q) The scalar factors of the elementary reflectors that define Q1. [out] WORK WORK is DOUBLE PRECISION 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 DORCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors.
See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
and DORGLQ.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 205 of file dorbdb1.f.

205 *
206 * -- LAPACK computational routine (version 3.6.1) --
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  DOUBLE PRECISION taup1(*), taup2(*), tauq1(*), work(*),
217  \$ x11(ldx11,*), x21(ldx21,*)
218 * ..
219 *
220 * ====================================================================
221 *
222 * .. Parameters ..
223  DOUBLE PRECISION one
224  parameter ( one = 1.0d0 )
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 dlarf, dlarfgp, dorbdb5, drot, xerbla
234 * ..
235 * .. External Functions ..
236  DOUBLE PRECISION dnrm2
237  EXTERNAL dnrm2
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. q .OR. m-p .LT. q ) THEN
252  info = -2
253  ELSE IF( q .LT. 0 .OR. m-q .LT. q ) 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-1, q-1 )
266  iorbdb5 = 2
267  lorbdb5 = q-2
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( 'DORBDB1', -info )
277  RETURN
278  ELSE IF( lquery ) THEN
279  RETURN
280  END IF
281 *
282 * Reduce columns 1, ..., Q of X11 and X21
283 *
284  DO i = 1, q
285 *
286  CALL dlarfgp( p-i+1, x11(i,i), x11(i+1,i), 1, taup1(i) )
287  CALL dlarfgp( m-p-i+1, x21(i,i), x21(i+1,i), 1, taup2(i) )
288  theta(i) = atan2( x21(i,i), x11(i,i) )
289  c = cos( theta(i) )
290  s = sin( theta(i) )
291  x11(i,i) = one
292  x21(i,i) = one
293  CALL dlarf( 'L', p-i+1, q-i, x11(i,i), 1, taup1(i), x11(i,i+1),
294  \$ ldx11, work(ilarf) )
295  CALL dlarf( 'L', m-p-i+1, q-i, x21(i,i), 1, taup2(i),
296  \$ x21(i,i+1), ldx21, work(ilarf) )
297 *
298  IF( i .LT. q ) THEN
299  CALL drot( q-i, x11(i,i+1), ldx11, x21(i,i+1), ldx21, c, s )
300  CALL dlarfgp( q-i, x21(i,i+1), x21(i,i+2), ldx21, tauq1(i) )
301  s = x21(i,i+1)
302  x21(i,i+1) = one
303  CALL dlarf( 'R', p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
304  \$ x11(i+1,i+1), ldx11, work(ilarf) )
305  CALL dlarf( 'R', m-p-i, q-i, x21(i,i+1), ldx21, tauq1(i),
306  \$ x21(i+1,i+1), ldx21, work(ilarf) )
307  c = sqrt( dnrm2( p-i, x11(i+1,i+1), 1 )**2
308  \$ + dnrm2( m-p-i, x21(i+1,i+1), 1 )**2 )
309  phi(i) = atan2( s, c )
310  CALL dorbdb5( 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 DORBDB1
321 *
subroutine dlarfgp(N, ALPHA, X, INCX, TAU)
DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: dlarfgp.f:106
subroutine drot(N, DX, INCX, DY, INCY, C, S)
DROT
Definition: drot.f:53
subroutine dlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition: dlarf.f:126
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dnrm2(N, X, INCX)
DNRM2
Definition: dnrm2.f:56
subroutine dorbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
DORBDB5
Definition: dorbdb5.f:158

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