LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ sorbdb4()

 subroutine sorbdb4 ( integer M, integer P, integer Q, real, dimension(ldx11,*) X11, integer LDX11, real, dimension(ldx21,*) X21, integer LDX21, real, dimension(*) THETA, real, dimension(*) PHI, real, dimension(*) TAUP1, real, dimension(*) TAUP2, real, dimension(*) TAUQ1, real, dimension(*) PHANTOM, real, dimension(*) WORK, integer LWORK, integer INFO )

SORBDB4

Purpose:
``` SORBDB4 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 SORBDB1, SORBDB2, and SORBDB3 handle cases in
which M-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 (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 REAL 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 REAL 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 REAL array, dimension (P) The scalar factors of the elementary reflectors that define P1.``` [out] TAUP2 ``` TAUP2 is REAL array, dimension (M-P) The scalar factors of the elementary reflectors that define P2.``` [out] TAUQ1 ``` TAUQ1 is REAL array, dimension (Q) The scalar factors of the elementary reflectors that define Q1.``` [out] PHANTOM ``` PHANTOM is REAL 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 REAL 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 SORCSD for details.

P1, P2, and Q1 are represented as products of elementary reflectors.
See SORCSD2BY1 for details on generating P1, P2, and Q1 using SORGQR
and SORGLQ.```
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Definition at line 216 of file sorbdb4.f.

216 *
217 * -- LAPACK computational routine (version 3.7.1) --
218 * -- LAPACK is a software package provided by Univ. of Tennessee, --
219 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
220 * July 2012
221 *
222 * .. Scalar Arguments ..
223  INTEGER info, lwork, m, p, q, ldx11, ldx21
224 * ..
225 * .. Array Arguments ..
226  REAL phi(*), theta(*)
227  REAL phantom(*), taup1(*), taup2(*), tauq1(*),
228  \$ work(*), x11(ldx11,*), x21(ldx21,*)
229 * ..
230 *
231 * ====================================================================
232 *
233 * .. Parameters ..
234  REAL negone, one, zero
235  parameter( negone = -1.0e0, one = 1.0e0, zero = 0.0e0 )
236 * ..
237 * .. Local Scalars ..
238  REAL c, s
239  INTEGER childinfo, i, ilarf, iorbdb5, j, llarf,
240  \$ lorbdb5, lworkmin, lworkopt
241  LOGICAL lquery
242 * ..
243 * .. External Subroutines ..
244  EXTERNAL slarf, slarfgp, sorbdb5, srot, sscal, xerbla
245 * ..
246 * .. External Functions ..
247  REAL snrm2
248  EXTERNAL snrm2
249 * ..
250 * .. Intrinsic Function ..
251  INTRINSIC atan2, cos, max, sin, sqrt
252 * ..
253 * .. Executable Statements ..
254 *
255 * Test input arguments
256 *
257  info = 0
258  lquery = lwork .EQ. -1
259 *
260  IF( m .LT. 0 ) THEN
261  info = -1
262  ELSE IF( p .LT. m-q .OR. m-p .LT. m-q ) THEN
263  info = -2
264  ELSE IF( q .LT. m-q .OR. q .GT. m ) THEN
265  info = -3
266  ELSE IF( ldx11 .LT. max( 1, p ) ) THEN
267  info = -5
268  ELSE IF( ldx21 .LT. max( 1, m-p ) ) THEN
269  info = -7
270  END IF
271 *
272 * Compute workspace
273 *
274  IF( info .EQ. 0 ) THEN
275  ilarf = 2
276  llarf = max( q-1, p-1, m-p-1 )
277  iorbdb5 = 2
278  lorbdb5 = q
279  lworkopt = ilarf + llarf - 1
280  lworkopt = max( lworkopt, iorbdb5 + lorbdb5 - 1 )
281  lworkmin = lworkopt
282  work(1) = lworkopt
283  IF( lwork .LT. lworkmin .AND. .NOT.lquery ) THEN
284  info = -14
285  END IF
286  END IF
287  IF( info .NE. 0 ) THEN
288  CALL xerbla( 'SORBDB4', -info )
289  RETURN
290  ELSE IF( lquery ) THEN
291  RETURN
292  END IF
293 *
294 * Reduce columns 1, ..., M-Q of X11 and X21
295 *
296  DO i = 1, m-q
297 *
298  IF( i .EQ. 1 ) THEN
299  DO j = 1, m
300  phantom(j) = zero
301  END DO
302  CALL sorbdb5( p, m-p, q, phantom(1), 1, phantom(p+1), 1,
303  \$ x11, ldx11, x21, ldx21, work(iorbdb5),
304  \$ lorbdb5, childinfo )
305  CALL sscal( p, negone, phantom(1), 1 )
306  CALL slarfgp( p, phantom(1), phantom(2), 1, taup1(1) )
307  CALL slarfgp( m-p, phantom(p+1), phantom(p+2), 1, taup2(1) )
308  theta(i) = atan2( phantom(1), phantom(p+1) )
309  c = cos( theta(i) )
310  s = sin( theta(i) )
311  phantom(1) = one
312  phantom(p+1) = one
313  CALL slarf( 'L', p, q, phantom(1), 1, taup1(1), x11, ldx11,
314  \$ work(ilarf) )
315  CALL slarf( 'L', m-p, q, phantom(p+1), 1, taup2(1), x21,
316  \$ ldx21, work(ilarf) )
317  ELSE
318  CALL sorbdb5( p-i+1, m-p-i+1, q-i+1, x11(i,i-1), 1,
319  \$ x21(i,i-1), 1, x11(i,i), ldx11, x21(i,i),
320  \$ ldx21, work(iorbdb5), lorbdb5, childinfo )
321  CALL sscal( p-i+1, negone, x11(i,i-1), 1 )
322  CALL slarfgp( p-i+1, x11(i,i-1), x11(i+1,i-1), 1, taup1(i) )
323  CALL slarfgp( m-p-i+1, x21(i,i-1), x21(i+1,i-1), 1,
324  \$ taup2(i) )
325  theta(i) = atan2( x11(i,i-1), x21(i,i-1) )
326  c = cos( theta(i) )
327  s = sin( theta(i) )
328  x11(i,i-1) = one
329  x21(i,i-1) = one
330  CALL slarf( 'L', p-i+1, q-i+1, x11(i,i-1), 1, taup1(i),
331  \$ x11(i,i), ldx11, work(ilarf) )
332  CALL slarf( 'L', m-p-i+1, q-i+1, x21(i,i-1), 1, taup2(i),
333  \$ x21(i,i), ldx21, work(ilarf) )
334  END IF
335 *
336  CALL srot( q-i+1, x11(i,i), ldx11, x21(i,i), ldx21, s, -c )
337  CALL slarfgp( q-i+1, x21(i,i), x21(i,i+1), ldx21, tauq1(i) )
338  c = x21(i,i)
339  x21(i,i) = one
340  CALL slarf( 'R', p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
341  \$ x11(i+1,i), ldx11, work(ilarf) )
342  CALL slarf( 'R', m-p-i, q-i+1, x21(i,i), ldx21, tauq1(i),
343  \$ x21(i+1,i), ldx21, work(ilarf) )
344  IF( i .LT. m-q ) THEN
345  s = sqrt( snrm2( p-i, x11(i+1,i), 1 )**2
346  \$ + snrm2( 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 slarfgp( q-i+1, x11(i,i), x11(i,i+1), ldx11, tauq1(i) )
356  x11(i,i) = one
357  CALL slarf( 'R', p-i, q-i+1, x11(i,i), ldx11, tauq1(i),
358  \$ x11(i+1,i), ldx11, work(ilarf) )
359  CALL slarf( 'R', q-p, q-i+1, x11(i,i), ldx11, tauq1(i),
360  \$ x21(m-q+1,i), ldx21, work(ilarf) )
361  END DO
362 *
363 * Reduce the bottom-right portion of X21 to [ 0 I ]
364 *
365  DO i = p + 1, q
366  CALL slarfgp( q-i+1, x21(m-q+i-p,i), x21(m-q+i-p,i+1), ldx21,
367  \$ tauq1(i) )
368  x21(m-q+i-p,i) = one
369  CALL slarf( 'R', q-i, q-i+1, x21(m-q+i-p,i), ldx21, tauq1(i),
370  \$ x21(m-q+i-p+1,i), ldx21, work(ilarf) )
371  END DO
372 *
373  RETURN
374 *
375 * End of SORBDB4
376 *
real function snrm2(N, X, INCX)
SNRM2
Definition: snrm2.f:76
subroutine sorbdb5(M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2, LDQ2, WORK, LWORK, INFO)
SORBDB5
Definition: sorbdb5.f:158
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
SLARF applies an elementary reflector to a general rectangular matrix.
Definition: slarf.f:126
subroutine slarfgp(N, ALPHA, X, INCX, TAU)
SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Definition: slarfgp.f:106
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:81
subroutine srot(N, SX, INCX, SY, INCY, C, S)
SROT
Definition: srot.f:94
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