LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ dlasd1()

subroutine dlasd1 ( integer  nl,
integer  nr,
integer  sqre,
double precision, dimension( * )  d,
double precision  alpha,
double precision  beta,
double precision, dimension( ldu, * )  u,
integer  ldu,
double precision, dimension( ldvt, * )  vt,
integer  ldvt,
integer, dimension( * )  idxq,
integer, dimension( * )  iwork,
double precision, dimension( * )  work,
integer  info 
)

DLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.

Download DLASD1 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
 where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called from DLASD0.

 A related subroutine DLASD7 handles the case in which the singular
 values (and the singular vectors in factored form) are desired.

 DLASD1 computes the SVD as follows:

               ( D1(in)    0    0       0 )
   B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
               (   0       0   D2(in)   0 )

     = U(out) * ( D(out) 0) * VT(out)

 where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
 with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
 elsewhere; and the entry b is empty if SQRE = 0.

 The left singular vectors of the original matrix are stored in U, and
 the transpose of the right singular vectors are stored in VT, and the
 singular values are in D.  The algorithm consists of three stages:

    The first stage consists of deflating the size of the problem
    when there are multiple singular values or when there are zeros in
    the Z vector.  For each such occurrence the dimension of the
    secular equation problem is reduced by one.  This stage is
    performed by the routine DLASD2.

    The second stage consists of calculating the updated
    singular values. This is done by finding the square roots of the
    roots of the secular equation via the routine DLASD4 (as called
    by DLASD3). This routine also calculates the singular vectors of
    the current problem.

    The final stage consists of computing the updated singular vectors
    directly using the updated singular values.  The singular vectors
    for the current problem are multiplied with the singular vectors
    from the overall problem.
Parameters
[in]NL
          NL is INTEGER
         The row dimension of the upper block.  NL >= 1.
[in]NR
          NR is INTEGER
         The row dimension of the lower block.  NR >= 1.
[in]SQRE
          SQRE is INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.

         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.
[in,out]D
          D is DOUBLE PRECISION array,
                        dimension (N = NL+NR+1).
         On entry D(1:NL,1:NL) contains the singular values of the
         upper block; and D(NL+2:N) contains the singular values of
         the lower block. On exit D(1:N) contains the singular values
         of the modified matrix.
[in,out]ALPHA
          ALPHA is DOUBLE PRECISION
         Contains the diagonal element associated with the added row.
[in,out]BETA
          BETA is DOUBLE PRECISION
         Contains the off-diagonal element associated with the added
         row.
[in,out]U
          U is DOUBLE PRECISION array, dimension(LDU,N)
         On entry U(1:NL, 1:NL) contains the left singular vectors of
         the upper block; U(NL+2:N, NL+2:N) contains the left singular
         vectors of the lower block. On exit U contains the left
         singular vectors of the bidiagonal matrix.
[in]LDU
          LDU is INTEGER
         The leading dimension of the array U.  LDU >= max( 1, N ).
[in,out]VT
          VT is DOUBLE PRECISION array, dimension(LDVT,M)
         where M = N + SQRE.
         On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
         vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
         the right singular vectors of the lower block. On exit
         VT**T contains the right singular vectors of the
         bidiagonal matrix.
[in]LDVT
          LDVT is INTEGER
         The leading dimension of the array VT.  LDVT >= max( 1, M ).
[in,out]IDXQ
          IDXQ is INTEGER array, dimension(N)
         This contains the permutation which will reintegrate the
         subproblem just solved back into sorted order, i.e.
         D( IDXQ( I = 1, N ) ) will be in ascending order.
[out]IWORK
          IWORK is INTEGER array, dimension( 4 * N )
[out]WORK
          WORK is DOUBLE PRECISION array, dimension( 3*M**2 + 2*M )
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, a singular value did not converge
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 202 of file dlasd1.f.

204*
205* -- LAPACK auxiliary routine --
206* -- LAPACK is a software package provided by Univ. of Tennessee, --
207* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
208*
209* .. Scalar Arguments ..
210 INTEGER INFO, LDU, LDVT, NL, NR, SQRE
211 DOUBLE PRECISION ALPHA, BETA
212* ..
213* .. Array Arguments ..
214 INTEGER IDXQ( * ), IWORK( * )
215 DOUBLE PRECISION D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
216* ..
217*
218* =====================================================================
219*
220* .. Parameters ..
221*
222 DOUBLE PRECISION ONE, ZERO
223 parameter( one = 1.0d+0, zero = 0.0d+0 )
224* ..
225* .. Local Scalars ..
226 INTEGER COLTYP, I, IDX, IDXC, IDXP, IQ, ISIGMA, IU2,
227 $ IVT2, IZ, K, LDQ, LDU2, LDVT2, M, N, N1, N2
228 DOUBLE PRECISION ORGNRM
229* ..
230* .. External Subroutines ..
231 EXTERNAL dlamrg, dlascl, dlasd2, dlasd3, xerbla
232* ..
233* .. Intrinsic Functions ..
234 INTRINSIC abs, max
235* ..
236* .. Executable Statements ..
237*
238* Test the input parameters.
239*
240 info = 0
241*
242 IF( nl.LT.1 ) THEN
243 info = -1
244 ELSE IF( nr.LT.1 ) THEN
245 info = -2
246 ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
247 info = -3
248 END IF
249 IF( info.NE.0 ) THEN
250 CALL xerbla( 'DLASD1', -info )
251 RETURN
252 END IF
253*
254 n = nl + nr + 1
255 m = n + sqre
256*
257* The following values are for bookkeeping purposes only. They are
258* integer pointers which indicate the portion of the workspace
259* used by a particular array in DLASD2 and DLASD3.
260*
261 ldu2 = n
262 ldvt2 = m
263*
264 iz = 1
265 isigma = iz + m
266 iu2 = isigma + n
267 ivt2 = iu2 + ldu2*n
268 iq = ivt2 + ldvt2*m
269*
270 idx = 1
271 idxc = idx + n
272 coltyp = idxc + n
273 idxp = coltyp + n
274*
275* Scale.
276*
277 orgnrm = max( abs( alpha ), abs( beta ) )
278 d( nl+1 ) = zero
279 DO 10 i = 1, n
280 IF( abs( d( i ) ).GT.orgnrm ) THEN
281 orgnrm = abs( d( i ) )
282 END IF
283 10 CONTINUE
284 CALL dlascl( 'G', 0, 0, orgnrm, one, n, 1, d, n, info )
285 alpha = alpha / orgnrm
286 beta = beta / orgnrm
287*
288* Deflate singular values.
289*
290 CALL dlasd2( nl, nr, sqre, k, d, work( iz ), alpha, beta, u, ldu,
291 $ vt, ldvt, work( isigma ), work( iu2 ), ldu2,
292 $ work( ivt2 ), ldvt2, iwork( idxp ), iwork( idx ),
293 $ iwork( idxc ), idxq, iwork( coltyp ), info )
294*
295* Solve Secular Equation and update singular vectors.
296*
297 ldq = k
298 CALL dlasd3( nl, nr, sqre, k, d, work( iq ), ldq, work( isigma ),
299 $ u, ldu, work( iu2 ), ldu2, vt, ldvt, work( ivt2 ),
300 $ ldvt2, iwork( idxc ), iwork( coltyp ), work( iz ),
301 $ info )
302*
303* Report the convergence failure.
304*
305 IF( info.NE.0 ) THEN
306 RETURN
307 END IF
308*
309* Unscale.
310*
311 CALL dlascl( 'G', 0, 0, one, orgnrm, n, 1, d, n, info )
312*
313* Prepare the IDXQ sorting permutation.
314*
315 n1 = k
316 n2 = n - k
317 CALL dlamrg( n1, n2, d, 1, -1, idxq )
318*
319 RETURN
320*
321* End of DLASD1
322*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
dlamrg
subroutine dlamrg(n1, n2, a, dtrd1, dtrd2, index)
DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition dlamrg.f:99
dlascl
subroutine dlascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
DLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition dlascl.f:143
dlasd2
subroutine dlasd2(nl, nr, sqre, k, d, z, alpha, beta, u, ldu, vt, ldvt, dsigma, u2, ldu2, vt2, ldvt2, idxp, idx, idxc, idxq, coltyp, info)
DLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.
Definition dlasd2.f:269
dlasd3
subroutine dlasd3(nl, nr, sqre, k, d, q, ldq, dsigma, u, ldu, u2, ldu2, vt, ldvt, vt2, ldvt2, idxc, ctot, z, info)
DLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and...
Definition dlasd3.f:217
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