LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ slasd1()

subroutine slasd1 ( integer  nl,
integer  nr,
integer  sqre,
real, dimension( * )  d,
real  alpha,
real  beta,
real, dimension( ldu, * )  u,
integer  ldu,
real, dimension( ldvt, * )  vt,
integer  ldvt,
integer, dimension( * )  idxq,
integer, dimension( * )  iwork,
real, dimension( * )  work,
integer  info 
)

SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.

Download SLASD1 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
 where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0.

 A related subroutine SLASD7 handles the case in which the singular
 values (and the singular vectors in factored form) are desired.

 SLASD1 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 SLASD2.

    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 SLASD4 (as called
    by SLASD3). 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 REAL array, dimension (NL+NR+1).
         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 REAL
         Contains the diagonal element associated with the added row.
[in,out]BETA
          BETA is REAL
         Contains the off-diagonal element associated with the added
         row.
[in,out]U
          U is REAL 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 REAL 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 REAL 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 slasd1.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 REAL ALPHA, BETA
212* ..
213* .. Array Arguments ..
214 INTEGER IDXQ( * ), IWORK( * )
215 REAL D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
216* ..
217*
218* =====================================================================
219*
220* .. Parameters ..
221*
222 REAL ONE, ZERO
223 parameter( one = 1.0e+0, zero = 0.0e+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 REAL ORGNRM
229* ..
230* .. External Subroutines ..
231 EXTERNAL slamrg, slascl, slasd2, slasd3, 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( 'SLASD1', -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 SLASD2 and SLASD3.
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 slascl( '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 slasd2( 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 slasd3( 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 possible convergence failure.
304*
305 IF( info.NE.0 ) THEN
306 RETURN
307 END IF
308*
309* Unscale.
310*
311 CALL slascl( '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 slamrg( n1, n2, d, 1, -1, idxq )
318*
319 RETURN
320*
321* End of SLASD1
322*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
slamrg
subroutine slamrg(n1, n2, a, strd1, strd2, index)
SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single...
Definition slamrg.f:99
slascl
subroutine slascl(type, kl, ku, cfrom, cto, m, n, a, lda, info)
SLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition slascl.f:143
slasd2
subroutine slasd2(nl, nr, sqre, k, d, z, alpha, beta, u, ldu, vt, ldvt, dsigma, u2, ldu2, vt2, ldvt2, idxp, idx, idxc, idxq, coltyp, info)
SLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.
Definition slasd2.f:269
slasd3
subroutine slasd3(nl, nr, sqre, k, d, q, ldq, dsigma, u, ldu, u2, ldu2, vt, ldvt, vt2, ldvt2, idxc, ctot, z, info)
SLASD3 finds all square roots of the roots of the secular equation, as defined by the values in D and...
Definition slasd3.f:217
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