slalsd()

subroutine slalsd	(	character	UPLO,
		integer	SMLSIZ,
		integer	N,
		integer	NRHS,
		real, dimension( * )	D,
		real, dimension( * )	E,
		real, dimension( ldb, * )	B,
		integer	LDB,
		real	RCOND,
		integer	RANK,
		real, dimension( * )	WORK,
		integer, dimension( * )	IWORK,
		integer	INFO
	)

SLALSD uses the singular value decomposition of A to solve the least squares problem.

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Purpose:

 SLALSD uses the singular value decomposition of A to solve the least
 squares problem of finding X to minimize the Euclidean norm of each
 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
 are N-by-NRHS. The solution X overwrites B.

 The singular values of A smaller than RCOND times the largest
 singular value are treated as zero in solving the least squares
 problem; in this case a minimum norm solution is returned.
 The actual singular values are returned in D in ascending order.

 This code makes very mild assumptions about floating point
 arithmetic. It will work on machines with a guard digit in
 add/subtract, or on those binary machines without guard digits
 which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.

Parameters

[in]	UPLO	UPLO is CHARACTER*1 = 'U': D and E define an upper bidiagonal matrix. = 'L': D and E define a lower bidiagonal matrix.
[in]	SMLSIZ	SMLSIZ is INTEGER The maximum size of the subproblems at the bottom of the computation tree.
[in]	N	N is INTEGER The dimension of the bidiagonal matrix. N >= 0.
[in]	NRHS	NRHS is INTEGER The number of columns of B. NRHS must be at least 1.
[in,out]	D	D is REAL array, dimension (N) On entry D contains the main diagonal of the bidiagonal matrix. On exit, if INFO = 0, D contains its singular values.
[in,out]	E	E is REAL array, dimension (N-1) Contains the super-diagonal entries of the bidiagonal matrix. On exit, E has been destroyed.
[in,out]	B	B is REAL array, dimension (LDB,NRHS) On input, B contains the right hand sides of the least squares problem. On output, B contains the solution X.
[in]	LDB	LDB is INTEGER The leading dimension of B in the calling subprogram. LDB must be at least max(1,N).
[in]	RCOND	RCOND is REAL The singular values of A less than or equal to RCOND times the largest singular value are treated as zero in solving the least squares problem. If RCOND is negative, machine precision is used instead. For example, if diag(S)*X=B were the least squares problem, where diag(S) is a diagonal matrix of singular values, the solution would be X(i) = B(i) / S(i) if S(i) is greater than RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to RCOND*max(S).
[out]	RANK	RANK is INTEGER The number of singular values of A greater than RCOND times the largest singular value.
[out]	WORK	WORK is REAL array, dimension at least (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
[out]	IWORK	IWORK is INTEGER array, dimension at least (3*N*NLVL + 11*N)
[out]	INFO	INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute a singular value while working on the submatrix lying in rows and columns INFO/(N+1) through MOD(INFO,N+1).

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

Definition at line 177 of file slalsd.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDB, N, NRHS, RANK, SMLSIZ
      REAL               RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      REAL               B( LDB, * ), D( * ), E( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE, TWO
      parameter( zero = 0.0e0, one = 1.0e0, two = 2.0e0 )
*     ..
*     .. Local Scalars ..
      INTEGER            BX, BXST, C, DIFL, DIFR, GIVCOL, GIVNUM,
     $                   GIVPTR, I, ICMPQ1, ICMPQ2, IWK, J, K, NLVL,
     $                   NM1, NSIZE, NSUB, NWORK, PERM, POLES, S, SIZEI,
     $                   SMLSZP, SQRE, ST, ST1, U, VT, Z
      REAL               CS, EPS, ORGNRM, R, RCND, SN, TOL
*     ..
*     .. External Functions ..
      INTEGER            ISAMAX
      REAL               SLAMCH, SLANST
      EXTERNAL           isamax, slamch, slanst
*     ..
*     .. External Subroutines ..
      EXTERNAL           scopy, sgemm, slacpy, slalsa, slartg, slascl,
     $                   slasda, slasdq, slaset, slasrt, srot, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, int, log, real, sign
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
*
      IF( n.LT.0 ) THEN
         info = -3
      ELSE IF( nrhs.LT.1 ) THEN
         info = -4
      ELSE IF( ( ldb.LT.1 ) .OR. ( ldb.LT.n ) ) THEN
         info = -8
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SLALSD', -info )
         RETURN
      END IF
*
      eps = slamch( 'Epsilon' )
*
*     Set up the tolerance.
*
      IF( ( rcond.LE.zero ) .OR. ( rcond.GE.one ) ) THEN
         rcnd = eps
      ELSE
         rcnd = rcond
      END IF
*
      rank = 0
*
*     Quick return if possible.
*
      IF( n.EQ.0 ) THEN
         RETURN
      ELSE IF( n.EQ.1 ) THEN
         IF( d( 1 ).EQ.zero ) THEN
            CALL slaset( 'A', 1, nrhs, zero, zero, b, ldb )
         ELSE
            rank = 1
            CALL slascl( 'G', 0, 0, d( 1 ), one, 1, nrhs, b, ldb, info )
            d( 1 ) = abs( d( 1 ) )
         END IF
         RETURN
      END IF
*
*     Rotate the matrix if it is lower bidiagonal.
*
      IF( uplo.EQ.'L' ) THEN
         DO 10 i = 1, n - 1
            CALL slartg( d( i ), e( i ), cs, sn, r )
            d( i ) = r
            e( i ) = sn*d( i+1 )
            d( i+1 ) = cs*d( i+1 )
            IF( nrhs.EQ.1 ) THEN
               CALL srot( 1, b( i, 1 ), 1, b( i+1, 1 ), 1, cs, sn )
            ELSE
               work( i*2-1 ) = cs
               work( i*2 ) = sn
            END IF
   10    CONTINUE
         IF( nrhs.GT.1 ) THEN
            DO 30 i = 1, nrhs
               DO 20 j = 1, n - 1
                  cs = work( j*2-1 )
                  sn = work( j*2 )
                  CALL srot( 1, b( j, i ), 1, b( j+1, i ), 1, cs, sn )
   20          CONTINUE
   30       CONTINUE
         END IF
      END IF
*
*     Scale.
*
      nm1 = n - 1
      orgnrm = slanst( 'M', n, d, e )
      IF( orgnrm.EQ.zero ) THEN
         CALL slaset( 'A', n, nrhs, zero, zero, b, ldb )
         RETURN
      END IF
*
      CALL slascl( 'G', 0, 0, orgnrm, one, n, 1, d, n, info )
      CALL slascl( 'G', 0, 0, orgnrm, one, nm1, 1, e, nm1, info )
*
*     If N is smaller than the minimum divide size SMLSIZ, then solve
*     the problem with another solver.
*
      IF( n.LE.smlsiz ) THEN
         nwork = 1 + n*n
         CALL slaset( 'A', n, n, zero, one, work, n )
         CALL slasdq( 'U', 0, n, n, 0, nrhs, d, e, work, n, work, n, b,
     $                ldb, work( nwork ), info )
         IF( info.NE.0 ) THEN
            RETURN
         END IF
         tol = rcnd*abs( d( isamax( n, d, 1 ) ) )
         DO 40 i = 1, n
            IF( d( i ).LE.tol ) THEN
               CALL slaset( 'A', 1, nrhs, zero, zero, b( i, 1 ), ldb )
            ELSE
               CALL slascl( 'G', 0, 0, d( i ), one, 1, nrhs, b( i, 1 ),
     $                      ldb, info )
               rank = rank + 1
            END IF
   40    CONTINUE
         CALL sgemm( 'T', 'N', n, nrhs, n, one, work, n, b, ldb, zero,
     $               work( nwork ), n )
         CALL slacpy( 'A', n, nrhs, work( nwork ), n, b, ldb )
*
*        Unscale.
*
         CALL slascl( 'G', 0, 0, one, orgnrm, n, 1, d, n, info )
         CALL slasrt( 'D', n, d, info )
         CALL slascl( 'G', 0, 0, orgnrm, one, n, nrhs, b, ldb, info )
*
         RETURN
      END IF
*
*     Book-keeping and setting up some constants.
*
      nlvl = int( log( real( n ) / real( smlsiz+1 ) ) / log( two ) ) + 1
*
      smlszp = smlsiz + 1
*
      u = 1
      vt = 1 + smlsiz*n
      difl = vt + smlszp*n
      difr = difl + nlvl*n
      z = difr + nlvl*n*2
      c = z + nlvl*n
      s = c + n
      poles = s + n
      givnum = poles + 2*nlvl*n
      bx = givnum + 2*nlvl*n
      nwork = bx + n*nrhs
*
      sizei = 1 + n
      k = sizei + n
      givptr = k + n
      perm = givptr + n
      givcol = perm + nlvl*n
      iwk = givcol + nlvl*n*2
*
      st = 1
      sqre = 0
      icmpq1 = 1
      icmpq2 = 0
      nsub = 0
*
      DO 50 i = 1, n
         IF( abs( d( i ) ).LT.eps ) THEN
            d( i ) = sign( eps, d( i ) )
         END IF
   50 CONTINUE
*
      DO 60 i = 1, nm1
         IF( ( abs( e( i ) ).LT.eps ) .OR. ( i.EQ.nm1 ) ) THEN
            nsub = nsub + 1
            iwork( nsub ) = st
*
*           Subproblem found. First determine its size and then
*           apply divide and conquer on it.
*
            IF( i.LT.nm1 ) THEN
*
*              A subproblem with E(I) small for I < NM1.
*
               nsize = i - st + 1
               iwork( sizei+nsub-1 ) = nsize
            ELSE IF( abs( e( i ) ).GE.eps ) THEN
*
*              A subproblem with E(NM1) not too small but I = NM1.
*
               nsize = n - st + 1
               iwork( sizei+nsub-1 ) = nsize
            ELSE
*
*              A subproblem with E(NM1) small. This implies an
*              1-by-1 subproblem at D(N), which is not solved
*              explicitly.
*
               nsize = i - st + 1
               iwork( sizei+nsub-1 ) = nsize
               nsub = nsub + 1
               iwork( nsub ) = n
               iwork( sizei+nsub-1 ) = 1
               CALL scopy( nrhs, b( n, 1 ), ldb, work( bx+nm1 ), n )
            END IF
            st1 = st - 1
            IF( nsize.EQ.1 ) THEN
*
*              This is a 1-by-1 subproblem and is not solved
*              explicitly.
*
               CALL scopy( nrhs, b( st, 1 ), ldb, work( bx+st1 ), n )
            ELSE IF( nsize.LE.smlsiz ) THEN
*
*              This is a small subproblem and is solved by SLASDQ.
*
               CALL slaset( 'A', nsize, nsize, zero, one,
     $                      work( vt+st1 ), n )
               CALL slasdq( 'U', 0, nsize, nsize, 0, nrhs, d( st ),
     $                      e( st ), work( vt+st1 ), n, work( nwork ),
     $                      n, b( st, 1 ), ldb, work( nwork ), info )
               IF( info.NE.0 ) THEN
                  RETURN
               END IF
               CALL slacpy( 'A', nsize, nrhs, b( st, 1 ), ldb,
     $                      work( bx+st1 ), n )
            ELSE
*
*              A large problem. Solve it using divide and conquer.
*
               CALL slasda( icmpq1, smlsiz, nsize, sqre, d( st ),
     $                      e( st ), work( u+st1 ), n, work( vt+st1 ),
     $                      iwork( k+st1 ), work( difl+st1 ),
     $                      work( difr+st1 ), work( z+st1 ),
     $                      work( poles+st1 ), iwork( givptr+st1 ),
     $                      iwork( givcol+st1 ), n, iwork( perm+st1 ),
     $                      work( givnum+st1 ), work( c+st1 ),
     $                      work( s+st1 ), work( nwork ), iwork( iwk ),
     $                      info )
               IF( info.NE.0 ) THEN
                  RETURN
               END IF
               bxst = bx + st1
               CALL slalsa( icmpq2, smlsiz, nsize, nrhs, b( st, 1 ),
     $                      ldb, work( bxst ), n, work( u+st1 ), n,
     $                      work( vt+st1 ), iwork( k+st1 ),
     $                      work( difl+st1 ), work( difr+st1 ),
     $                      work( z+st1 ), work( poles+st1 ),
     $                      iwork( givptr+st1 ), iwork( givcol+st1 ), n,
     $                      iwork( perm+st1 ), work( givnum+st1 ),
     $                      work( c+st1 ), work( s+st1 ), work( nwork ),
     $                      iwork( iwk ), info )
               IF( info.NE.0 ) THEN
                  RETURN
               END IF
            END IF
            st = i + 1
         END IF
   60 CONTINUE
*
*     Apply the singular values and treat the tiny ones as zero.
*
      tol = rcnd*abs( d( isamax( n, d, 1 ) ) )
*
      DO 70 i = 1, n
*
*        Some of the elements in D can be negative because 1-by-1
*        subproblems were not solved explicitly.
*
         IF( abs( d( i ) ).LE.tol ) THEN
            CALL slaset( 'A', 1, nrhs, zero, zero, work( bx+i-1 ), n )
         ELSE
            rank = rank + 1
            CALL slascl( 'G', 0, 0, d( i ), one, 1, nrhs,
     $                   work( bx+i-1 ), n, info )
         END IF
         d( i ) = abs( d( i ) )
   70 CONTINUE
*
*     Now apply back the right singular vectors.
*
      icmpq2 = 1
      DO 80 i = 1, nsub
         st = iwork( i )
         st1 = st - 1
         nsize = iwork( sizei+i-1 )
         bxst = bx + st1
         IF( nsize.EQ.1 ) THEN
            CALL scopy( nrhs, work( bxst ), n, b( st, 1 ), ldb )
         ELSE IF( nsize.LE.smlsiz ) THEN
            CALL sgemm( 'T', 'N', nsize, nrhs, nsize, one,
     $                  work( vt+st1 ), n, work( bxst ), n, zero,
     $                  b( st, 1 ), ldb )
         ELSE
            CALL slalsa( icmpq2, smlsiz, nsize, nrhs, work( bxst ), n,
     $                   b( st, 1 ), ldb, work( u+st1 ), n,
     $                   work( vt+st1 ), iwork( k+st1 ),
     $                   work( difl+st1 ), work( difr+st1 ),
     $                   work( z+st1 ), work( poles+st1 ),
     $                   iwork( givptr+st1 ), iwork( givcol+st1 ), n,
     $                   iwork( perm+st1 ), work( givnum+st1 ),
     $                   work( c+st1 ), work( s+st1 ), work( nwork ),
     $                   iwork( iwk ), info )
            IF( info.NE.0 ) THEN
               RETURN
            END IF
         END IF
   80 CONTINUE
*
*     Unscale and sort the singular values.
*
      CALL slascl( 'G', 0, 0, one, orgnrm, n, 1, d, n, info )
      CALL slasrt( 'D', n, d, info )
      CALL slascl( 'G', 0, 0, orgnrm, one, n, nrhs, b, ldb, info )
*
      RETURN
*
*     End of SLALSD
*

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