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OpenBLAS/lapack-netlib/SRC/slalsd.f

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*> \brief \b SLALSD uses the singular value decomposition of A to solve the least squares problem.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLALSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slalsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slalsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slalsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
* RANK, WORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ
* REAL RCOND
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* REAL B( LDB, * ), D( * ), E( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': D and E define an upper bidiagonal matrix.
*> = 'L': D and E define a lower bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] SMLSIZ
*> \verbatim
*> SMLSIZ is INTEGER
*> The maximum size of the subproblems at the bottom of the
*> computation tree.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the bidiagonal matrix. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of columns of B. NRHS must be at least 1.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*> E is REAL array, dimension (N-1)
*> Contains the super-diagonal entries of the bidiagonal matrix.
*> On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B in the calling subprogram.
*> LDB must be at least max(1,N).
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*> 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).
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*> RANK is INTEGER
*> The number of singular values of A greater than RCOND times
*> the largest singular value.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> 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).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension at least
*> (3*N*NLVL + 11*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> 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).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup realOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
*> California at Berkeley, USA \n
*> Osni Marques, LBNL/NERSC, USA \n
*
* =====================================================================
SUBROUTINE SLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
$ RANK, WORK, IWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. 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
*
END
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