SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
$ RANK, WORK, IWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION B( LDB, * ), D( * ), E( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DLALSD 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.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* = 'U': D and E define an upper bidiagonal matrix.
* = 'L': D and E define a lower bidiagonal matrix.
*
* SMLSIZ (input) INTEGER
* The maximum size of the subproblems at the bottom of the
* computation tree.
*
* N (input) INTEGER
* The dimension of the bidiagonal matrix. N >= 0.
*
* NRHS (input) INTEGER
* The number of columns of B. NRHS must be at least 1.
*
* D (input/output) DOUBLE PRECISION array, dimension (N)
* On entry D contains the main diagonal of the bidiagonal
* matrix. On exit, if INFO = 0, D contains its singular values.
*
* E (input/output) DOUBLE PRECISION array, dimension (N-1)
* Contains the super-diagonal entries of the bidiagonal matrix.
* On exit, E has been destroyed.
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
* On input, B contains the right hand sides of the least
* squares problem. On output, B contains the solution X.
*
* LDB (input) INTEGER
* The leading dimension of B in the calling subprogram.
* LDB must be at least max(1,N).
*
* RCOND (input) DOUBLE PRECISION
* 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).
*
* RANK (output) INTEGER
* The number of singular values of A greater than RCOND times
* the largest singular value.
*
* WORK (workspace) DOUBLE PRECISION 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).
*
* IWORK (workspace) INTEGER array, dimension at least
* (3*N*NLVL + 11*N)
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: The algorithm failed to compute an singular value while
* working on the submatrix lying in rows and columns
* INFO/(N+1) through MOD(INFO,N+1).
*
* Further Details
* ===============
*
* Based on contributions by
* Ming Gu and Ren-Cang Li, Computer Science Division, University of
* California at Berkeley, USA
* Osni Marques, LBNL/NERSC, USA
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
* ..
* .. 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
DOUBLE PRECISION CS, EPS, ORGNRM, R, RCND, SN, TOL
* ..
* .. External Functions ..
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DLANST
EXTERNAL IDAMAX, DLAMCH, DLANST
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGEMM, DLACPY, DLALSA, DLARTG, DLASCL,
$ DLASDA, DLASDQ, DLASET, DLASRT, DROT, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, INT, LOG, 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( 'DLALSD', -INFO )
RETURN
END IF
*
EPS = DLAMCH( '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 DLASET( 'A', 1, NRHS, ZERO, ZERO, B, LDB )
ELSE
RANK = 1
CALL DLASCL( '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 DLARTG( 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 DROT( 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 DROT( 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 = DLANST( 'M', N, D, E )
IF( ORGNRM.EQ.ZERO ) THEN
CALL DLASET( 'A', N, NRHS, ZERO, ZERO, B, LDB )
RETURN
END IF
*
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
CALL DLASCL( '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 DLASET( 'A', N, N, ZERO, ONE, WORK, N )
CALL DLASDQ( '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( IDAMAX( N, D, 1 ) ) )
DO 40 I = 1, N
IF( D( I ).LE.TOL ) THEN
CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
ELSE
CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS, B( I, 1 ),
$ LDB, INFO )
RANK = RANK + 1
END IF
40 CONTINUE
CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO,
$ WORK( NWORK ), N )
CALL DLACPY( 'A', N, NRHS, WORK( NWORK ), N, B, LDB )
*
* Unscale.
*
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
CALL DLASRT( 'D', N, D, INFO )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
*
RETURN
END IF
*
* Book-keeping and setting up some constants.
*
NLVL = INT( LOG( DBLE( N ) / DBLE( 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 DCOPY( 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 DCOPY( NRHS, B( ST, 1 ), LDB, WORK( BX+ST1 ), N )
ELSE IF( NSIZE.LE.SMLSIZ ) THEN
*
* This is a small subproblem and is solved by DLASDQ.
*
CALL DLASET( 'A', NSIZE, NSIZE, ZERO, ONE,
$ WORK( VT+ST1 ), N )
CALL DLASDQ( '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 DLACPY( 'A', NSIZE, NRHS, B( ST, 1 ), LDB,
$ WORK( BX+ST1 ), N )
ELSE
*
* A large problem. Solve it using divide and conquer.
*
CALL DLASDA( 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 DLALSA( 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( IDAMAX( 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 DLASET( 'A', 1, NRHS, ZERO, ZERO, WORK( BX+I-1 ), N )
ELSE
RANK = RANK + 1
CALL DLASCL( '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 DCOPY( NRHS, WORK( BXST ), N, B( ST, 1 ), LDB )
ELSE IF( NSIZE.LE.SMLSIZ ) THEN
CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,
$ WORK( VT+ST1 ), N, WORK( BXST ), N, ZERO,
$ B( ST, 1 ), LDB )
ELSE
CALL DLALSA( 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 DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
CALL DLASRT( 'D', N, D, INFO )
CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
*
RETURN
*
* End of DLALSD
*
END