ddisna.f

*> \brief \b DDISNA
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE DDISNA( JOB, M, N, D, SEP, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOB
*       INTEGER            INFO, M, N
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   D( * ), SEP( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DDISNA computes the reciprocal condition numbers for the eigenvectors
*> of a real symmetric or complex Hermitian matrix or for the left or
*> right singular vectors of a general m-by-n matrix. The reciprocal
*> condition number is the 'gap' between the corresponding eigenvalue or
*> singular value and the nearest other one.
*>
*> The bound on the error, measured by angle in radians, in the I-th
*> computed vector is given by
*>
*>        DLAMCH( 'E' ) * ( ANORM / SEP( I ) )
*>
*> where ANORM = 2-norm(A) = max( abs( D(j) ) ).  SEP(I) is not allowed
*> to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of
*> the error bound.
*>
*> DDISNA may also be used to compute error bounds for eigenvectors of
*> the generalized symmetric definite eigenproblem.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOB
*> \verbatim
*>          JOB is CHARACTER*1
*>          Specifies for which problem the reciprocal condition numbers
*>          should be computed:
*>          = 'E':  the eigenvectors of a symmetric/Hermitian matrix;
*>          = 'L':  the left singular vectors of a general matrix;
*>          = 'R':  the right singular vectors of a general matrix.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          If JOB = 'L' or 'R', the number of columns of the matrix,
*>          in which case N >= 0. Ignored if JOB = 'E'.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
*>                              dimension (min(M,N)) if JOB = 'L' or 'R'
*>          The eigenvalues (if JOB = 'E') or singular values (if JOB =
*>          'L' or 'R') of the matrix, in either increasing or decreasing
*>          order. If singular values, they must be non-negative.
*> \endverbatim
*>
*> \param[out] SEP
*> \verbatim
*>          SEP is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
*>                               dimension (min(M,N)) if JOB = 'L' or 'R'
*>          The reciprocal condition numbers of the vectors.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup disna
*
*  =====================================================================
      SUBROUTINE ddisna( JOB, M, N, D, SEP, INFO )
*
*  -- 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          JOB
      INTEGER            INFO, M, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   D( * ), SEP( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      parameter( zero = 0.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            DECR, EIGEN, INCR, LEFT, RIGHT, SING
      INTEGER            I, K
      DOUBLE PRECISION   ANORM, EPS, NEWGAP, OLDGAP, SAFMIN, THRESH
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           lsame, dlamch
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      info = 0
      eigen = lsame( job, 'E' )
      left = lsame( job, 'L' )
      right = lsame( job, 'R' )
      sing = left .OR. right
      IF( eigen ) THEN
         k = m
      ELSE IF( sing ) THEN
         k = min( m, n )
      END IF
      IF( .NOT.eigen .AND. .NOT.sing ) THEN
         info = -1
      ELSE IF( m.LT.0 ) THEN
         info = -2
      ELSE IF( k.LT.0 ) THEN
         info = -3
      ELSE
         incr = .true.
         decr = .true.
         DO 10 i = 1, k - 1
            IF( incr )
     $         incr = incr .AND. d( i ).LE.d( i+1 )
            IF( decr )
     $         decr = decr .AND. d( i ).GE.d( i+1 )
   10    CONTINUE
         IF( sing .AND. k.GT.0 ) THEN
            IF( incr )
     $         incr = incr .AND. zero.LE.d( 1 )
            IF( decr )
     $         decr = decr .AND. d( k ).GE.zero
         END IF
         IF( .NOT.( incr .OR. decr ) )
     $      info = -4
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DDISNA', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( k.EQ.0 )
     $   RETURN
*
*     Compute reciprocal condition numbers
*
      IF( k.EQ.1 ) THEN
         sep( 1 ) = dlamch( 'O' )
      ELSE
         oldgap = abs( d( 2 )-d( 1 ) )
         sep( 1 ) = oldgap
         DO 20 i = 2, k - 1
            newgap = abs( d( i+1 )-d( i ) )
            sep( i ) = min( oldgap, newgap )
            oldgap = newgap
   20    CONTINUE
         sep( k ) = oldgap
      END IF
      IF( sing ) THEN
         IF( ( left .AND. m.GT.n ) .OR. ( right .AND. m.LT.n ) ) THEN
            IF( incr )
     $         sep( 1 ) = min( sep( 1 ), d( 1 ) )
            IF( decr )
     $         sep( k ) = min( sep( k ), d( k ) )
         END IF
      END IF
*
*     Ensure that reciprocal condition numbers are not less than
*     threshold, in order to limit the size of the error bound
*
      eps = dlamch( 'E' )
      safmin = dlamch( 'S' )
      anorm = max( abs( d( 1 ) ), abs( d( k ) ) )
      IF( anorm.EQ.zero ) THEN
         thresh = eps
      ELSE
         thresh = max( eps*anorm, safmin )
      END IF
      DO 30 i = 1, k
         sep( i ) = max( sep( i ), thresh )
   30 CONTINUE
*
      RETURN
*
*     End of DDISNA
*
      END