*> \brief \b DDISNA
*
* =========== DOCUMENTATION ===========
*
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* http://www.netlib.org/lapack/explore-html/
*
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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.
*
*> \date December 2016
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DDISNA( JOB, M, N, D, SEP, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. 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