*> \brief \b DDISNA * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DDISNA + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * 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 auxOTHERcomputational * * ===================================================================== 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