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SUBROUTINE DDISNA( JOB, M, N, D, SEP, INFO )
*
* -- LAPACK routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER JOB
INTEGER INFO, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), SEP( * )
* ..
*
* Purpose
* =======
*
* 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.
*
* Arguments
* =========
*
* JOB (input) 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.
*
* M (input) INTEGER
* The number of rows of the matrix. M >= 0.
*
* N (input) INTEGER
* If JOB = 'L' or 'R', the number of columns of the matrix,
* in which case N >= 0. Ignored if JOB = 'E'.
*
* D (input) 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.
*
* SEP (output) DOUBLE PRECISION array, dimension (M) if JOB = 'E'
* dimension (min(M,N)) if JOB = 'L' or 'R'
* The reciprocal condition numbers of the vectors.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* =====================================================================
*
* .. 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
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