ddisna()

subroutine ddisna	(	character	job,
		integer	m,
		integer	n,
		double precision, dimension( * )	d,
		double precision, dimension( * )	sep,
		integer	info
	)

DDISNA

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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.

Parameters

[in]	JOB	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.
[in]	M	M is INTEGER The number of rows of the matrix. M >= 0.
[in]	N	N is INTEGER If JOB = 'L' or 'R', the number of columns of the matrix, in which case N >= 0. Ignored if JOB = 'E'.
[in]	D	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.
[out]	SEP	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.
[out]	INFO	INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 116 of file ddisna.f.

*
*  -- 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
*

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