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*> \brief \b DPTCON
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPTCON + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptcon.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptcon.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptcon.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPTCON computes the reciprocal of the condition number (in the
*> 1-norm) of a real symmetric positive definite tridiagonal matrix
*> using the factorization A = L*D*L**T or A = U**T*D*U computed by
*> DPTTRF.
*>
*> Norm(inv(A)) is computed by a direct method, and the reciprocal of
*> the condition number is computed as
*> RCOND = 1 / (ANORM * norm(inv(A))).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The n diagonal elements of the diagonal matrix D from the
*> factorization of A, as computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> The (n-1) off-diagonal elements of the unit bidiagonal factor
*> U or L from the factorization of A, as computed by DPTTRF.
*> \endverbatim
*>
*> \param[in] ANORM
*> \verbatim
*> ANORM is DOUBLE PRECISION
*> The 1-norm of the original matrix A.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal of the condition number of the matrix A,
*> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the
*> 1-norm of inv(A) computed in this routine.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \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 September 2012
*
*> \ingroup doublePTcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The method used is described in Nicholas J. Higham, "Efficient
*> Algorithms for Computing the Condition Number of a Tridiagonal
*> Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DPTCON( N, D, E, ANORM, RCOND, WORK, INFO )
*
* -- LAPACK computational routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
INTEGER INFO, N
DOUBLE PRECISION ANORM, RCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IX
DOUBLE PRECISION AINVNM
* ..
* .. External Functions ..
INTEGER IDAMAX
EXTERNAL IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( ANORM.LT.ZERO ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DPTCON', -INFO )
RETURN
END IF
*
* Quick return if possible
*
RCOND = ZERO
IF( N.EQ.0 ) THEN
RCOND = ONE
RETURN
ELSE IF( ANORM.EQ.ZERO ) THEN
RETURN
END IF
*
* Check that D(1:N) is positive.
*
DO 10 I = 1, N
IF( D( I ).LE.ZERO )
$ RETURN
10 CONTINUE
*
* Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
*
* m(i,j) = abs(A(i,j)), i = j,
* m(i,j) = -abs(A(i,j)), i .ne. j,
*
* and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
*
* Solve M(L) * x = e.
*
WORK( 1 ) = ONE
DO 20 I = 2, N
WORK( I ) = ONE + WORK( I-1 )*ABS( E( I-1 ) )
20 CONTINUE
*
* Solve D * M(L)**T * x = b.
*
WORK( N ) = WORK( N ) / D( N )
DO 30 I = N - 1, 1, -1
WORK( I ) = WORK( I ) / D( I ) + WORK( I+1 )*ABS( E( I ) )
30 CONTINUE
*
* Compute AINVNM = max(x(i)), 1<=i<=n.
*
IX = IDAMAX( N, WORK, 1 )
AINVNM = ABS( WORK( IX ) )
*
* Compute the reciprocal condition number.
*
IF( AINVNM.NE.ZERO )
$ RCOND = ( ONE / AINVNM ) / ANORM
*
RETURN
*
* End of DPTCON
*
END
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