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version 1.2, 2010/04/21 13:45:16 version 1.19, 2023/08/07 08:38:54
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   *> \brief \b DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DLAGTF + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlagtf.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlagtf.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlagtf.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, N
   *       DOUBLE PRECISION   LAMBDA, TOL
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IN( * )
   *       DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
   *> tridiagonal matrix and lambda is a scalar, as
   *>
   *>    T - lambda*I = PLU,
   *>
   *> where P is a permutation matrix, L is a unit lower tridiagonal matrix
   *> with at most one non-zero sub-diagonal elements per column and U is
   *> an upper triangular matrix with at most two non-zero super-diagonal
   *> elements per column.
   *>
   *> The factorization is obtained by Gaussian elimination with partial
   *> pivoting and implicit row scaling.
   *>
   *> The parameter LAMBDA is included in the routine so that DLAGTF may
   *> be used, in conjunction with DLAGTS, to obtain eigenvectors of T by
   *> inverse iteration.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the matrix T.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (N)
   *>          On entry, A must contain the diagonal elements of T.
   *>
   *>          On exit, A is overwritten by the n diagonal elements of the
   *>          upper triangular matrix U of the factorization of T.
   *> \endverbatim
   *>
   *> \param[in] LAMBDA
   *> \verbatim
   *>          LAMBDA is DOUBLE PRECISION
   *>          On entry, the scalar lambda.
   *> \endverbatim
   *>
   *> \param[in,out] B
   *> \verbatim
   *>          B is DOUBLE PRECISION array, dimension (N-1)
   *>          On entry, B must contain the (n-1) super-diagonal elements of
   *>          T.
   *>
   *>          On exit, B is overwritten by the (n-1) super-diagonal
   *>          elements of the matrix U of the factorization of T.
   *> \endverbatim
   *>
   *> \param[in,out] C
   *> \verbatim
   *>          C is DOUBLE PRECISION array, dimension (N-1)
   *>          On entry, C must contain the (n-1) sub-diagonal elements of
   *>          T.
   *>
   *>          On exit, C is overwritten by the (n-1) sub-diagonal elements
   *>          of the matrix L of the factorization of T.
   *> \endverbatim
   *>
   *> \param[in] TOL
   *> \verbatim
   *>          TOL is DOUBLE PRECISION
   *>          On entry, a relative tolerance used to indicate whether or
   *>          not the matrix (T - lambda*I) is nearly singular. TOL should
   *>          normally be chose as approximately the largest relative error
   *>          in the elements of T. For example, if the elements of T are
   *>          correct to about 4 significant figures, then TOL should be
   *>          set to about 5*10**(-4). If TOL is supplied as less than eps,
   *>          where eps is the relative machine precision, then the value
   *>          eps is used in place of TOL.
   *> \endverbatim
   *>
   *> \param[out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (N-2)
   *>          On exit, D is overwritten by the (n-2) second super-diagonal
   *>          elements of the matrix U of the factorization of T.
   *> \endverbatim
   *>
   *> \param[out] IN
   *> \verbatim
   *>          IN is INTEGER array, dimension (N)
   *>          On exit, IN contains details of the permutation matrix P. If
   *>          an interchange occurred at the kth step of the elimination,
   *>          then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
   *>          returns the smallest positive integer j such that
   *>
   *>             abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
   *>
   *>          where norm( A(j) ) denotes the sum of the absolute values of
   *>          the jth row of the matrix A. If no such j exists then IN(n)
   *>          is returned as zero. If IN(n) is returned as positive, then a
   *>          diagonal element of U is small, indicating that
   *>          (T - lambda*I) is singular or nearly singular,
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>          = 0:  successful exit
   *>          < 0:  if INFO = -k, the kth 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 DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )        SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
 *  *
 *  -- LAPACK routine (version 3.2) --  *  -- LAPACK computational routine --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            INFO, N        INTEGER            INFO, N
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       DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * )        DOUBLE PRECISION   A( * ), B( * ), C( * ), D( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n  
 *  tridiagonal matrix and lambda is a scalar, as  
 *  
 *     T - lambda*I = PLU,  
 *  
 *  where P is a permutation matrix, L is a unit lower tridiagonal matrix  
 *  with at most one non-zero sub-diagonal elements per column and U is  
 *  an upper triangular matrix with at most two non-zero super-diagonal  
 *  elements per column.  
 *  
 *  The factorization is obtained by Gaussian elimination with partial  
 *  pivoting and implicit row scaling.  
 *  
 *  The parameter LAMBDA is included in the routine so that DLAGTF may  
 *  be used, in conjunction with DLAGTS, to obtain eigenvectors of T by  
 *  inverse iteration.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  N       (input) INTEGER  
 *          The order of the matrix T.  
 *  
 *  A       (input/output) DOUBLE PRECISION array, dimension (N)  
 *          On entry, A must contain the diagonal elements of T.  
 *  
 *          On exit, A is overwritten by the n diagonal elements of the  
 *          upper triangular matrix U of the factorization of T.  
 *  
 *  LAMBDA  (input) DOUBLE PRECISION  
 *          On entry, the scalar lambda.  
 *  
 *  B       (input/output) DOUBLE PRECISION array, dimension (N-1)  
 *          On entry, B must contain the (n-1) super-diagonal elements of  
 *          T.  
 *  
 *          On exit, B is overwritten by the (n-1) super-diagonal  
 *          elements of the matrix U of the factorization of T.  
 *  
 *  C       (input/output) DOUBLE PRECISION array, dimension (N-1)  
 *          On entry, C must contain the (n-1) sub-diagonal elements of  
 *          T.  
 *  
 *          On exit, C is overwritten by the (n-1) sub-diagonal elements  
 *          of the matrix L of the factorization of T.  
 *  
 *  TOL     (input) DOUBLE PRECISION  
 *          On entry, a relative tolerance used to indicate whether or  
 *          not the matrix (T - lambda*I) is nearly singular. TOL should  
 *          normally be chose as approximately the largest relative error  
 *          in the elements of T. For example, if the elements of T are  
 *          correct to about 4 significant figures, then TOL should be  
 *          set to about 5*10**(-4). If TOL is supplied as less than eps,  
 *          where eps is the relative machine precision, then the value  
 *          eps is used in place of TOL.  
 *  
 *  D       (output) DOUBLE PRECISION array, dimension (N-2)  
 *          On exit, D is overwritten by the (n-2) second super-diagonal  
 *          elements of the matrix U of the factorization of T.  
 *  
 *  IN      (output) INTEGER array, dimension (N)  
 *          On exit, IN contains details of the permutation matrix P. If  
 *          an interchange occurred at the kth step of the elimination,  
 *          then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)  
 *          returns the smallest positive integer j such that  
 *  
 *             abs( u(j,j) ).le. norm( (T - lambda*I)(j) )*TOL,  
 *  
 *          where norm( A(j) ) denotes the sum of the absolute values of  
 *          the jth row of the matrix A. If no such j exists then IN(n)  
 *          is returned as zero. If IN(n) is returned as positive, then a  
 *          diagonal element of U is small, indicating that  
 *          (T - lambda*I) is singular or nearly singular,  
 *  
 *  INFO    (output) INTEGER  
 *          = 0   : successful exit  
 *          .lt. 0: if INFO = -k, the kth argument had an illegal value  
 *  
 * =====================================================================  * =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.2  
changed lines
  Added in v.1.19

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