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*> \brief \b SLAGTF 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 SLAGTF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slagtf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slagtf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slagtf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, N
* REAL LAMBDA, TOL
* ..
* .. Array Arguments ..
* INTEGER IN( * )
* REAL A( * ), B( * ), C( * ), D( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLAGTF 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 SLAGTF may
*> be used, in conjunction with SLAGTS, 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 REAL 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 REAL
*> On entry, the scalar lambda.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is REAL 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 REAL 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 REAL
*> 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 REAL 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) ).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,
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0 : successful exit
*> .lt. 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.
*
*> \date December 2016
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE SLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, N
REAL LAMBDA, TOL
* ..
* .. Array Arguments ..
INTEGER IN( * )
REAL A( * ), B( * ), C( * ), D( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER K
REAL EPS, MULT, PIV1, PIV2, SCALE1, SCALE2, TEMP, TL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
REAL SLAMCH
EXTERNAL SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'SLAGTF', -INFO )
RETURN
END IF
*
IF( N.EQ.0 )
$ RETURN
*
A( 1 ) = A( 1 ) - LAMBDA
IN( N ) = 0
IF( N.EQ.1 ) THEN
IF( A( 1 ).EQ.ZERO )
$ IN( 1 ) = 1
RETURN
END IF
*
EPS = SLAMCH( 'Epsilon' )
*
TL = MAX( TOL, EPS )
SCALE1 = ABS( A( 1 ) ) + ABS( B( 1 ) )
DO 10 K = 1, N - 1
A( K+1 ) = A( K+1 ) - LAMBDA
SCALE2 = ABS( C( K ) ) + ABS( A( K+1 ) )
IF( K.LT.( N-1 ) )
$ SCALE2 = SCALE2 + ABS( B( K+1 ) )
IF( A( K ).EQ.ZERO ) THEN
PIV1 = ZERO
ELSE
PIV1 = ABS( A( K ) ) / SCALE1
END IF
IF( C( K ).EQ.ZERO ) THEN
IN( K ) = 0
PIV2 = ZERO
SCALE1 = SCALE2
IF( K.LT.( N-1 ) )
$ D( K ) = ZERO
ELSE
PIV2 = ABS( C( K ) ) / SCALE2
IF( PIV2.LE.PIV1 ) THEN
IN( K ) = 0
SCALE1 = SCALE2
C( K ) = C( K ) / A( K )
A( K+1 ) = A( K+1 ) - C( K )*B( K )
IF( K.LT.( N-1 ) )
$ D( K ) = ZERO
ELSE
IN( K ) = 1
MULT = A( K ) / C( K )
A( K ) = C( K )
TEMP = A( K+1 )
A( K+1 ) = B( K ) - MULT*TEMP
IF( K.LT.( N-1 ) ) THEN
D( K ) = B( K+1 )
B( K+1 ) = -MULT*D( K )
END IF
B( K ) = TEMP
C( K ) = MULT
END IF
END IF
IF( ( MAX( PIV1, PIV2 ).LE.TL ) .AND. ( IN( N ).EQ.0 ) )
$ IN( N ) = K
10 CONTINUE
IF( ( ABS( A( N ) ).LE.SCALE1*TL ) .AND. ( IN( N ).EQ.0 ) )
$ IN( N ) = N
*
RETURN
*
* End of SLAGTF
*
END
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