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SUBROUTINE DLAGTF( N, A, LAMBDA, B, C, TOL, D, IN, INFO )
*
* -- LAPACK routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
INTEGER INFO, N
DOUBLE PRECISION LAMBDA, TOL
* ..
* .. Array Arguments ..
INTEGER IN( * )
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 ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER K
DOUBLE PRECISION EPS, MULT, PIV1, PIV2, SCALE1, SCALE2, TEMP, TL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
CALL XERBLA( 'DLAGTF', -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 = DLAMCH( '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 DLAGTF
*
END
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