bisect.f
SUBROUTINE BISECT (N, EPS1, D, E, E2, LB, UB, MM, M, W, IND, IERR,
+ RV4, RV5)
C***BEGIN PROLOGUE BISECT
C***PURPOSE Compute the eigenvalues of a symmetric tridiagonal matrix
C in a given interval using Sturm sequencing.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A5, D4C2A
C***TYPE SINGLE PRECISION (BISECT-S)
C***KEYWORDS EIGENVALUES, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the bisection technique
C in the ALGOL procedure TRISTURM by Peters and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 418-439(1971).
C
C This subroutine finds those eigenvalues of a TRIDIAGONAL
C SYMMETRIC matrix which lie in a specified interval,
C using bisection.
C
C On INPUT
C
C N is the order of the matrix. N is an INTEGER variable.
C
C EPS1 is an absolute error tolerance for the computed
C eigenvalues. If the input EPS1 is non-positive,
C it is reset for each submatrix to a default value,
C namely, minus the product of the relative machine
C precision and the 1-norm of the submatrix.
C EPS1 is a REAL variable.
C
C D contains the diagonal elements of the input matrix.
C D is a one-dimensional REAL array, dimensioned D(N).
C
C E contains the subdiagonal elements of the input matrix
C in its last N-1 positions. E(1) is arbitrary.
C E is a one-dimensional REAL array, dimensioned E(N).
C
C E2 contains the squares of the corresponding elements of E.
C E2(1) is arbitrary. E2 is a one-dimensional REAL array,
C dimensioned E2(N).
C
C LB and UB define the interval to be searched for eigenvalues.
C If LB is not less than UB, no eigenvalues will be found.
C LB and UB are REAL variables.
C
C MM should be set to an upper bound for the number of
C eigenvalues in the interval. WARNING - If more than
C MM eigenvalues are determined to lie in the interval,
C an error return is made with no eigenvalues found.
C MM is an INTEGER variable.
C
C On OUTPUT
C
C EPS1 is unaltered unless it has been reset to its
C (last) default value.
C
C D and E are unaltered.
C
C Elements of E2, corresponding to elements of E regarded
C as negligible, have been replaced by zero causing the
C matrix to split into a direct sum of submatrices.
C E2(1) is also set to zero.
C
C M is the number of eigenvalues determined to lie in (LB,UB).
C M is an INTEGER variable.
C
C W contains the M eigenvalues in ascending order.
C W is a one-dimensional REAL array, dimensioned W(MM).
C
C IND contains in its first M positions the submatrix indices
C associated with the corresponding eigenvalues in W --
C 1 for eigenvalues belonging to the first submatrix from
C the top, 2 for those belonging to the second submatrix, etc.
C IND is an one-dimensional INTEGER array, dimensioned IND(MM).
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C 3*N+1 if M exceeds MM. In this case, M contains the
C number of eigenvalues determined to lie in
C (LB,UB).
C
C RV4 and RV5 are one-dimensional REAL arrays used for temporary
C storage, dimensioned RV4(N) and RV5(N).
C
C The ALGOL procedure STURMCNT contained in TRISTURM
C appears in BISECT in-line.
C
C Note that subroutine TQL1 or IMTQL1 is generally faster than
C BISECT, if more than N/4 eigenvalues are to be found.
C
C Questions and comments should be directed to B. S. Garbow,
C Applied Mathematics Division, ARGONNE NATIONAL LABORATORY
C ------------------------------------------------------------------
C
C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C system Routines - EISPACK Guide, Springer-Verlag,
C 1976.
C***ROUTINES CALLED R1MACH
C***REVISION HISTORY (YYMMDD)
C 760101 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE BISECT