tsturm.f
SUBROUTINE TSTURM (NM, N, EPS1, D, E, E2, LB, UB, MM, M, W, Z,
+ IERR, RV1, RV2, RV3, RV4, RV5, RV6)
C***BEGIN PROLOGUE TSTURM
C***PURPOSE Find those eigenvalues of a symmetric tridiagonal matrix
C in a given interval and their associated eigenvectors by
C Sturm sequencing.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A5, D4C2A
C***TYPE SINGLE PRECISION (TSTURM-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine finds those eigenvalues of a TRIDIAGONAL
C SYMMETRIC matrix which lie in a specified interval and their
C associated eigenvectors, using bisection and inverse iteration.
C
C On Input
C
C NM must be set to the row dimension of the two-dimensional
C array parameter, Z, as declared in the calling program
C dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrix. N is an INTEGER variable.
C N must be less than or equal to NM.
C
C EPS1 is an absolute error tolerance for the computed eigen-
C values. It should be chosen so that the accuracy of these
C eigenvalues is commensurate with relative perturbations of
C the order of the relative machine precision in the matrix
C elements. If the input EPS1 is non-positive, it is reset
C for each submatrix to a default value, namely, minus the
C product of the relative machine precision and the 1-norm of
C the submatrix. EPS1 is a REAL variable.
C
C D contains the diagonal elements of the symmetric tridiagonal
C matrix. D is a one-dimensional REAL array, dimensioned D(N).
C
C E contains the subdiagonal elements of the symmetric
C tridiagonal matrix in its last N-1 positions. E(1) is
C arbitrary. E is a one-dimensional REAL array, dimensioned
C 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. MM is an INTEGER variable.
C WARNING - If more than MM eigenvalues are determined to lie
C in the interval, an error return is made with no values or
C vectors found.
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 as
C negligible, have been replaced by zero causing the matrix to
C split into a direct sum of submatrices. E2(1) is also set
C 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 if the matrix
C does not split. If the matrix splits, the eigenvalues are
C in ascending order for each submatrix. If a vector error
C exit is made, W contains those values already found. W is a
C one-dimensional REAL array, dimensioned W(MM).
C
C Z contains the associated set of orthonormal eigenvectors.
C If an error exit is made, Z contains those vectors already
C found. Z is a one-dimensional REAL array, dimensioned
C Z(NM,MM).
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C 3*N+1 if M exceeds MM no eigenvalues or eigenvectors
C are computed,
C 4*N+J if the eigenvector corresponding to the J-th
C eigenvalue fails to converge in 5 iterations, then
C the eigenvalues and eigenvectors in W and Z should
C be correct for indices 1, 2, ..., J-1.
C
C RV1, RV2, RV3, RV4, RV5, and RV6 are temporary storage arrays,
C dimensioned RV1(N), RV2(N), RV3(N), RV4(N), RV5(N), and
C RV6(N).
C
C The ALGOL procedure STURMCNT contained in TRISTURM
C appears in TSTURM in-line.
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 890531 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 TSTURM