57 research outputs found
Accurate polynomial root-finding methods for symmetric tridiagonal matrix eigenproblems
In this paper we consider the application of polynomial root-finding methods to the
solution of the tridiagonal matrix eigenproblem. All considered solvers are based on evaluating the Newton correction. We show that the use of scaled three-term recurrence relations complemented with error free transformations yields some compensated schemes which significantly improve the accuracy of computed results at a modest increase in computational cost. Numerical experiments illustrate that under some restriction on the conditioning the novel iterations can approximate and/or refine the eigenvalues of a tridiagonal matrix with high relative accuracy
The Brown-Colbourn conjecture on zeros of reliability polynomials is false
We give counterexamples to the Brown-Colbourn conjecture on reliability
polynomials, in both its univariate and multivariate forms. The multivariate
Brown-Colbourn conjecture is false already for the complete graph K_4. The
univariate Brown-Colbourn conjecture is false for certain simple planar graphs
obtained from K_4 by parallel and series expansion of edges. We show, in fact,
that a graph has the multivariate Brown-Colbourn property if and only if it is
series-parallel.Comment: LaTeX2e, 17 pages. Version 2 makes a few small improvements in the
exposition. To appear in Journal of Combinatorial Theory
Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems
In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of
arbitrarily high order of accuracy are introduced in closed form. The stability
domain of RKG polynomials extends in the the real direction with the square of
polynomial degree, and in the imaginary direction as an increasing function of
Gegenbauer parameter. Consequently, the polynomials are naturally suited to the
construction of high order stabilized Runge-Kutta (SRK) explicit methods for
systems of PDEs of mixed hyperbolic-parabolic type.
We present SRK methods composed of ordered forward Euler stages, with
complex-valued stepsizes derived from the roots of RKG stability polynomials of
degree . Internal stability is maintained at large stage number through an
ordering algorithm which limits internal amplification factors to .
Test results for mildly stiff nonlinear advection-diffusion-reaction problems
with moderate () mesh P\'eclet numbers are provided at second,
fourth, and sixth orders, with nonlinear reaction terms treated by complex
splitting techniques above second order.Comment: 20 pages, 7 figures, 3 table
Quasiperiodic patterns of the complex dimensions of nonlattice self-similar strings, via the LLL algorithm
The Lattice String Approximation algorithm (or LSA algorithm) of M. L.
Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex
dimensions of a nonlattice self-similar fractal string by the complex
dimensions of a lattice self-similar fractal string. The implication of this
procedure is that the set of complex dimensions of a nonlattice string has a
quasiperiodic pattern. Using the LSA algorithm, together with the
multiprecision polynomial solver MPSolve which is due to D. A. Bini, G.
Fiorentino and L. Robol, we give a new and significantly more powerful
presentation of the quasiperiodic patterns of the sets of complex dimensions of
nonlattice self-similar fractal strings. The implementation of this algorithm
requires a practical method for generating simultaneous Diophantine
approximations, which in some cases we can accomplish by the continued fraction
process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use
the LLL algorithm of A. K. Lenstra, H. W. Lenstra, and L. Lov\'asz.Comment: 38 pages, 11 figures, 7 table
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