52,895 research outputs found
Practical implementation and error bounds of integer-type general algorithm for higher order differential equations
In our preceding paper, we have proposed an algorithm for obtaining
finite-norm solutions of higher-order linear ordinary differential equations of
the Fuchsian type [\sum_m p_m (x) (d/dx)^m] f(x) = 0 (where p_m is a polynomial
with rational-number-valued coefficients), by using only the four arithmetical
operations on integers, and we proved its validity. For any nonnegative integer
k, it is guaranteed mathematically that this method can produce all the
solutions satisfying \int |f(x)|^2 (x^2+1)^k dx < \infty, under some
conditions. We materialize this algorithm in practical procedures. An
interger-type quasi-orthogonalization used there can suppress the explosion of
calculations. Moreover, we give an upper limit of the errors. We also give some
results of numerical experiments and compare them with the corresponding exact
analytical solutions, which show that the proposed algorithm is successful in
yielding solutions with high accuracy (using only arithmetical operations on
integers).Comment: Comparison with existing method is adde
Numerical techniques for lattice QCD in the --regime
In lattice QCD it is possible, in principle, to determine the parameters in
the effective chiral lagrangian (including weak interaction couplings) by
performing numerical simulations in the --regime, i.e. at quark
masses where the physical extent of the lattice is much smaller than the
Compton wave length of the pion. The use of a formulation of the lattice theory
that preserves chiral symmetry is attractive in this context, but the numerical
implementation of any such approach requires special care in this kinematical
situation due to the presence of some very low eigenvalues of the Dirac
operator. We discuss a set of techniques (low-mode preconditioning and
adapted-precision algorithms in particular) that make such computations
numerically safe and more efficient by a large factor.Comment: Plain TeX source, 32 pages, figures include
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