Non-Computability and Intractability: Does It Matter to Physics?

Should the impossibility results of theoretical computer science be of concern to physics

Associated Polynomials and Uniform Methods for the Solution of Linear Problems

To every polynomial P of degree n we associate a sequence of n-1 polynomials of increasing degree which we call the associated polynomials of P. The associated polynomials depend in a particularly simple way on the coefficients of P. These polynomials have appeared in many guises in the literature, usually related to some particular application and most often going unrecognized. They have been called Horner polynomials and Laguerre polynomials. Often what occurs is not an associated polynomial itself but a number which is an associated polynomial evaluated at a zero of P. The properties of associated polynomials have never been investigated in themselves. We shall try to demonstrate that associated polynomials provide a useful unifying concept. Although many of the results of this paper are new, we shall also present known results in our framework

Construction of Globally Convergent Iteration Functions for the Solution of Polynomial Equations

Iteration functions for the approximation of zeros of a polynomial P are usually given as explicit functions of P and its derivatives. We introduce a class of iteration functions which are themselves constructed according to a certain algorithm given below. The construction of the iteration functions requires only simple polynomial manipulation which may be performed on a computer

Variational Calculations of the 2³S State of Helium

With a 12-parameter Hylleraas-type wave function containing only positive powers, a new calculation has been carried out for the 2³S state of helium by the Ritz variational principle. The energy was minimized by a descent process. A nonrelativistic energy of -1.0876088 Hylleraas units was reached as compared with the best previously published value of -1.0876015 Hylleraas units from a 6-parameter function. When masspolarization and a2Ry corrections are included, the 12-parameter function gives an ionization potential 2.52 cm-1 less than the experimental value of 38 454.64 cm-1. The electron density at the nucleus is also calculated and compared with the experimental hyperfine-spectrum value. All numerical work was carried out on an I.B.M. 650 computer

Computational Complexity of Iterative Processes

The theory of optimal algorithmic processes is part of computational complexity. This paper deals with analytic computational complexity. The relation between the goodness of an iteration algorithm and its new function evaluation and memory requirements are analyzed. A new conjecture is stated

Variational Calculations of Energy and Fine Structure for the 2³P State of Helium

ERV accurate calculations of the energies of the 1s2 1S and 1s2s3S states of helium have been made recently. It is now possible to consider the contributions of relativistic and electrodynamic effects to the observed ionization energies in a meaningful way. These small effects must still be added to the large calculated nonrelativistic energy, however, before a comparison with experiment can be made. In the 3P states, the spin-dependent part of the theoretical electromagnetic interaction of the two electrons can be compared directly with observed fine-structure splittings. Recent improvements in the experimental accuracy of the fine-structure measurements, 45 particularly the direct observation of the splittings as radio-frequency transitions, have made such a comparison possible to an order of accuracy including higher order electrodynamic corrections to the usual fine structure formulas

09391 Abstracts Collection -- Algorithms and Complexity for Continuous Problems

From 20.09.09 to 25.09.09, the Dagstuhl Seminar 09391 Algorithms and Complexity for Continuous Problems was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available