New Fast Arctangent Approximation Algorithm for Generic Real-Time Embedded Applications

Fast and accurate arctangent approximations are used in several contemporary applications, including embedded systems, signal processing, radar, and power systems. Three main approximation techniques are well-established in the literature, varying in their accuracy and resource utilization levels. Those are the iterative coordinate rotational digital computer (CORDIC), the lookup tables (LUTs)-based, and the rational formulae techniques. This paper presents a novel technique that combines the advantages of both rational formulae and LUT approximation methods. The new algorithm exploits the pseudo-linear region around the tangent function zero point to estimate a reduced input arctangent through a modified rational approximation before referring this estimate to its original value using miniature LUTs. A new 2nd order rational approximation formula is introduced for the first time in this work and benchmarked against existing alternatives as it improves the new algorithm performance. The eZDSP-F28335 platform has been used for practical implementation and results validation of the proposed technique. The contributions of this work are summarized as follows: (1) introducing a new approximation algorithm with high precision and application-based flexibility; (2) introducing a new rational approximation formula that outperforms literature alternatives with the algorithm at higher accuracy requirement; and (3) presenting a practical evaluation index for rational approximations in the literature.

The Newton product of polynomial projectors Part 1: Construction and algebraic properties

We propose a new way of combining two polynomial projectors on spaces of functions of few variables to obtain a polynomial projector on a space of functions of many variables. We present various algebraic properties of our construction and study the main approximation properties of the new projectors.

Atomic and Molecular Structure

This chapter assesses some applications drawn from atomic and molecular structure. It deals with the structures of the lighter atoms and the simplest molecule, molecular hydrogen. The main approximation method used here is the energy variational principle, which is a powerful technique for computing the low-lying energies of a system such as an atom or molecule. The chapter then introduces the Pauli exclusion principle, which governs the symmetry of the state vector for a system of identical particles such as electrons. Two general features of the exclusion principle are worth noting. First, although the spins make only a very weak contribution to the Hamiltonians for helium, the lowest energy state with spin one is above the spin zero ground state, which is a considerable difference. Second, an electron arriving as a cosmic ray particle from a distant galaxy has to have a wave function antisymmetric with respect to the local electrons, even though the new electron has been away from us for a long time.

On the self-propulsion of an -sphere micro-robot

The aim of this paper is to describe the self-propulsion of a micro-robot (or micro-swimmer) consisting of $N$ spheres moving along a fixed line. The spheres are linked to each other by arms with their lengths changing periodically. We use the asymptotic procedure containing the two-timing method and a distinguished limit. We show that self-propulsion velocity appears (in the main approximation) as a linear combination of velocities of all possible triplets of spheres. Velocities and efficiencies of three-, four- and five-sphere swimmers are calculated.

Approximate Implicitization Using Linear Algebra

We consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating-point implementation in computer-aided geometric design (CAGD) systems. We unify the approaches under the names of commonly known polynomial basis functions and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.

Excitation of forbidden lines in gaseous nebulae I. Formulation and calculations for 2pq ions

A formulation is given for electron collisions with ions in configurations s22s22pq and The main approximation is neglect of coupling to other configurations. Hartree-Fock functions are used for the ion states and the complete wave functions are expressed as sums of vector-coupled anti-symmetrized products of ion functions and orbitals for the colliding electron. Variational principles are used to obtain coupled integro-differential equations for the radial functions for the colliding electron, and to correct results obtained from approximate solutions of these equations. All algebraic reductions are carried out without the introduction of subsidiary approximations, and conservation and reciprocity theorems are therefore satisfied exactly. Expressions are tabulated for all algebraic coefficients. Numerical calculations are made in two approximations: in the exact resonance approximation, used only for />-waves, the wave functions are calculated with quadrupole interactions neglected; and in the distorted wave approximation the wave functions are calculated from static central potentials. Variational corrections are calculated and are found to be reasonably small. It is concluded that the final corrected results should agree closely with results which would be obtained from exact solutions of the coupled equations. Collision strengths are calculated for all inelastic collisions in configurations 1 to 5, for at least three different energies, and for values of the residual charge z = 1, 2, 3, 4, 5 and the limit of z-> oo. Results may be interpolated for all members of the iso-electronic sequences. Results for energies such that some channels are open and others closed are obtained by means of extrapolation techniques.

Collision Strengths for Excitation of Forbidden Lines

New calculations of collision strengths are being made for ions in configurations npq, n = 2 and 3 and q = 1–5. Wave functions for the entire system are represented by sums over anti-symmetrized products of ion functions and orbitals for the colliding electron, and sets of coupled integro-differential equations are obtained for the colliding electron radial functions. The main approximation is neglect of coupling to configurations other than npq. Calculations are made using improved forms of the exact resonance and distorted wave approximations. For the 2pq ions it is found that the dominant contributions come from the p-waves; the exact resonance approximation is used for the p-waves and the distorted wave approximation is used for all other partial waves. It is considered that the results obtained should approximate closely to those which would be obtained from full solutions of the coupled equations. For the 3pq ions the dominant contributions come from the p-waves; for these ions the distorted wave approximation is used for all partial waves.

ONE-PARTICLE ENERGY SPECTRUM OF AN ELECTRON SYSTEM IN A PERIODIC FIELD

The energy spectrum of an electron in the vicinity of the Fermi surface of an electron system in a periodic field is determined approximately by applying the generalized self-consistent field method of Cohen. The main approximation involved may be regarded as an extension of the random phase approximation usually applied to the interelectronic interaction of an electron gas, to the case including a periodic field. The effect of the periodic field is treated by a secondorder perturbation.The result is expressed as[Formula: see text]The correction terms are roughly estimated numerically and are shown to be small for a simple model of a metal, metallic hydrogen. A brief discussion of the extension of the approximation is given.