Variable Precision Arithmetic Research Articles

A Matlab package computing simultaneous Gaussian quadrature rules for multiple orthogonal polynomials

The aim of this paper is to describe a Matlab package for computing the simultaneous Gaussian quadrature rules associated with a variety of multiple orthogonal polynomials.Multiple orthogonal polynomials can be considered as a generalization of classical orthogonal polynomials, satisfying orthogonality constraints with respect to r different measures, with r≥1. Moreover, they satisfy (r+2)-term recurrence relations. In this manuscript, without loss of generality, r is considered equal to 2. The so-called simultaneous Gaussian quadrature rules associated with multiple orthogonal polynomials can be computed by solving a banded lower Hessenberg eigenvalue problem. Unfortunately, computing the eigendecomposition of such a matrix turns out to be strongly ill-conditioned and the Matlab function balance.m does not improve the condition of the eigenvalue problem. Therefore, most procedures for computing simultaneous Gaussian quadrature rules are implemented with variable precision arithmetic. Here, we propose a Matlab package that allows to reliably compute the simultaneous Gaussian quadrature rules in floating point arithmetic. It makes use of a variant of a new balancing procedure, recently developed by the authors of the present manuscript, that drastically reduces the condition of the Hessenberg eigenvalue problem.

The Numerical Assembly Technique for arbitrary planar beam structures based on an improved homogeneous solution

AbstractThe Numerical Assembly Technique (NAT) is extended to investigate arbitrary planar frame structures with a focus on the computation of natural frequencies. This allows us to obtain highly accurate results without resorting to spatial discretization. To this end, we systematically introduce a frame structure as a set of nodes, beams, bearings, springs, and external loads and formulate the corresponding boundary and coupling conditions. As the underlying homogeneous solution of the governing equations, we use a novel approach recently presented in the literature. This greatly improves the numerical stability and allows the stable computation of very high natural frequencies accurately. We show this improvement by investigating the condition number of the system matrix and also by the use of variable precision arithmetic.

An Accurate and Numerically Stable Formulation for Computing the Electromagnetic Fields in Uniform Bend Rectangular Waveguides

This article describes a numerically stable formulation for the analysis of electromagnetic fields in rectangular cross section waveguides with a curved longitudinal axis. A novel set of scaled Hankel functions for real-valued arguments and complex-valued orders is introduced for rescaling the characteristic equations associated with the transverse electric (TE) and magnetic (TM) fields of the exact boundary value problem. The exponentially scaled cylindrical functions presented here prevent numerical underflow and overflow errors associated with large real and large imaginary orders without sacrificing accuracy. The proposed methodology is validated against variable-precision arithmetic (VPA) results. Numerical results are also presented for waveguides with large radii of curvature, where the present methodology is compared with perturbation ones in several examples. Dielectric-filled waveguides with small radii of curvature are investigated, and our solutions are compared with finite-integration technique (FIT) results.

A Reconfigurable Posit Tensor Unit with Variable-Precision Arithmetic and Automatic Data Streaming

A Reconfigurable Posit Tensor Unit with Variable-Precision Arithmetic and Automatic Data Streaming

Representation and evaluation of the Arrhenius and general temperature integrals by special functions

The non-isothermal analysis of reactions with a constant heating rate involves a temperature integral. This integral is popularized as the Arrhenius integral, but when the pre-exponential factor of the Arrhenius equation depends on temperature with a power-law relationship, the integral is known as the general temperature integral. In this paper, it is shown that when the activation energy and exponent of power-law assumed to be constant, these integrals can be expressed and evaluated by several kinds of special functions and using commercial or free softwares. The special functions include exponential integral function, incomplete gamma function, confluent hypergeometric function, Whittaker function and generalized hypergeometric function. New rational approximations can be obtained by expansion of the special functions. MATLAB and GNU Octave were used to compute the special function models as well as numerical integration. The accuracy of these methods was examined by variable precision arithmetic. For both softwares, the special function models showed higher accuracies compared to the numerical integration methods and the approximation functions. In addition, new iterative isoconversional method for constant activation energy and exponent of power-law has been proposed that shows very high accuracy in calculation of the activation energy.

Effectiveness of Floating-Point Precision on the Numerical Approximation by Spectral Methods

With the fast advances in computational sciences, there is a need for more accurate computations, especially in large-scale solutions of differential problems and long-term simulations. Amid the many numerical approaches to solving differential problems, including both local and global methods, spectral methods can offer greater accuracy. The downside is that spectral methods often require high-order polynomial approximations, which brings numerical instability issues to the problem resolution. In particular, large condition numbers associated with the large operational matrices, prevent stable algorithms from working within machine precision. Software-based solutions that implement arbitrary precision arithmetic are available and should be explored to obtain higher accuracy when needed, even with the higher computing time cost associated. In this work, experimental results on the computation of approximate solutions of differential problems via spectral methods are detailed with recourse to quadruple precision arithmetic. Variable precision arithmetic was used in Tau Toolbox, a mathematical software package to solve integro-differential problems via the spectral Tau method.

Design and implementation of multiple-precision arithmetic environment in MATLAB for reliable numerical computations

In the present study a new multiple-precision arithmetic environment in MATLAB is developed based on exflib, a fast multiple-precision arithmetic in the programming language C, C++ and FORTRAN. We discuss design and implementation of interface between MATLAB and exflib without any modifications to exflib. Although the proposed design incurs the overhead in execution, it is more accurate and faster than Variable Precision Arithmetic (VPA), which is the official multiple-precision arithmetic environment in MATLAB. We also exhibit an efficiency of the proposed environment for reliable computation to overcome the difficulties caused by rounding errors.

Numerical Solution for the Extrapolation Problem of Analytic Functions

In this work, a numerical solution for the extrapolation problem of a discrete set of n values of an unknown analytic function is developed. The proposed method is based on a novel numerical scheme for the rapid calculation of higher order derivatives, exhibiting high accuracy, with error magnitude of O(10−100) or less. A variety of integrated radial basis functions are utilized for the solution, as well as variable precision arithmetic for the calculations. Multiple alterations in the function's direction, with no curvature or periodicity information specified, are efficiently foreseen. Interestingly, the proposed procedure can be extended in multiple dimensions. The attained extrapolation spans are greater than two times the given domain length. The significance of the approximation errors is comprehensively analyzed and reported, for 5832 test cases.

Three effective inverse Laplace transform algorithms for computing time-domain electromagnetic responses

The inverse Laplace transform is one of the methods used to obtain time-domain electromagnetic (EM) responses in geophysics. The Gaver-Stehfest algorithm has so far been the most popular technique to compute the Laplace transform in the context of transient electromagnetics. However, the accuracy of the Gaver-Stehfest algorithm, even when using double-precision arithmetic, is relatively low at late times due to round-off errors. To overcome this issue, we have applied variable-precision arithmetic in the MATLAB computing environment to an implementation of the Gaver-Stehfest algorithm. This approach has proved to be effective in terms of improving accuracy, but it is computationally expensive. In addition, the Gaver-Stehfest algorithm is significantly problem dependent. Therefore, we have turned our attention to two other algorithms for computing inverse Laplace transforms, namely, the Euler and Talbot algorithms. Using as examples the responses for central-loop, fixed-loop, and horizontal electric dipole sources for homogeneous and layered mediums, these two algorithms, implemented using normal double-precision arithmetic, have been shown to provide more accurate results and to be less problem dependent than the standard Gaver-Stehfest algorithm. Furthermore, they have the capacity for yielding more accurate time-domain responses than the cosine and sine transforms for which the frequency-domain responses are obtained by interpolation between a limited number of explicitly computed frequency-domain responses. In addition, the Euler and Talbot algorithms have the potential of requiring fewer Laplace- or frequency-domain function evaluations than do the other transform methods commonly used to compute time-domain EM responses, and thus of providing a more efficient option.

Polynomials orthogonal with respect to exponential integrals

Moment-based methods and related Matlab software are provided for generating orthogonal polynomials and associated Gaussian quadrature rules having as weight function the exponential integral E? of arbitrary positive order ? supported on the positive real line or on a finite interval [0,c], c>0. By using the symbolic capabilities of Matlab, allowing for variable-precision arithmetic, the codes provided can be used to obtain as many of the recurrence coefficients for the orthogonal polynomials as desired, to any given accuracy, by choosing d-digit arithmetic with d large enough to compensate for the underlying ill-conditioning.

High-accuracy algorithms for solving the discrete-time periodic Riccati equation

The paper proposes an algorithm for solving the discrete-time periodic Riccati equation, which arises, for instance, in designing a stabilization system for a hopping robot. This algorithm does not have a number of constraints on the matrix coefficients peculiar to other algorithms. It is significant that variable-precision arithmetic can be used in numerical implementation of the algorithm. This makes it possible to solve discrete-time periodic Riccati equations with high accuracy. Some examples are given

A Micropower Programmable DSP Using Approximate Signal Processing Based on Distributed Arithmetic

A recent trend in low-power design has been the employment of reduced precision processing methods for decreasing arithmetic activity and average power dissipation. Such designs can trade off power and arithmetic precision as system requirements change. This work explores the potential of distributed arithmetic (DA) computation structures for low-power precision-on-demand computation. We present an ultralow-power DSP which uses variable precision arithmetic, low-voltage circuits, and conditional clocks to implement a biomedical detection and classification algorithm using only 560 nW. Low energy consumption enables self-powered operation using ambient mechanical vibrations, converted to electric energy by a MEMS transducer and accompanying power electronics. The MEMS energy scavenging system is estimated to deliver 4.3 to 5.6 /spl mu/W of power to the DSP load.

Software for high radix on-line arithmetic

High radix on-line arithmetic provides an efficient method for performing variable-precision arithmetic. It can be implemented on conventional microprocessors using sequences of three operand instructions. This paper presents software support for high radix on-line arithmetic. This software includes emulation modules for high radix operations, and a precision analysis program for setting the input and intermediate variable tolerances necessary for guaranteed result accuracy over a specified domain.

Iterative refinement for linear systems in variable-precision arithmetic

We define a binary-cascade iterative-refinement process (BCIR) for solving a nonsingular linear system with prescribed relative accuracy. For very high accuracy requirements BCIR uses a software-implemented variable-precision arithmetic. The time-cost and accuracy of BCIR are deduced under various conditions.

Computational aspects of deciding if all roots of a polynomial lie within the unit circle

In this paper we discuss the computational aspects of two algorithms due to E. I. Jury for determining if all the zeros of a polynomial with integer coefficients lie within the unit circle. We show that Jury's original algorithm asymptotically requires an exponential amount of computing time when variable-precision arithmetic is employed. We show that his modified algorithm requires only a polynomially bounded amount of computing time when variable-precision arithmetic is employed. Finally we produce a congruence arithmetic algorithm analogous to Jury's modified algorithm which requires less computing time than Jury's modified algorithm.

Application of MATLAB Symbolic Maths with Variable Precision Arithmetic (VPA) to Compute Some High Order Gauss Legendre Quadrature Rules

Gauss Legendre Quadrature rules are extremely accurate and they should be considered seriously when many integrals of similar nature are to be evaluated. This paper is concerned with the derivation and computation of numerical integration rules for the three integrals: (See text for formulae) which are dependent on the zeros and the squares of the zeros of Legendre Polynomial and is quite well known in the Gaussian Quadrature theory. We have developed the necessary MATLAB programs based on symbolic maths which can compute the sampling points and the weight coefficients and are reported here upto 32 – digits accuracy and we believe that they are reported to this accuracy for the first time. The MATLAB programs appended here are based on symbolic maths. They are very sophisticated and they can compute Quadrature rules of high order, whereas one of the recent MATLAB program appearing in reference [21] can compute Gauss Legendre Quadrature rules upto order twenty, because the zeros of Legendre polynomials cannot be computed to desired accuracy by MATLAB routine roots (……..). Whereas we have used the MATLAB routine solve (……..) to find zeros of polynomials which is very efficient. This is worth noting in the present context. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 117-125 DOI: http://dx.doi.org/10.3329/ganit.v29i0.8521