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

José A O Matos,Paulo B Vasconcelos

doi:10.3390/mca26020042

José A O Matos, Paulo B Vasconcelos

Open Access

https://doi.org/10.3390/mca26020042

Copy DOI

Abstract

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.

Highlights

Two of the main goals when implementing numerical algorithms are correctness and speed—that is, to have the results with the required precision and as fast as possible
There is a cost in computing performance when using higher precision numerical types, because software-based solutions are being used, e.g., studies indicated that quadruple precision can be up to four orders of magnitude slower than double precision
Depending on the type of the problem being studied, the higher precision only needs to be applied at selected precision bottlenecks and, only paying the speed penalty for precision where strictly required. This already happens in libraries, such as the C math library where some of the calculations for the functions are done internally in extended double precision or higher with the results of the calculations being returned in double precision

Summary

Introduction

Two of the main goals when implementing numerical algorithms are correctness and speed—that is, to have the results with the required precision and as fast as possible. Depending on the type of the problem being studied, the higher precision only needs to be applied at selected precision bottlenecks and, only paying the speed penalty for precision where strictly required This already happens in libraries, such as the C math library (like the GNU libc) where some of the calculations for the functions are done internally in extended double precision or higher with the results of the calculations being returned in double precision. In comparison to our approach, accuracy is improved only when needed, not affecting the overall computational effort as much For this approach, the Infinity Computer is an adequate computational environment where this arithmetic can be implemented [8], and in [9], some work on the solution of initial value problems by Taylor-series-based methods have already been made within this setup. Experimental results on the computation of approximate solutions of differential problems via spectral methods will be exposed with recourse to multiprecision arithmetic via the variable-precision arithmetic (arbitrary-precision arithmetic) freely available in MATLAB and Octave

The Tau Spectral Method

Numerical Experiments

Example 1

Example 2

Example 3

Conclusions