Polar Affine Arithmetic: Optimal Affine Approximation and Operation Development for Computation in Polar Form Under Uncertainty

Uncertainties practically arise from numerous factors, such as ambiguous information, inaccurate model, and environment disturbance. Interval arithmetic has emerged to solve problems with uncertain parameters, especially in the computational process where only the upper and lower bounds of parameters can be ascertained. In rectangular coordinate systems, the basic interval operations and improved interval algorithms have been developed in the numerical analysis. However, in polar coordinate systems, interval arithmetic still suffers from issues of complex computation and overestimation. This article defines a polar affine variable and develops a polar affine arithmetic (PAA) that extends affine arithmetic to the polar coordinate systems, which performs better in many aspects than the corresponding polar interval arithmetic (PIA). Basic arithmetic operations are developed based on the complex affine arithmetic. The Chebyshev approximation theory and the min-range approximation theory are used to identify the best affine approximation. PAA can accurately keep track of the interdependency among multiple variables throughout the calculation procedure, which prominently reduces the solution conservativeness. Numerical examples implemented in MATLAB programs show that, compared with benchmark results from the Monte Carlo method, the proposed PAA ensures completeness of the exact solution and presents a more compact solution region than PIA when dependency exists in the calculation process. Meanwhile, a comparison of affine arithmetic in polar and rectangular coordinates is presented. An application of PAA in circuit analysis is quantitatively presented and potential applications in other research fields involving complex variables in polar form will be gradually developed.