Connections of accuracy of least‐square polynomial approximation with degree of polynomials

Abstract

AbstractThe least‐square approximation, the minimax approximation, the approximation by partial sum of series expansion and its modifications, are generally used methods of functional approximation, and each has proper advantage for each use. In the case of the least‐square polynomial approximation, the accuracy is expected intuitively to increase as the degree of polynomial is increased, and then reach a ceiling. However, actually, a decrease in accuracy occurs at a certain degree of polynomial when it is run from low to high. As one of its causes, the effect of ill‐condition of simultaneous linear equations in the power series approximation is well known. However, there are phenomena of decrease in accuracy which cannot be explained by the foregoing. To elucidate their causes, this paper shows that the decrease in accuracy as the degree of polynomial increases also occurs in the approximation by orthogonal polynomial in which results are obtained without solving simultaneous linear equations. Also, the differences of the two representative approximations are shown quantitatively by numerical examples. Moreover, the analysis of the cause and experiments on predicting theoretical approximation accuracy by simple formulas are reported.