Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

Abstract

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals $I^{(1)}[g]=\int ^b_a \frac{g(x)}{x-t}\,dx$ and $I^{(2)}[g]=\int ^b_a \frac{g(x)}{(x-t)^2}\,dx$. These integrals are not defined in the regular sense; $I^{(1)}[g]$ is defined in the sense of Cauchy Principal Value while $I^{(2)}[g]$ is defined in the sense of Hadamard Finite Part. With $h=(b-a)/n, \,n=1,2,\ldots$, and $t=a+kh$ for some $k\in \{1,\ldots ,n-1\}, \,t$ being fixed, the numerical quadrature formulas ${Q}^{(1)}_n[g]$ for $I^{(1)}[g]$ and $Q^{(2)}_n[g]$ for $I^{(2)}[g]$ are

$$\begin{aligned} {Q}^{(1)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2),\quad f(x)=\frac{g(x)}{x-t}, \end{aligned}$$

and

$$\begin{aligned} Q^{(2)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2)-\pi ^2g(t)h^{-1},\quad f(x)=\frac{g(x)}{(x-t)^2}. \end{aligned}$$

We provided a complete analysis of the errors in these formulas under the assumption that $g\in C^\infty [a,b]$. We actually show that

$$\begin{aligned} I^{(k)}[g]-{Q}^{(k)}_n[g]\sim \sum ^\infty _{i=1} c^{(k)}_ih^{2i}\quad \text {as}\,n \rightarrow \infty , \end{aligned}$$

the constants $c^{(k)}_i$ being independent of $h$. In this work, we apply the Richardson extrapolation to ${Q}^{(k)}_n[g]$ to obtain approximations of very high accuracy to $I^{(k)}[g]$. We also give a thorough analysis of convergence and numerical stability (in finite-precision arithmetic) for them. In our study of stability, we show that errors committed when computing the function $g(x)$, which form the main source of errors in the rest of the computation, propagate in a relatively mild fashion into the extrapolation table, and we quantify their rate of propagation. We confirm our conclusions via numerical examples.