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

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Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals $$I^{(1)}[g]=\int ^b_a \frac{g(x)}{x-t}\,dx$$ I ( 1 ) [ g ] = ? a b g ( x ) x - t d x and $$I^{(2)}[g]=\int ^b_a \frac{g(x)}{(x-t)^2}\,dx$$ I ( 2 ) [ g ] = ? a b g ( x ) ( x - t ) 2 d x . These integrals are not defined in the regular sense; $$I^{(1)}[g]$$ I ( 1 ) [ g ] is defined in the sense of Cauchy Principal Value while $$I^{(2)}[g]$$ I ( 2 ) [ g ] is defined in the sense of Hadamard Finite Part. With $$h=(b-a)/n, \,n=1,2,\ldots $$ h = ( b - a ) / n , n = 1 , 2 , ? , and $$t=a+kh$$ t = a + k h for some $$k\in \{1,\ldots ,n-1\}, \,t$$ k ? { 1 , ? , n - 1 } , t being fixed, the numerical quadrature formulas $${Q}^{(1)}_n[g]$$ Q n ( 1 ) [ g ] for $$I^{(1)}[g]$$ I ( 1 ) [ g ] and $$Q^{(2)}_n[g]$$ Q n ( 2 ) [ g ] for $$I^{(2)}[g]$$ I ( 2 ) [ g ] are $$\begin{aligned} {Q}^{(1)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2),\quad f(x)=\frac{g(x)}{x-t}, \end{aligned}$$ Q n ( 1 ) [ g ] = h ? j = 1 n f ( a + j h - h / 2 ) , f ( x ) = g ( x ) x - t , and $$\begin{aligned} Q^{(2)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2)-\pi ^2g(t)h^{-1},\quad f(x)=\frac{g(x)}{(x-t)^2}. \end{aligned}$$ Q n ( 2 ) [ g ] = h ? j = 1 n f ( a + j h - h / 2 ) - ? 2 g ( t ) h - 1 , f ( x ) = g ( x ) ( x - t ) 2 . We provided a complete analysis of the errors in these formulas under the assumption that $$g\in C^\infty [a,b]$$ g ? C ? [ a , b ] . We actually show that $$\begin{aligned} I^{(k)}[g]-{Q}^{(k)}_n[g]\sim \sum ^\infty _{i=1} c^{(k)}_ih^{2i}\quad \text {as}\,n \rightarrow \infty , \end{aligned}$$ I ( k ) [ g ] - Q n ( k ) [ g ] ~ ? i = 1 ? c i ( k ) h 2 i as n ? ? , the constants $$c^{(k)}_i$$ c i ( k ) being independent of $$h$$ h . In this work, we apply the Richardson extrapolation to $${Q}^{(k)}_n[g]$$ Q n ( k ) [ g ] to obtain approximations of very high accuracy to $$I^{(k)}[g]$$ 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 $$g(x)$$ 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.