Abstract
Applying the aliasing asymptotics on the coefficients of the Chebyshev expansions, the convergence rate of Clenshaw–Curtis quadrature for Jacobi weights is presented for functions with algebraic endpoint singularities. Based upon a new constructed symmetric Jacobi weight, the optimal error bound is derived for this kind of function. In particular, in this case, the Clenshaw–Curtis quadrature for a new constructed Jacobi weight is exponentially convergent. Numerical examples illustrate the theoretical results.
Highlights
In 1960, whose formulae is usually derived from polynomial interpolation by a finite sum n
In the case the function with algebraic endpoint singularities is known f ( x ) = (1 − x )γ (1 + x )δ g( x ) where γ, δ > 0, and g( x ) is analytic in Bernstein ellipse including [−1, 1], we will see that the Clenshaw-Curtis quadrature is exponentially convergent from a new constructed Jacobi weight function
We confirm the convergence rate of Clenshaw–Curtis quadrature for Jacobi weights and compare it with Gauss–Jacobi quadrature All the numerical results are computed in an Apple laptop with 1.6 GHz Intel Core i5 and 4 GB 1600 MHz DDR3
Summary
During the past few decades, the convergence rates of Gauss–Legendre quadrature for integrals with singularities at one or both endpoints have received considerable attention [14,15,16,17,18,19,20]. We study the convergence rate of the Clenshaw-Curtis quadrature rule for I [ f ] for functions with algebraic endpoint singularities. In the case the function with algebraic endpoint singularities is known f ( x ) = (1 − x )γ (1 + x )δ g( x ) where γ, δ > 0, and g( x ) is analytic in Bernstein ellipse including [−1, 1], we will see that the Clenshaw-Curtis quadrature is exponentially convergent from a new constructed Jacobi weight function
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