Construction of conformal maps based on the locations of singularities for improving the double exponential formula
For researchers in numerical integration, this work incrementally improves upon existing methods for handling singularities in DE formulas.
The paper proposes a new transformation formula for the double exponential (DE) integration formula that is less sensitive to singularities near the real axis, generalizing DE transformations via explicit Schwarz-Christoffel maps, and demonstrates effectiveness through numerical experiments.
The double exponential formula, or the DE formula, is a high-precision integration formula using a change of variables called a DE transformation; whereas there is a disadvantage that it is sensitive to singularities of an integrand near the real axis. To overcome this disadvantage, Slevinsky and Olver (SIAM J. Sci. Comput., 2015) attempted to improve it by constructing conformal maps based on the locations of singularities. Based on their ideas, we construct a new transformation formula. Our method employs special types of the Schwarz-Christoffel transformations for which we can derive their explicit form. Then, the new transformation formula can be regarded as a generalization of the DE transformations. We confirm its effectiveness by numerical experiments.