# NUMERICAL QUADRATURE: THEORY AND COMPUTATION

Ye, Lingyun (2006) NUMERICAL QUADRATURE: THEORY AND COMPUTATION. Masters thesis, Dalhousie University.

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The double exponential `tanh-sinh' transformation has proved to be a very useful tool in numerical integration. It can often exhibit a remarkable convergence speed even for the integrands with end-point singularities. Under certain circumstances, it can be shown that the tanh-sinh method has a quadratic convergence ratio of $O\left(\exp\left(-CN/\ln N\right)\right)$, where $N$ is the number of abscissas. The tanh-sinh scheme is also well-suited for parallel implementation. These merits make the tanh-sinh scheme the best quadrature rule when high precision results are desired. This thesis surveys the development of the double exponential transformation and its (parallel) implementations in one and two dimensions. We examine the application of the tanh-sinh methods in relation to many other scientific fields. We also provide a proof to the quadratic convergence property of the tanh-sinh method over the Hardy space. This proof can be generalized to analyze the performance of other transformations.