Tanh-sinh Quadrature
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High Quality Content by WIKIPEDIA articles! Tanh-sinh quadrature is a method for numerical integration introduced by Hidetosi Takahasi and Masatake Mori in 1974. It uses the change of variables x = tanh(tfrac12 pi sinh t), to transform an integral on ( 1, 1) to an integral on the entire real line. After this transformation, the integrand decays with a double exponential rate, and thus, this method is also known as the double exponential (DE) formula. For a given step size h, the integral is approximated by the sum int_{-1}^1 f(x) , dx approx sum_{k=-infty}^infty w_k f(x_k). with the abscissas ...
High Quality Content by WIKIPEDIA articles! Tanh-sinh quadrature is a method for numerical integration introduced by Hidetosi Takahasi and Masatake Mori in 1974. It uses the change of variables x = tanh(tfrac12 pi sinh t), to transform an integral on ( 1, 1) to an integral on the entire real line. After this transformation, the integrand decays with a double exponential rate, and thus, this method is also known as the double exponential (DE) formula. For a given step size h, the integral is approximated by the sum int_{-1}^1 f(x) , dx approx sum_{k=-infty}^infty w_k f(x_k). with the abscissas x_k = tanh(tfrac12 pi sinh kh) and the weights w_k = frac{tfrac12 h pi cosh kh}{cosh^2(tfrac12 pi sinh kh)}.
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