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Mapped Quadrature
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[QUICK DESCRIPTION] Many quadrature rules are derived from others by first making a change-of-variable in the integral and then applying an existing quadrature rule. [maths] I=\int_a^b f(x) \, dx =\int_{g(a)}^{g(b)} f(g(u)) g'(u) \, du \approx Q = \sum w_i f(g(u_i)) [/maths] Quadrature error can be represented by a contour integral over a curve enclosing the interval. This error can be estimated using residues around poles, integrals along branch cuts, or saddle points. [maths] I-Q_n = \frac{1}{2\pi i} \int_C G(w) f(g(w)) g'(w) dw [/maths] where $C$ encloses $(g(a),g(b))$ and $G(w)$ has poles at the quadrature points $u_i$. My idea is to unify the existing cases by defining a Riemann surface by the change-of-variable and then the error is represented by an integral over the Riemann surface. [Prerequisites] *Riemann Surfaces *Integral Representation of Error - eg Donaldson and Elliott *Residues and Steepest Descents *Riemann-Hilbert Problems - Percy Deift [EXAMPLE] Simple $x=e^u$, $x=\sqrt{u}$ [EXAMPLE] Chebyshev $x=\cos(\theta)$ [[http://mathworld.wolfram.com/Chebyshev-GaussQuadrature.html]] [EXAMPLE] Trapezoidal Rule - Sinc Stenger [[http://www.cs.utah.edu/~stenger/]] [EXAMPLE] Trapezoidal Rule - Double Exponential Mori [[http://mathworld.wolfram.com/DoubleExponentialIntegration.html]] [EXAMPLE] Sigmoidal Elliott [[http://anziamj.austms.org.au/V46/E/Elliott/home.html]] [EXAMPLE] Rational Basis Functions Boyd [[http://portal.acm.org/citation.cfm?id=33099]] [EXAMPLE] Elliott and Johnson $x=a+\sin(u-b)...$ Gauss Legendre [[http://anziamj.austms.org.au/V40/E006/home.html]] [EXAMPLE] My PhD [[http://arxiv.org/abs/math/0512347]] [GENERAL DISCUSSION]
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