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Revision of Mapped Quadrature from Mon, 20/07/2009 - 07:40

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.

 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))

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.

 I-Q_n = \frac{1}{2\pi i} \int_C G(w) f(g(w)) g'(w) dw

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 1

Simple x=e^u, x=\sqrt{u}

Example 3

Trapezoidal Rule - Sinc Stenger http://www.cs.utah.edu/~stenger/

Example 4

Trapezoidal Rule - Double Exponential Mori

http://mathworld.wolfram.com/DoubleExponentialIntegration.html

Example 6

Rational Basis Functions

Boyd

His Book

Example 7

Elliott and Johnson

x=a+\sin(u-b)...

Gauss Legendre

http://anziamj.austms.org.au/V40/E006/home.html

General discussion