a repository of mathematical know-how

Revision of Mapped Quadrature from Mon, 20/07/2009 - 06:13

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_n = \sum_{i=1}^n w_{i,n} f(g(u_{i,n}))

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

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.

Riemann Surfaces

Integral Representation of Error Donaldson and Elliott

Residues and Steepest Descents

Riemann-Hilbert Problems

Percy Deift

Example 1

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

Example 2

Chebyshev x=\cos(\theta)

Example 3

Trapezoidal Rule - Sinc, Double Exponential Stenger, Mori

Example 4

Sigmoidal Elliott

Example 5

Rational Basis Functions Boyd

Example 6

Elliott and Johnson x=a+\sin(u-b)... Gauss Legendre

General discussion