quadrature_variety

quadrature_variety, covers a variety of topics involving the estimating of integrals.

The notes:

• quadrature_variety.pdf
• quadrature_variety.tex

Scripts and functions:

• circle_mc.m, circle_mc(nmax) estimates an integral over the unit circle using the Monte Carlo method with nmax points.
• clenshaw_curtis_ab.m, clenshaw_curtis_ab ( n, a, b ) returns a Clenshaw-Curtis rule for [a,b] using n points.
• clenshaw_curtis_m1p1.m, clenshaw_curtis_m1p1 ( n ) returns a Clenshaw-Curtis rule for [-1,+1] using n points.
• gauss_ab.m, gauss_ab ( n, a, b ) returns a Gauss rule for [a,b] using n points.
• gauss_m1p1.m, gauss_m1p1 ( n ) returns a Gauss rule for [-1,+1] using n points.
• hat.m, hat ( x1, x2, x3, x ) evaluates at x the hat function defined by nodes x1, x2, x3.
• hat_integral.m, hat_integral ( n ) estimates the integral of phi_i(x) * f(x) where phi_i(x) is a hat function associated with the i-th node of a mesh of [a,b].
• hump.m, hump(x) evaluates the hump() function.
• hump_int.m, hump_int(a,b) returns the exact integral of the hump() function over [a,b].
• hump_mc.m, hump_mc(n) estimates the integral of the hump() function over [0,2] using an n-point Monte Carlo method.
• hump_simpson.m, hump_simpson applies a sequence of composite Simpson rules to estimate the integral of the hump() function over [0,2].
• quadrature_playoff.m, quadrature_playoff compares the performance of several quadrature rules on a single problem.
• rectangle_gauss.m, rectangle_gauss(n1,n2) uses an n1 x n2 product Gauss rule to estimate an integral over a rectangle.
• rectangle_integral2.m, rectangle_integral2( ) uses MATLAB's integral2() function to estimate an integral over a rectangle.
• sinexp.m, a 2D test integrand sinexp(x,y)=sin(x)*exp(x+y).

Images:

• chebyshev_points.png, illustrates how the chebyshev points are defined.
• maple_leaf.png, suggests how the area of an irregular region can be estimated by a Monte Carlo procedure.