gibson:teaching:fall-2016:math753:quadrature

Quadrature is the numerical approximation of definite integrals of functions evaluated at discrete gridpoints.

There are many, many quadrature formulae. We will start with the Trapezoid Rule and Simpson's Rule.

In both cases we will assume we have an evenly spaced set of gridpoints on the interval , with , , and , where is the gridspacing. We also assume we know the value of the function evaluated at the gridpoints, i.e. we have a vector of values .

The Trapezoid rule approximates the definite integral of over using piecewise linear interpolation between each pair of datapoints.

for some where .

For we would have

Here we have collapsed the error term into .

Simpson's Rule pproximates the definite integral of over using piecewise quadratic interpolation between triplets of datapoints.

for some where , and where is the fourth derivative of . Simpson's rule requires that is even, so that the total number of gridpoints, , is odd.

For example, for and gridpoints, we have

- Trapezoidal rule (wikipedia)
- Trapezoidal rule (math world)
- Simpson's rule (wikipedia)
- Simpson's rule (math world)
- Newton-Cotes formulae (wikipedia)
- Newton-Cotes formulae (math world)
- Trapezoidal rule (math world)

gibson/teaching/fall-2016/math753/quadrature.txt · Last modified: 2016/11/17 07:04 by gibson