Numerical Integration

[Namir]

Hi All,

I posted an article (as a pdf file) on my web site. You can download it if you click here.

The article is a result of some basic investigation to enhance the Trapezoidal method by using additional points inside each integration interval. My new approach produces some improvements over the basic Trapezoidal rule using weights with points inside the interval. Interestingly, the new method does very well when these weights match those of Simpson's rule, Simpson's 3/8 rule, and other similar methods. The article explains how you can mix the Trapezoidal method with better methods when the scheme of the latter are applied to each integration interval in an isolated or compartmentalized manner. The article shows the source code for VBA functions that implement the various algorithms covered by the article.

Namir

[Allen]

Thank you for the interesting article! You are the quintessential quadrature expert.

[Namir]

Glad you enjoyed the article. I was just asking some basic questions about numerical integration.

Namir

[Namir]

I updated the article to correct a mistake and clarify a few things.

Namir

[Mike Morrow]

How does it compare to Gauss-Kronrod quadrature? The implementation of G-K quadrature on the fx-115/991ES seems to give much better and faster results than the quadrature methods of the various HP calculators, including the HP 50g and my beloved HP 42S.

[Namir]

The Gauss-Kronrod quadrature is a better method.

[Allen]

I wonder if you have considered using something other than the monomial basis for your objective function for your numeric integration??

For example, whether using uniform weighting or individual weights, you can use the method of undetermined coefficients to solve the abscissa locations. For example to find a three point, equal-weighted solution on the following range:

With 4 unknowns, you can just use this equation:

to find +/-inv(sqrt(2)), and 0 are the nodes, with the weight w=2/3.

Any number of weights or points can be calculated this way, but this only uses the monomial basis. With a different (perhaps higher-order?) basis function, Could it be more accurate??

Edited: 22 Nov 2011, 10:07 p.m.

[Namir]

I really have not seen any numerical integration algorithm with a single weight. Being a pragmatic person, if you can find a method with a constant weight and it works well, that is fine.

Namir

[Allen]

I was just reading about this method last week. The key is using the monomial basis to actually move the nodes around to compensate for the equal weight. I first heard about Chebyshev Quadrature in the book I am reading. It's simpler numerically than Simpson and a few other methods because you save n-multiplications by factoring out the weighting function from the summation. It's interesting to solve, too.

I hope you all have a Merry Thanksgiving!!