Gauß Integration

There are lots of mathematical theories opening deep and interesting worlds for the inclined mathematician. But for the public I prefer those smart little ideas, the non-obvious ones. One of my favorites is the Gauß integration.

It is a matter of routine to apply linear algebra to get a formula for integration of the form

\(\int_a^b f(t) \,dt = \sum_{k=0}^n a_k f(x_k) \)

which is exact for polynomials of degree n. You just have to choose the points x, and derive the coefficients with a linear system.

If you allow the points x to float in the interval, you have suddenly 2n+2 degrees of freedom. So it can be guesses that it might be possible to select points and coefficients so that the integration is exact for polynomials of degree 2n+1 (which have 2n+2 coefficients). To find these points, is a much tougher step.

I could not find out if the method actually goes back to Gauß. The main theorem of the history of math says that nothing is named after its inventor. But in this case, the method is so clever, it could be Gauß’s.

The trick is to use the zeros of the (n+1)-th orthogonal polynomial q for the points x. This will work, since we can divide any polynomial p of degree 2n+1 by q, and get

\(\int_a^b p(t) \,dt = \int_a^b h(t)q(t) \,dt + \int_a^b r(t) \,dt \)

with h and r being polynomials of degree at most n. Since the first term on the right hand side vanishes, we need only a formula exact for the remainder r. Moreover, for the summation we get

\(\sum_k a_k p(x_k) = \sum_k a_k h(x_k)q(x_k) + \sum_k a_k r(x_k) \)

Again the first term on the right vanishes. I consider these both equations a coincidence, which needs a brilliant mind to be observed.

The method can also be implied for integration with a positive weight function, even on infinite intervals. It is a very useful and impressive method. If you need to demonstrate the usefulness of a little bit of math, this is one candidate. The other might be Newton’s method for the solution of equations.

I have

an Euler notebook, which computes the points x and the coefficients a.

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