Direct Method for Finding Formulas

Direct method for finding formulas

Pre requisites: Calculus, College level Algebra, Linear Algebra, and Analytic Geometry, or at least some basic Trigonometry.

This next technique similarly demonstrates yet another derivation of the Simpson formula, but also goes beyond classical interpolation in that one can find through defining equations substituted into a given formula, solutions to a given function.

So this goes as follows. Let an a formula be written as

You may notice this being similar to the Simpson formula except having undetermined coefficients a, b, and c. What we do next is set up a series of linear equations by substituting three different defining equations and solving both sides with the given values, x = -1, 0, 1 as indicated above.

The defining equations are:

If we solve for the coefficients through substitution here we end up with the Simpson's formula yet again.

Demonstrating this in application.

Finding the formula for

Where the undefined coefficients are w_-1, w₀, and w₁.

We construct the defining equations similarly as in above, but the results are different.

So If you'd forgotten your integration techniques here, I'll provide a brief refresher here in terms of techniques. In this case we'll us a well known integration by parts method.

For Equation (4) we have using substitutions 

 

For equation (5) to evaluate the left hand side of the equation we use, integration by parts with substitutions.

 

For equation (6) we use integration by parts as we did before above,

Substituting eq.s (8) into (4), (10) into (5), and (13) into (6), the defining equations can be written.

 

Now we have a set of linear equations that can be solved via substitution (Gaussian elimination) or other preferred method.

Using substitutions, we find

Thus the given formula looks as follows:

This will eventually lead into the another technique which is doing much the same, except using sample points as parameters.   Basically, rather then using x = -1, 0, and 1 as above, we define different sample points in determining the formula for such function.   I might also mention alongside this Gauss Quadrature Integration which is merely an abstraction of the method of determining formulas from sample points as parameters method...just applied in the nth case.


In case you are wondering where all the readings were taken from I'd cite credit here to Numerical methods for Scientists and Engineers, R.W. Hamming, Dover

Just recent past post is also a credit to topics considered here.  Much of this could be considered supplementary to his writings.