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StoneTemplePython

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your second question seems like an excellent introduction to Vandermonde Matrices. I wouldn't recommend solving this 5x5 matrix by hand. With some cleverness you can get it down to a 4x4 matrix but still a lot of work to do by hand. A good computer exercise. It would be worth learning to derive the determinant formula for Vandermonde Matrix though.The other question I have is about deriving a polynomial to approximate a function. The example given is the function sin((pi * x^2)/ 2). The textbook says you can't integrate that (I don't remember that much calculus), so it shows finding a polynomial that approximates it and integrating that instead.

The text shows finding: f(0)=0, f(.25)=.098017, f(.5)=.382683, f(.75)=.77301, f(1)=1

Then it somehow uses linear algebra (matrices representing a system of equations or something?) to find the approximating polynomial p(x)=.098796x + .762356x^2 + 2.14429x^3 - 2.00544x^4

Could someone show how to do this?

(You can see its effect in your first question but sketching the equation and solving it the 'regular way' seems preferable for intuition I think.)

The big idea is that a degree ##n## single variable polynomial (that isn't identically zero) is completely specified by ##n+1## distinct data points.