February 20, 2017

Homework Help: math

Posted by mathstudent on Friday, January 18, 2008 at 12:13am.

Find the least squares approximation of x over the interval [0,1] by a polynomial of the form a + b*e^x

The polynomial produces an output space with two linearly independent basis vectors: u1 = 1, u2 = e^x

I believe these are the steps to solve the problem

1) Select a valid inner product that makes steps 2-3 simple.
2) Compute two orthonormal basis vectors (g1, g2) from the linearly independent basis vectors (u1, u2)
3) Calculate the projection of f(x) = x onto the output space of the polynomial represented by (g1, g2) by

<x, g1>*g1 + <x, g2>*g2

I'm not sure I'm picking a good inner product, because the numbers aren't very clean.

I pick for an inner product
<f,g> = Definite integral over [0,1] of f(x)*g(x) dx

Via Gram-Schmidt:
Linearly Independent Vector u1 = 1
Orthogonal Vector v1 = 1
Orthonormal vector g1 = 1

Linearly Independent Vector u2 = e^x
Orthogonal Vector v2 = e^x - e + 1
Orthonormal vector g2 = (e^x - e + 1)/sqrt(1/2*(e-5)*(e+1)

Projection of f(x)=x onto {g1,g2}=
<f,g1>*g1 + <f,g2>*g2
= 1/2 + 1/4(e^x-e-1)/(e-5)
Which can be rewritten in the a + b*e^x form as:
(1/2 - (e+1)/(e-5)) + 1/(4(e-5)) * e^x

This answer isn't close to the book answer:
-1/2 + 1/(e-1) * e^x

Where did I go wrong?

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