 Solution

ID: past-exam-of-the-numerical-analysis-course-of-the-university-of-oxford/2014/2/d/solution

p_{2} (x) = β x

: the constant and quadratic coefficients vanish. The remaining normal equation is

0 = \int_{- 1}^{1} (x^{1969} - β x) x d x = \frac{2}{1971} - β \frac{2}{3} .

Therefore

p_{2} (x) = \frac{3}{1971} x = \frac{x}{657} .

The residual is automatically orthogonal to

1

and

x^{2}

by parity and to

x

by construction, so it is orthogonal to every element of

Π_{2}

. This checks the full linear least squares condition, rather than only a necessary coefficient equation.

 New to topics? Read the docs here!