E5.1.
Consider the following quadrature rule for approximating \(\int_{-1}^1 f(x) \mathrm{d} x\): \[ Q(f) = w_0 f(-1)+w_1 f(1) + w_2 f'(-1)+w_3 f'(1). \] Here, the quadrature nodes are \(-1,1\) and \(Q\) evaluates \(f\) and also its derivative \(f'\) at these nodes. Determine the weights \(w_i\) so that \(Q\) has degree of precision 3.
By applying a change of variable, write this quadrature rule over the general interval \([a,b]\).
\(\star\)E5.2.
Consider the 3-point Gauss quadrature rule:
\[Q_{\mathrm{Gauss},2}^{[-1,1]}(f) = \frac{5}{9} f \left(-\sqrt{\frac{3}{5}}\right) + \frac{8}{9} f(0) + \frac{5}{9} f \left(\sqrt{\frac{3}{5}}\right).\]
Integrate the polynomial \[\int_{-1}^{1} 2x^5 + 16x^4 - 3x^2 +9x - 2\,dx\] analytically, and subsequently, show that \(Q_{\mathrm{Gauss},2}^{[-1,1]}\) integrates this exactly.
Compute \(E^{[-1,1]}(x^{6})\) and the degree of precision for this rule.
\(\star\)E5.3. When deriving the Gauss quadrature rules in Section 3.3, we have full freedom of the weights and nodes. Suppose we are required to choose the end points i.e. \(x_0 = -1\), and \(x_N = 1\).
What is the maximum degree of precision achievable using this quadrature rule with \(N+1\) points?
Consider the quadrature rule \[Q_{\mathrm{R},3}^{[-1,1]}(f) = w_0 f(-1) + w_1 f(x_1) + w_2 f(x_2) + w_3 f(1).\] Write down equations which must be satisfied for the unknowns and solve them.
HINT: Use symmetry as in Section 3.3.