6A (Numerical Analysis)

Exactness for

1

and

x

gives

a_{0} + a_{1} = \frac{1}{2}, a_{0} x_{0} + a_{1} x_{1} = \frac{1}{3} .

Hence

a_{0} = \frac{x _{1} /2 - 1/3}{x _{1} - x _{0}}, a_{1} = \frac{1/3 - x _{0} /2}{x _{1} - x _{0}} .

b

A two-node Gaussian rule is exact through degree three, the maximal degree

2 n - 1

. Its nodes are the zeros of the monic quadratic orthogonal to

1, x

x

[0, 1]

p = x^{2} + ux + v

gives

\int_{0}^{1} p x d x = 0, \int_{0}^{1} p x^{2} d x = 0,

u = - 6/5

v = 3/10

. Therefore

x_{0} = \frac{3}{5} - \frac{6}{10}, x_{1} = \frac{3}{5} + \frac{6}{10} .

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