User:Yongtao-Li/HW2.10

=Problem 2.10: "Proof of Simpson's Rule(simple)"=

Given
In the interval $$ \left[ {a,b} \right] $$ there are three nodes $$ {x_0} = a, {x_1} = \frac{2},  x{_2} = b $$ satisfy that

Find
Use (10.1) to find expression for $$ \left\{ \right\} $$ in terms $$ \left( {{x_i},f({x_i})} \right) $$.

Proof
From (10.1) we can have three equations for 3 unknown $$ \left\{ \right\} $$

$$ {c_2}{x_0}^2 + {c_1}{x_0} + {c_0} = f\left( \right) $$

$$ {c_2}{x_1}^2 + {c_1}{x_1} + {c_0} = f\left( \right) $$

$$ {c_2}{x_2}^2 + {c_1}{x_2} + {c_0} = f\left( \right) $$

And then we write them in matrix form:

where

$$ \left[ K \right] = \left[ {\begin{array}{ccccccccccccccc} 1&&{x_0^2} \\ 1&&{x_1^2} \\ 1&&{x_2^2} \end{array}} \right] $$

Because the determine of the matrix K is

$$ \det \left[ K \right] = - \left( {{x_0} - {x_1}} \right)\left( {{x_0} - {x_2}} \right)\left( {{x_1} - {x_2}} \right) $$

we then compute the inverse of matrix K is

$${\left[ K \right]^{ - 1}} = \left[ {\begin{array}{ccccccccccccccc} {\frac}&{ - \frac}&{\frac} \\ { - \frac}&{\frac}&{ - \frac} \\ {\frac{1}}&{ - \frac{1}}&{\frac{1}} \end{array}} \right]\; $$

Then we can get $$ \left\{ \right\} $$ by multiple $$  {\left[ K \right]^{ - 1}} $$ on both sides of (10.2):

Consider that $$ {x_0} = a, {x_1} = \frac{2},  x{_2} = b $$ we can rewrite $$ \left\{ \right\} $$:

If integrate (10,1) on $$ \left[ {a,b} \right] $$, then

(10.9) is exactly Simpson's Rule for numerical integration.