User:Egm6322.s12.team2.Xia/RP12.2

= R 12.2 - Alternative formula for weights of GL quad =

Given
Using the following fomula

$$\displaystyle (1-x^{2})P_{n}^{'}=(n+1)xP_{n}-(n+1)P_{n+1} $$ (12.2.1)

Find
Show that

$$\displaystyle (n+1)P_{n+1}(x_{j})=-(1-x_{j}^{2})P_{n}^{'}(x_{j})$$ (12.2.2) and

$$\displaystyle w_{j}=\frac{2}{(1-x_{j}^{2})[P_{n}^{'}(x_{j})]^{2}}$$ (12.2.3)

Solution
From the 12.2.1, we have

$$\displaystyle (1-x_{j}^{2})P_{n}^{'}(x_{j})=(n+1)x_{j}P_{n}(x_{j})-(n+1)P_{n+1}(x_{j})$$ (12.2.4) Since $$\displaystyle P_{n}(x_{j})=0$$,(The proof is @ p.45-17) then

$$\displaystyle (n+1)P_{n+1}(x_{j})=-(1-x_{j}^{2})P_{n}^{'}(x_{j})$$ (12.2.5) In terms of (5)p.45b-6

$$\displaystyle w_{j}=\frac{-2}{(n+1)P_{n+1}(x_{j})P_{n}^{'}(x_{j})}$$ (12.2.6) then substitute 12.2.5 into 12.2.6, we obtain

$$\displaystyle w_{j}=\frac{2}{(1-x_{j}^{2})[P_{n}^{'}(x_{j})]^{2}}$$ (12.2.3)