University of Florida/Egm6321/f09.team1.gzc/Mtg12

$L e t f_{n}^{L} (t) b e L a g r a n g e i n t e r p o l a t i o n o f o r d e r n t o f (t) . \Rightarrow f_{n}^{L} (.) \in \underset{s e t o f p o l y o f d e g r e e ⩽ n}{\underset{⏟}{P_{n}}}$

$n = 1 f_{1}^{L} (.) i n t e r p o l a t i o n e x a c t l y$

$P_{1} (x) \to s t r a i g h t l i n e s (l i n e a r f u n c t i o n s) i . e ., e_{1} (P_{1}, t) \equiv 0 \forall t$

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$n = 2 f_{2}^{L} (.) i n t e r p o l a t i o n e x a c t l y p a r a b o l a s (p o l y n o m i a l o f \underset{d e g r e e}{\underset{⏟}{o r d e r}}) 2$

$n = 3 c u b i c p o l y n o m i a l$

$⋮$

$n f_{n}^{L} (.) i n t e r p o l a t i o n e x a t c l y \overset{\in P_{n}}{\overset{⏞}{p o l y n o m i a l o f \underset{a c t u a l l y d e g r e e ⩽ n}{\underset{⏟}{d e g r e e n}}}}$

$i . e ., e_{n} (p_{n}; t) = 0 \forall t$

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$I F f = p \in P_{n} \Rightarrow f^{(n + 1)} \equiv 0$

$(2) p . 11 - 3 \Rightarrow e_{n} (p, t) = 0 \forall t$

$i . e ., f_{n}^{L} (.) i n t e r p o l a t i o n e x a c t l y p \in P_{n}$

$M e t h o d t o c o m p . {w_{i, n}; i = 0, . ., n} g i v e n {x_{i}; i = 0, . ., n} \underline{n o t} n e c e s s a r i l y e q u i d i s t a n t .$

$\underline{j = 0} c o n s i d e r f = p_{j} = p_{0} \in P_{0} c o n s t a n t$

$e_{n} (\underset{= 1}{\underset{⏟}{p_{0}}; t}) = \underset{1}{\underset{⏟}{f (t)}} - f_{n (t)}^{L} = 1 - f_{n}^{L} (t) = 0$

$\Rightarrow \underset{\sum_{i = 0}^{n} l_{i, n} (t) \underset{1}{\underset{⏟}{f (x_{i})}}}{\underset{⏟}{f_{n}^{L} (t)}} = 1 \Rightarrow \int_{a}^{b} (\underset{f_{n}^{L} (t)}{\underset{⏟}{\sum_{i = 0}^{n} l_{i, n} (t) . 1}}) d t = \underset{b - a}{\underset{⏟}{\int_{a}^{b} 1 d t}}$

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$\Rightarrow \underset{w_{i, n}}{\underset{⏟}{(\int_{a}^{b} l_{i, n} (t) d t)}} . 1 = b - a$

$\Rightarrow$

\sum_{i = 0}^{n} w_{i, n} = b - a

$\underline{j = 1} l e t f = p_{j} = p_{1} \in P_{1}$

$c h o o s e f (x) = p_{1} (x) = x \in P_{1}$

$e_{n} (\underset{= x}{\underset{⏟}{p_{1}}}; t) = \underset{t}{\underset{⏟}{f (t)}} - f_{n}^{L} (t) = t - f_{n} (t) = 0$

$\Rightarrow \underset{\sum_{i = 0}^{n} l_{i, n} (t) \underset{x_{i}}{\underset{⏟}{f (x_{i})}}}{\underset{⏟}{f_{n}^{L} (t)}} = t \Rightarrow \int_{a}^{b} \underset{f_{n}^{L} (t)}{\underset{⏟}{(\sum_{i = 0}^{n} l_{i, n} (t) . x_{i})} d t} = \underset{(b^{2} - a^{2}) / 2}{\underset{⏟}{\int_{a}^{b} t d t}}$

$\Rightarrow$

\sum_{i = 0}^{n} w_{i, n} x_{i} = (b^{2} - a^{2}) / 2

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University of Florida/Egm6321/f09.team1.gzc/Mtg12

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