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128a_su09_hw5

# 128a_su09_hw5 - MATH 128A SUMMER 2009 HOMEWORK 5(1(This is...

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Unformatted text preview: MATH 128A, SUMMER 2009: HOMEWORK 5 (1) (This is a slightly modified version of 4.7.8.) Show that the quadrature rule R b a f ( x ) dx ≈ ∑ n i =0 c i f ( x i ) cannot have degree of precision greater than 2 n + 1, regardless of the choice of c ,...,c n and x ,...,x n . Hint: Suppose the nodes ( x i ) and coefficients ( c i ) are fixed. Construct a polynomial f ( x ) of degree 2( n + 1) such that f ( x i ) = 0 for each i , but f ( x ) > 0 for other values of x . (2) 4.9 : 6–7a [reworded] The Laguerre polynomials { L ( x ) ,L 1 ( x ) ... } form an orthogonal set on [0 , ∞ ) with respect to the inner product h f,g i = Z ∞ e- x f ( x ) g ( x ) dx (i.e., h L i ,L j i = 0 for i 6 = j ). The polynomial L n ( x ) has n distinct zeros x 1 ,...,x n in [0 , ∞ ). Let c n,i = Z ∞ e- x n Y j =1 j 6 = i x- x j x i- x j dx. Show that the quadrature formula Z ∞ e- x f ( x ) dx ≈ n X i =1 c n,i f ( x i ) has degree of precision 2 n- 1. ( Hint : Follow the steps in the proof of Theorem 4.7.): Follow the steps in the proof of Theorem 4....
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128a_su09_hw5 - MATH 128A SUMMER 2009 HOMEWORK 5(1(This is...

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