NumericalAnalysis-683-2008aug

# NumericalAnalysis-683-2008aug - (c What is the expected...

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MAT 683 Numerical Analysis August, 2008 NAME: 1. For which values of s [0 , 1], will there exist a unique quadratic polynomial p that satisﬁes the following conditions p (0) = p 0 , p (1) = p 1 , p 0 ( s ) = p 2 ? 2. Consider the equation 3 x + g ( x ) = 0 (1) with g : R R and g 0 ( x * ) = 0, where x * is the unique solution of (1). Assume the following iteration is used to obtain the solution x * x n +1 = - g ( x n ) 3 (a) What is the rate of convergence of this iteration? (b) State and prove a local convergence theorem for this iteration? 3. Consider the integration formula Z 1 - 1 f ( x ) dx f ( α 1 ) β + f ( α 2 ) β. (a) Determine α 1 , α 2 , and β so that this formula is exact for all quadratic polynomials. (b) What is the minimal degree polynomial for which the formula with the coeﬃcients derived in (a) is not exact?
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Unformatted text preview: (c) What is the expected order of a composite integration method based upon the formula with coeﬃcients derived in (a)? 4. Let Π n denote the space of all polynomials of degree ≤ n . Given a function f deﬁned on [-1 , 1], we denote dist( f, Π n ) = inf p ∈ Π n k f-p k , with k f-p k = sup x ∈ [-1 , 1] | f ( x )-p ( x ) | . (a) Given f ( x ) = x n +1 , ﬁnd p ∈ Π n such that k f-p k = dist( f, Π n ) . (2) Prove that your p indeed satisﬁes (2). (b) For the same f , compute dist( f, Π n ). (c) With f and p as above, discuss brieﬂy the use of the error function f-p in poly-nomial interpolation. 1...
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