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Unformatted text preview: x ) = x ln x . Fix the two nodes x = 1 and x 1 = 3. (So n = 1.) (a) Use the linear interpolant and the Hermite interpolant to approximate the value of f (1 . 5). Which estimate is more accurate? (b) Use the error estimate theorems on pages 348 and 410 to obtain rigorous bounds on the error in each interpolant at x = 1 . 5. 4. (Other types of interpolation) Many types of interpolants can be computed by solving linear systems. This problem explores two examples. (a) Find a polynomial of least degree that satisﬁes the following ﬁve conditions: p (1) =1 p (1) = 2 p 00 (1) = 0 p (2) = 1 p (2) =2 . (b) A trigonometric polynomial has the form p ( x ) = a + N X j =1 [ a j cos( jx ) + b j sin( jx )] . The coeﬃcients { a j } and { b j } are arbitrary real numbers. Interpolate the function f ( x ) = ex sin x at the points x = 0 , 1 ,..., 10 using a trigonometric polynomial with N = 5. 2...
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 Winter '08
 LinZhi
 Polynomials, Numerical Analysis, Hermite interpolant, chebyshev polynomials, polynomial interpolant pn

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