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homework-04 - H OMEWORK 4 120202 ESM4A N UMERICAL M ETHODS...

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Unformatted text preview: H OMEWORK 4 120202: ESM4A - N UMERICAL M ETHODS Spring 2009 Prof. Dr. Lars Linsen Zymantas Darbenas School of Engineering and Science Jacobs University Due: Friday, March 13, 2009 at noon (in the mailbox labeled "Linsen" in the entrance hall of Research I). Problem 10: System of Nonlinear Equations. Solve the system of nonlinear equations x2 + 2y 2 x + 5y = 10 = -2 (8 points) by applying two iterations of the generalized Newton's method with starting point (2, -1). Problem 11: Polynomial Interpolation. Given knots ui and points pi , i = 0, . . . , 4, by ui pi 0 3 1 1 2 4 4 5 8 1 (12 points) (a) Use polynomial interpolation with respect to the monomial basis to determine a polynomial p(u) of minimum degree that interpolates the points at the knots. (b) Use Lagrange interpolation to determine a polynomial p(u) of minimum degree that interpolates the points at the knots. Problem 12: Polynomial Interpolation. (10 points) (a) Given a knot sequence u0 < u1 < . . . < un and points p0 , . . . , pn . Show that there is a unique polynomial p(u) of degree n such that p(ui ) = pi for i = 0, . . . , n. (b) Show that the Lagrange polynomials have the property of partition of unity, i.e., show that n Ln (u) 1 for all u R. i i=0 ...
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