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# pset_3 - 2.29 Numerical Fluid Mechanics Problem Set 3...

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2.29: Numerical Fluid Mechanics Problem Set 3 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.29 NUMERICAL FLUID MECHANICS— SPRING 2007 Problem Set 3 Totally 120 points Posted 04/03/07, due Thursday 4 p.m. 04/19/07, Focused on Lecture 8 to 17 Problem 3.1 (15 points): Consider the following system of equations: Ax = b , A = 1 2 ! 1 2 8 0 ! 1 0 4 " # \$ \$ \$ % & , b = 0 8 4 " # \$ \$ \$ % & a) Cholesky factorize A (Note that A is positive definite). b) Find an LU factorization form for A. c) Use LU factorization of A to find x. d) Compute the x ! = 1.5 by two iterations of successive over-relaxation scheme. Use relaxation parameter and initial guess of zero. e) Compute the solution by 4 iterations of conjugate gradient method. Problem 3.2 (10 points): Polynomial Interpolation Consider the below (x,y) pairs: x = ! 2 0 1 2 " # \$ \$ \$ \$ % & , y = f ( x ) = 2 0 1 ! 2 " # \$ \$ \$ \$ % & a) Find the Lagrange polynomial for above points. b) Interpolate that polynomial at x=-1.

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