Prac_Final.pdf - Numerical Analysis Section 42 Final Exam...

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Numerical Analysis Section 42 Final Exam Fall 3077 (the future) Name: This exam is scheduled for 110 minutes. It is closed book and but you are allowed one double sided notecard. No calculator should be necessary. By signing your name, you agree to the NYU honor code. The exam is worth all of the points (lol, jk, it’s just sorta for goofs). 1
MATH-UA.252 1 Basic understanding questions [XX pts] Answer the following statements with true or false (no justification necessary), or provide a short answer for non true/false questions. 1 Let I = R b a f ( x ) dx , I n be the result of the composite trapezoid rule approximation to R b a f ( x ) dx with n + 1 points quadrature points, and e n = | I - I n | . For n large, what is e n / e 2 n ? 2 True or false: Let p n be the Lagrange interpolant to a function f with n + 1 interpolation points, and e n ( x ) = | p n ( x ) - f ( x ) | . The interpolation error k e n k always gets arbitrarily small for large n , i.e., k e n k 0 as n . 3 If λ is an eigenvalue of A , then | λ - a i , i | ≤ || A || for some i , where 1 i n . 4 True or false: Let 1 x 0 x 2 0 ... x n 0 κ 2 ( V ) >> 1 for large n . (Things to think about: where does this matrix come up in terms of polynomial fitting and what does it being ill conditioned mean? What is the solution to the matrix being ill-conditioned?) 9 Give the result obtained with one step of the Forward (explicit)
MATH-UA.252 (for additional space)
MATH-UA.252 2 [ODEs, 3+7 points] You are given the Initial Value Problem (IVP) y 0 ( x ) = ( x - 1 ) y ( x ) , y ( 0 ) = 2 . you are given the following two methods y n + 1 = y n + Δ t f ( x n , y n ) y n + 1 = y n + Δ t 2 ( f ( x n , y n )+ f ( x n + 1 , y n + 1 )) (a) Verify that y ( x ) = 2 e x 2 ( x - 2 ) satisfies the IVP. (b) State whether each method is implicit or explicit and compute the order of accuracy.

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