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Numerical Analysis Prelim Exam August 23, 2008 1. Consider polynomial interpolation using Newton’s divided diﬀerence formula. (a) Use the Newton divided diﬀerence method to obtain a quadratic polynomial that takes these values: p(0) = 2, p (0) = −9, p(1) = −4, p (1) = 4, p(2) = 44. (b) What condition will have to be placed on the nodes x0 and x1 if the interpolation problem p(xi ) = ci0 , p (xi ) = ci2 , i = 0, 1 it to be solvable by a cubic polynomial (for arbitrarycij )? 2. Consider the iterative method for solving a nonlinear system f (x) = 0, where f : Rn → Rn : xk+1 = xk − Af (xk ), (1) where A is an n × n matrix. (a) Derive necessary and suﬃcient conditions on A such that the iterations deﬁned in (1) is locally convergent to a root of f and determine the rate of convergence. (b) Formulate Newton’s method for root ﬁnding in the above form by allowing A to depend on x. 3. Consider the linear system U = δ2F where U is a K × K matrix in block form A I U = I .. . I , I A 1 2 A and I are two n × n matrices, K = n2 . I is the identity matrix and −4 1 1 −4 1 A= .. .. .. . . . . 1 −4 1 1 −4 (a) If the linear system is solved with LU factorization, how many non-zero elements are in the matrices L and U ? What is the number of operations required to solve the system this way? Does one need to perform any pivoting? (b) Write down the Jacobi iterations scheme for solving the linear system and derive the convergence rate. (c) Do the same as in (b) using Gauss-Seidel iterations. (d) What are the eigenvectors of the matrix? Find an algorithm that solves the problem in O(N 2 log N ) time? Justify your answer. 3 4. The following elliptic problem is approximated by a ﬁnite element method: · (a(x) u(x)) = f (x), x ∈ Ω ⊂ R2 , 0 < a0 ≤ a(x) < a1 u(x) = g(x), x ∈ ∂Ω1 ∂u(x) + u(x) = 0, x ∈ ∂Ω2 ∂x1 ∂u(x) = 0, x ∈ ∂Ω3 ∂x2 Ω = [0, 1]2 , ∂Ω1 = {(x1 , x2 ) : x1 = 0, 0 ≤ x2 ≤ 1}, ∂Ω2 = {(x1 , x2 ) : x1 = 1, 0 ≤ x2 ≤ 1}, ∂Ω3 = {(x1 , x2 ) : 0 ≤ x1 ≤ 1, x2 = 0 or 1}. (a) Determine an appropriate weak formulation of the problem. (b) Prove conditions on the corresponding bilinear form which are needed for existence, uniqueness, and convergence of a ﬁnite element method. (You may assume that Poincaré inequality is valid and may use it in your estimates.) (c) Describe a ﬁnite element method mesh and a set of basis functions such that the linear system from the ﬁnite element approximation is sparse and of band structure. 4 5. Consider linear multistep methods for y = f (y) : p q αj yn+j = βk fn+k . j=0 k=0 (a) Describe Dahlquist’s root condition for the stability for this scheme and use it to determine the stability of the scheme yn+2 + 4yn+1 − 5yn = (4hfn+1 + 2hfn ). ¨ (b) The Störmer-Verlet scheme for q = f (q) is given by qn+1 − 2qn + qn−1 = h2 f (qn ). q = 1 is the repeated root for the corresponding characteristic polynomial. Is the scheme stable? Justify your answer. 6. Consider ut + ux = 0 to be solved in [0, 1) for t ≥ 0 with periodic bounary conditions and smooth initial data. (a) Construct a second order accurate and stable ﬁnite diﬀerence scheme which is of the form un+1 = aun + bun + cun . j j j−1 j−2 (b) Construct an unconditionally stable, convergent, second order spatially accurate scheme of the form aun+1 + bun+1 + cun+1 = dun + eun + f un . j+1 j j−1 j+1 j i−1 Justify your answer.

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posted: | 3/11/2010 |

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