Numerical Analysis Prelim Exam

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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.