Numerical Analysis Candidacy Test by rma97348

VIEWS: 7 PAGES: 8

									          Numerical Analysis Candidacy Test


                                       Fall 2004




Name: .............................................................


    In the following you must show your work.
   You should have worked out 7 problems when you are done: one from
each of section 1, 2, 3, and 4, and three from section 4.




                                             1
1     Interpolation
Do Problem 1 or Problem 2.

Problem 1 Let p(x) denote the polynomial of degree ≤ 4 with p(0) = 0,
p(π/2) = 1, p(π) = 0, p(3π/2) = −1, and p(2π) = 0. Prove that we get an
error bound
                                               π          π
                                 (x + π) x +   2
                                                   x x−   2
                                                              (x − π)
             |p(x) − sin(x)| ≤
                                                   5!
for x ∈ [0, 2π].




Problem 2 Find the Newton form of the unique polynomial p(x) of degree
≤ 5 such that
                           p(2) = 3
                           p (2) = 13
                           p(1) = 2
                           p (1) = 0
                           p(0) = 1
                           p (0) = 1.




                                       2
2     Solving Nonlinear Equations
Do Problem 3 or 4.

Problem 3 You want to solve x2 − 2 = 0 on the interval [1, 2].

    1. Why does the bisection method apply on [1, 2].

    2. Find the approximate solution using two iterations of the bisection
       method on[1, 2].

    3. What upperbound for the error is given by the bisection method?




Problem 4 You want to solve x2 − 2 = 0 on the interval [1, 2].

    1. Find the approximate solution using two iterations of Newton’s method
       starting at x = 2.

    2. Explain what quadratic convergence means and give a result making pre-
       cise the statement “Newton’s method is quadratically convergent under
       good circumstances.”




                                      3
3    Singular value decomposition
Do either Problem 5 or Problem 6.

Problem 5 a) Give a statement of the singular value decomposition for a
square matrix.
   b) What is the singular value decomposition of
                                           
                     1                      1
            B := 5  2  1 1 1 +  1  1 2 −3 .
                     3                   −1




Problem 6 a) Give a statement of the singular value decomposition for a
square matrix.
   b) What are the singular values of

                                    1 0
                                          .
                                    1 1




                                     4
4      Norms and Eigenvalues
Do Problem 7 or Problem 8.

Problem 7 Let
                                         a b
                                 A :=           .
                                         c d
                                                               v
Regard A as the map from R2 to R2 which sends x :=                  to v :=
                                                               w
   av + bw
           . Let the norm ||x||∞ on both of the R2 s be the ∞-norm, i.e.,
   cv + dw
||x||∞ = max{|v|, |w|}.

    1. Give the definition of the associated matrix norm ||A||∞ for A.

    2. prove that ||A||∞ = max{|a| + |b|, |c| + |d|}.




Problem 8 You have a complex matrix,
                                           
                            a1,1 · · · a1,n
                     A= .         .    . .
                          .       .    . 
                             .     .    .
                            an,1 · · · an,n

    1. Explain what the Gerschgorin disks associated to A are and what their
       relation to eigenvalues of A are.

    2. Use this relation to prove that if A is diagonally dominant, i.e., if
       |ai,i | >  |ai,j | for all i, then A is invertible.
               j=i




                                        5
5      Miscellaneous
Do three of the following problems

Problem 9 Consider a real valued 3 times differentiable function on an in-
terval (0, 2). Using Taylor’s theorem we have

                  f (1 + h) − f (1)                h
                                    = f (1) + f (1) + O(h2 )            (1)
                          h                        2
for all sufficiently small h. This gives the approximation

                                  f (1 + h) − f (1)
                        f (1) =                     + O(h)
                                          h
for all sufficiently small h. Use Richardson extrapolation on Eq.1 to find an
O(h2 ) approximation to f (1).




Problem 10 Perform the following computations in a) 3 digit rounding
arithmetic; and compute b) the absolute error, and c) the relative error in
each case:

    1. (13.4 − 0.04) − 12.8;

    2. (13.4 − 12.8) − 0.04.

Put your answers in the table below.


     expression to       3 digit rounding answer absolute error relative error
     be evaluated             (1 point each)     (1 point each) (1 point each)
 (13.4 − 0.04) − 12.8
 (13.4 − 12.8) − 0.04



                                         6
Problem 11 Consider the following initial value problem:

                        y = −y 2 ; 0 ≤ x ≤ 1; y(0) = 1.

Use Taylor’s method of order two to approximate the solution at

  1. with h = 0.5; and

  2. with h = 0.25;
                                                                       1
  3. The actual solution of the above differential equation is y(x) =      .
                                                                    1+x
      Compute the absolute error of the approximate solution at x = 0.5 for
      the approximations with h = 0.5 and h = 0.25.




Problem 12 Let S(x) be a free cubic spline defined on [−1, 1] with nodes
x0 = −1, x1 = 0, x2 = 1, and S(−1) = 0, S(0) = 1 and S(1) = 2. Assume
that S(x) is given by the polynomial S0 (x) = a(x + 1) + b(x + 1)3 on [−1, 0]
and by the polynomial S1 (x) = 1+cx+3dx2 −dx3 on [0, 1]. Find the relations
that the real numbers a, b, c, d satisfy, and use them to find a, b, c, d, i.e., find
the actual real numbers. [Recall the free splines satisfy the conditions that
S0 (−1) = 0 = S1 (1).]




                                        7
Problem 13 Prove:
Neville’s Recursion Formula: Let x0 , . . . , xn be distinct real numbers in
the interval, [a, b]. Let f be a real valued function on [a, b], and let P (x)
denote the unique polynomial of degree ≤ n such that f (xi ) = P (xi ) for
i = 0, . . . , n. For each c = 0, . . . , n, let Pb(x) denote the unique polynomial
                                                  c
of degree ≤ n − 1 such that Pb(xi ) = f (xi ) for 0 ≤ i ≤ n with i = c. Then
                                  c
for two distinct points xj and xk in the set {x0 , . . . , xn }:

                              (x − xj )Pb(x) − (x − xk )Pb (x)
                                        j                k
                    P (x) =
                                          xk − xj

or equivalently:
                              x − xj          x − xk
                    P (x) =           Pb(x) +
                                       j             Pb (x).
                              xk − xj         xj − xk k




                                         8

								
To top