# Numerical Analysis Candidacy Test by rma97348

VIEWS: 7 PAGES: 8

• pg 1
```									          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