Foundations of Computational Math II Exam 2
In-class Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
Monday April 11, 2011
Question
Points
Points
Possible Awarded
1. Approximation
20
2. Newton Cotes
25
3. Extrapolation
25
4. General

Foundations of Computational Math II Exam 1
In-class Exam
Open Notes, Textbook, Homework Solutions Only
Calculators Allowed
Wednesday February 23, 2011
Question
Points
Points
Possible Awarded
1. Interpolation
40
2. Piecewise Interpolation
30
3. Splines an

MAD 5404 Program 1
Morgan Weiss
Problem 1 Exercises
Let Fn Cnn be the unitary matrix representing the discrete Fourier transform of length n
and so FnH Cnn is the inverse DFT of length n. For example, for n = 4
1 1 1 1
1 1
1
1
2
3
1 1 2 3
and F4H = 1 1

Homework 11 Foundations of Computational Math 2
Spring 2011
Solutions will be posted Wednesday, 4/20/11
Problem 11.1
Consider the Runge Kutta method called the implicit midpoint rule given by:
h
y1 = yn1 + f1
2
h
f1 = f (tn1 + , y1)
2
yn = yn1 + hf1
An al

Sir James Lighthill Distinguished Lectureship
Provost Lawrence G. Abele established the Sir James Lighthill Distinguished Lectureship Award to honor leaders in mathematical sciences
and to attract them to visit the Florida State University to give lecture

Program 1 Foundations of Computational Math 2 Spring
2011
Due date: 11:59 PM Monday, 2/7/11 via email
Problem 1.1
Assume you are given distinct points x0 , . . . , xn , and a function f (x).
1.1.a. Implement a routine that computes the divided dierence ta

Program 2 Foundations of Computational Math 2 Spring
2011
Due date: 11:59 PM Friday, 2/18/11
Problem 2.1
In this exercise you will implement two interpolating spline algorithms as code and demonstrate their use.
Assume you are given distinct points x0 , .

Program 3 Foundations of Computational Math 2 Spring
2011
Due date: 11:59 PM Wednesday, 3/16/11
Problem 3.1
In this exercise you will implement three parametric curve drawing algorithms as code and
investigate their use.
The algorithms are:
parametric in

INTERNATIONAL
FINANCE
18
CHAPTER
Objectives
After studying this chapter, you will be able to
Explain how international trade is financed
Describe a countrys balance of payments accounts
Explain what determines the amount of international
borrowing and

Homework 10 Foundations of Computational Math 2
Spring 2011
Solutions will be posted Monday, 4/11/11
Problem 10.1
Consider the following linear multistep method:
yn = 4yn1 + 5yn2 + h(4fn1 + 2fn2 )
The method is not 0-stable.
10.1.a. Determine, p, the orde

Homework 9 Foundations of Computational Math 2 Spring
2011
Solutions will be posted Friday, 3/25/11
Problem 9.1
In this problem we consider the numerical approximation of the integral
1
I=
f (x)dx
1
with f (x) = ex . In particular, we use a priori error e

Homework 1 Foundations of Computational Math 2 Spring
2011
Solutions will be posted Monday, 1/17/11
Written homework problems are study questions. You need not turn in solutions but you
are strongly encouraged to do the problems and read the posted soluti

Homework 2 Foundations of Computational Math 2 Spring
2011
Solutions will be posted Wednesday, 1/26/10
Written homework problems are study questions. You need not turn in solutions but you
are strongly encouraged to do the problems and read the posted sol

Homework 3 Foundations of Computational Math 2 Spring
2011
Solutions will be posted Wednesday, 2/2/11
Problem 3.1
Show that given a set of points
x0 , x1 , . . . , xn
a Leja ordering can be computed in O (n2 ) operations.
Problem 3.2
Consider a polynomial

Homework 4 Foundations of Computational Math 2 Spring
2011
Solutions will be posted Monday, 2/7/11
Problem 4.1
Suppose we want to approximate a functionf (x) on the interval [a, b] with a piecewise
quadratic interpolating polynomial with a constant spacin

Homework 5 Foundations of Computational Math 2 Spring
2011
Solutions will be posted Monday, 2/14/11
Problem 5.1
Recall that we have derived dierent sets of linear equations for the coecients of an interpolating cubic spline.
Assume that f (x) = x3 and ana

Homework 6 Foundations of Computational Math 2 Spring
2011
Solutions will be posted Monday, 2/28/11
Problem 6.1
Consider a minimax approximation to a function f (x) on [a, b]. Assume that f (x) is continuous with continuous rst and second order derivative

Homework 7 Foundations of Computational Math 2 Spring
2011
Solutions will be posted 3/7/11
Problem 7.1
For this problem, consider the space L2 [1, 1] with inner product and norm
1
(f, g ) =
f (x)g (x)dx and f
2
= (f, f )
1
Let Pi (x), for i = 0, 1, . . .

Homework 8 Foundations of Computational Math 2 Spring
2011
Solutions will be posted Monday, 3/21/10
Problem 8.1
This is not a programming assignment and you need not turn in any code. This problem
considers the used of discrete least squares for approxima

Homework 1 Foundations of Computational Math 2 Spring
2013
Problem 1.1
Consider the data points
(x, y ) = cfw_(0, 2), (0.5, 5), (1, 8)
Write the interpolating polynomial in both Lagrange and Newton form for the given
data.
Solution:
For the Lagrange form

Solutions for Homework 2 Foundations of Computational
Math 2 Spring 2013
Problem 2.1
Let pn (x) be the unique polynomial that interpolates the data
(x0 , f0 ), . . . , (xn , fn )
Suppose that we assume the form
pn (x) = 0 + 1 (x x0 ) + + n (x x0 )(x x1 )

Solutions for Homework 6 Foundations of Computational
Math 2 Spring 2014
Problem 6.1
For this problem, consider the space L2 [1, 1] with inner product and norm
1
(f, g) =
f (x)g(x)dx and f
2
= (f, f )
1
Let f (x) = x3 + x2 . Determine the best linear leas

Homework 2 Foundations of Computational Math 2 Spring
2014
Problem 2.1
Show that given a set of points
x0 , x1 , . . . , xn
a Leja ordering can be computed in O(n2 ) operations.
Solution:
The following MATLAB code due to Higham (see Higham 2002 Accuracy a

Solutions for Homework 4 Foundations of Computational
Math 2 Spring 2014
Problem 4.1
Consider a set of equidistant mesh points, xk = x0 + kh, 0 k m.
4.1.a. Determine a cubic spline bi (x) that satises the following conditions:
bi (xj ) =
0 if j < i 1 or j

Solutions Homework 5 Foundations of Computational
Math 2 Spring 2014
Problem 5.1
For this problem, consider the space L2 [1, 1] with inner product and norm
1
(f, g) =
f (x)g(x)dx and f
2
= (f, f )
1
Let Pi (x), for i = 0, 1, . . . be the Legendre polynomi

Homework 1 Foundations of Computational Math 2 Spring
2014
Problem 1.1
Consider the data points
(x, y) = cfw_(0, 2), (0.5, 5), (1, 8)
Write the interpolating polynomial in both Lagrange and Newton form for the given
data.
Solution:
For the Lagrange form w

Solutions for Homework 7 Foundations of Computational
Math 2 Spring 2014
Problem 7.1
In this problem we consider the numerical approximation of the integral
1
I=
f (x)dx
1
with f (x) = ex . In particular, we use a priori error estimation to choose a step

Homework 9 Foundations of Computational Math 2 Spring
2014
Problem 9.1
Consider the Runge Kutta method called the implicit midpoint rule given by:
h
y1 = yn1 + f1
2
h
f1 = f (tn1 + , y1)
2
yn = yn1 + hf1
An alternate form of the the method is given by:
yn

Homework 8 Foundations of Computational Math 2 Spring
2014
Problem 8.1
Consider the following linear multistep method:
yn = 4yn1 + 5yn2 + h(4fn1 + 2fn2 )
The method is not 0-stable.
8.1.a. Determine, p, the order of consistency of the method.
8.1.b. Deter

Solutions for Homework 3 Foundations of Computational
Math 2 Spring 2014
Problem 3.1
Suppose we want to approximate a functionf (x) on the interval [a, b] with a piecewise
quadratic interpolating polynomial, g2 (x), with a constant spacing, h, of the inte