__Math 4340 - Numerical Analysis__

__Homework #6, Date Due: March 20, 2007__

**Problem 1**

Construct a contraction mapping
function *g(x)*, using the method
presented in class, for finding the root of the function *f(x) = exp(x) + 2*ln(x)* in the interval *(0.3,06)*. Using
this function, write a computer code to find this root with a tolerance of
1.0E-7 (i.e. your function value should be less than 1.0E-7 at the root that
you found).

**Problem 2**

Find the root of the same function
using both

**Problems 3-5**

Solve problems number 5.2, 5.3 and 5.4 page 246 in the textbook.

__Homework #5, Date Due: March 1st, 2007__

**Problems 1-4**

Do problems 7.11, 7.15, 7.19 and
7.21 in the textbook. For 7.15, also evaluate the integrals you obtain using a quadrature method of your choice (with at least two function
evaluation points) between those presented in class. DO NOT solve 7.14, i.e.
don’t try to evaluate the integral using Gauss-Hermite
quadrature.

**Problem #5**

Using Monte-Carlo simulation, estimate the integral over the circle in the *x-y *plane of radius 1 and with center at
(*x*=1, *y*=1) of the function *f(x) =
sin(x+y)*.
Use the same values of N as in your previous Monte-Carlo integral
evaluations and print the approximate values you obtain, as well as the
relative error, for each N.

__Homework #4, Date Due: Feb. 15 ^{th},
2007__

**Problems 1-4**

Solve problems 7.1, 7.4, 7.5, 7.10
in the textbook.

**Problem #5**

Again using Monte-Carlo simulation, estimate the integral from 1 to 3 of the
function *f(x) = sin(x)*. Use the same values of N as in your previous
Monte-Carlo integral evaluation and print the approximate values you obtain, as
well as the relative error, for each N.

__Homework #3, Date Due: February 6, 2007__

__Homework #2, Date Due: January 25, 2007__

**Problem #1**

Using Monte-Carlo simulation,
estimate the number “Pi” as four times the ratio of the area of the
unit circle over that of the square of side length two, centered at the origin.
Do this for N=1,000; N=10,000; N=100,000; N=200,000; N=400,000; N=800,000; and
N=1,000,000. Plot the relative error in your approximations versus N in one
plot, and the relative error versus square root of N on a different plot. Turn
in the two plots, as well as a printout of your program and a list with the
approximate values you got for Pi for the different N values.

**Problem #2**

Again using Monte-Carlo simulation, estimate the integral from 0 to 1 of the
function *f(x) = sin(x)*. Use the same values of N as above and again
plot the relative errors in the two ways and turn in the plots, the program and
a list of the approximate values.

__Homework #1, Date Due: January 18, 2007__

**Chapter 1 Problems, Pages 34-35: **

** **Solve: 1.1, 1.2, 1.6, 1.7,
1.14

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__NOTE:__** In case
some of the problems have the answer provided at the end of the book, you need
to show the details of how that answer is obtained. Otherwise, just writing the
answer will bring you no credit. I hope you all agree that this is just fair!
**