MAT 402 - WINTER 2012
Homework 3
Ryan Szypowski
Due March 4, 2012
1. Consider the Initial Value Problem
y = (1 + t2 ) arctan y,
y(0) = 1.
0t1
(a) Show that the given initial value problem is well-posed by showing that f (t, y)
satises a Lipschitz conditio
MAT 402 - WINTER 2012
Homework 2 Solutions
Ryan Szypowski
Due January 24, 2012
1. Setup: In this problem, we will work through building the rst few orthogonal polynomials in the family of Jacobi polynomials with = = 1. What this means is that
we are makin
MAT 402 - WINTER 2012
Homework 4
Ryan Szypowski
Due February 14, 2012
1. Using the denition, show that the trapezoid rule has degree of precision 1. That is,
show that
I(1) = I1,closed (1), I(x) = I1,closed (x), but I(x2 ) = I1,closed (x2 ).
2. Approximat
MAT 402 - WINTER 2012
Homework 5
Ryan Szypowski
Due February 28, 2012
1. First, use Romberg Integration using the error estimate
|Rn,n Rn1,n1 |
2n1
to approximate
1
2
ex dx
1
with error less than 101 .
Then, use Adaptive Simpsons Rule using the error esti
MAT 402 - WINTER 2012
Homework 5
Ryan Szypowski
Due February 28, 2012
1. First, use Romberg Integration using the error estimate
|Rn,n Rn1,n1 |
2n1
to approximate
1
2
ex dx
1
with error less than 101 .
Then, use Adaptive Simpsons Rule using the error esti
MAT 402 - WINTER 2012
Homework 6
Ryan Szypowski
Due March 8, 2012
1. Prove that the so-called Optimal RK2 Method, given by
w0 =
wi+1 = wi + h
1
3
f (ti , wi ) + f
4
4
ti +
2h
2h
, wi + f (ti , wi )
3
3
,
is consistent and stable.
2. Derive the three step
MAT 402 - WINTER 2012
Homework 6 Solutions
Ryan Szypowski
Due March 8, 2012
1. Prove that the so-called Optimal RK2 Method, given by
w0 =
wi+1 = wi + h
1
3
f (ti , wi ) + f
4
4
ti +
2h
2h
, wi + f (ti , wi )
3
3
,
is consistent and stable.
Solution: Firs
MAT 402 WINTER 2012
Name:
Test #1 Solutions
Please show all of your work. Answers without justifaction may be worth 0.
Please make your answers easy to read. This means it should be clear what you are doing
from one step to the next and your work should b
MAT 402 WINTER 2012
Name:
Test #2
Please show all of your work. Answers without justifaction may be worth 0.
Please make your answers easy to read. This means it should be clear what you are doing
from one step to the next and your work should be legible.
MAT 402 - WINTER 2012
Homework 3 Solutions
Ryan Szypowski
Due January 31, 2012
1. Let f (x) = ex . Produce a polynomial interpolant of f which interpolates the following
information: f (0), f (0), f (0), f (ln 2). Use this interpolant to approximate f (0.
MAT 402 - WINTER 2012
Homework 2
Ryan Szypowski
Due January 24, 2012
1. Setup: In this problem, we will work through building the rst few orthogonal polynomials in the family of Jacobi polynomials with = = 1. What this means is that
we are making them ort
MAT 402 - WINTER 2014
Homework 4
Ryan Szypowski
Due March 13, 2012
1. Kuttas method is a three stage Runge-Kutta method with tableau
0
1
2
1
2
1 1 2
1
6
2
3
1
6
Prove that it is consistent and stable.
2. Consider the two step Adams-Moulton method (also ca
MAT 402 - WINTER 2014
Homework 4
Ryan Szypowski
Due March 13, 2012
1. Kuttas method is a three stage Runge-Kutta method with tableau
0
1
2
1
2
1 1 2
1
6
2
3
1
6
Prove that it is consistent and stable.
Solution: For this tableau, we can identify that
1
2
1
MAT 402 - WINTER 2014
Homework 1
Ryan Szypowski
Due January 28, 2014
1. Let f (x) = sin x . Compute the cubic interpolant of f , P0,1,2,3 (x), which interpolates
2
at x0 = 2, x1 = 1, x2 = 1, x3 = 2. Do this using Newton form and provide the
divided dieren
MAT 402 - WINTER 2014
Homework 2
Ryan Szypowski
Due February 11, 2014
1. Let f (x) = ex . Use the second-order centered dierence forumla to approximate f (0)
using h = 0.1, 0.05, 0.025. Verify that the method seems to be second order by computing the erro
MAT 402 - WINTER 2012
Homework 3
Ryan Szypowski
Due March 4, 2012
1. Consider the Initial Value Problem
y = (1 + t2 ) arctan y,
y(0) = 1.
0t1
(a) Show that the given initial value problem is well-posed by showing that f (t, y)
satises a Lipschitz conditio
MAT 402 - WINTER 2012
Homework 1 Solutions
Ryan Szypowski
Due January 17, 2012
1. Let f (x) = sin x . Compute the cubic interpolant of f , P0,1,2,3 (x), which interpolates
2
at x0 = 2, x1 = 1, x2 = 1, x3 = 2. Do this using Newton form and provide the
divi
MAT 402 - WINTER 2012
Homework 1
Ryan Szypowski
Due January 17, 2012
1. Let f (x) = sin x . Compute the cubic interpolant of f , P0,1,2,3 (x), which interpolates
2
at x0 = 2, x1 = 1, x2 = 1, x3 = 2. Do this using Newton form and provide the
divided dieren
MAT 402 - WINTER 2012
Homework 3
Ryan Szypowski
Due January 31, 2012
1. Let f (x) = ex . Produce a polynomial interpolant of f which interpolates the following
information: f (0), f (0), f (0), f (ln 2). Use this interpolant to approximate f (0.5).
What i
MAT 402 - WINTER 2012
Homework 4 Solution
Ryan Szypowski
Due February 14, 2012
1. Using the denition, show that the trapezoid rule has degree of precision 1. That is,
show that
I(1) = I1,closed (1), I(x) = I1,closed (x), but I(x2 ) = I1,closed (x2 ).
Solu