Assignment # 3
Math 456 - Spring 2006
Due: Tuesday, March 7
1. (30 pts) Find the eigenvalues, eigenvectors, k · k1and k · k∞for the following,
A=1 3
4 2 , B =2 1
1 2 , C =2 3
−1−2, D =2 5
−1−2
2. (10 pts) Find k · k2for each of the following matrices,
A=2 5
−1−2, B =
570
−3−5 0
0 0 −2
3. (10 pts) Solve y0=yexactly assuming that y(0) = 1.
4. (10 pts) Use Euler’s method to solve y0=ywith y(0) = 1 and a step size
of h=.1. Produce a table with the values of x, y and F(x, y) for each x=
0, .1, .2,...,.9,1 (as done in the notes).
5. (10 pts) Use Runge-Kutta method to solve y0= 5x4with y(0) = 0 on the
interval [0,1] with h=.1. Produce a table with the values of x, y and F(x, y)
for each x(as in problem 3 above).
6. (20 pts) Use Euler’s method to solve y000 −2xy00 + 4y0−x2y= 1 with initial
conditions y(0) = 1, y0(0) = 2, y 00(0) = 3. Produce a table with the values of x
and yin the interval [0,1] using a step size of h=.1.
7. (10 pts) Use Runge-Kutta method to solve 3x2y00 −xy0+y= 0 with initial
conditions y(1) = 4, y0(1) = 2 on the interval [1,2] with h=.2. Print out a
table with all the values for x, y and zfor all the points x= 1,1.2,1.4,1.6,1.8
and 2. (Here zrefers to V- notation provided in the notes and in class).
Note:
1) You must show ALL work for credit.
2) You should use a calculator or a computer and write your own numerical algo-
rithms to produce the solutions to problems 4, 5, 6 and 7. You must submit/write
the code for your Euler and Runge-Kutta methods together with your assignment
for any credit to be given.