Homework 9
Spring 2016 MA-UY 4423
Prof. M. O’Neil
Homework 9
Due: 2:00pm April 14, 2016
Each problem is worth 10 points.
Exercise 1 : Consider the initial value problem y 0 = λy, y(0) = 1. Show that the local truncation
error of the classical fourth-order Runge-Kutta method is O(h4 ):
h3 λ3
h4 λ4
h2 λ2
+
+
yk
yk+1 = 1 + hλ +
2
6
24
Exercise 2 : Show that the classical fourth-order Runge-Kutta method is stable by showing that
when it is written in the form
yk+1 = yk + hΨ(tk , yk , h)
the function Ψ satisfies a Lipschitz condition.
Exercise 3 : Show that the local truncation error in the multi-step method
5
5
13
yk+2 − 3yk+1 + 2yk = h
f (tk+2 , yk+2 ) − f (tk+1 , yk+1 ) − f (tk , yk )
12
3
12
is O(h2 ).
Exercise 4: Let the n × n matrix A be defined as:

0
1

..
.
..

.
A=

0
−α0 · · · −αn−2

1
−αn−1


.

Show that the characteristic polynomial ρ(λ) = det(A − λI) of the matrix A is
!
n−1
X
n
n
`
ρ(λ) = (−1)
λ +
α` λ .
`=0
Hint: Expand the determinant using the last row.
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