CAAM 336
·
DIFFERENTIAL EQUATIONS
Problem Set 8
Posted Thursday 21 October 2010. Due Wednesday 27 October 2010, 5pm.
1. [50 points]
Consider the following three matrices:
(i)
A
=
0
1
1
0
(ii)
A
=

50
49
49

50
(iii)
A
=
0
1

1
0
.
(a) For each of the matrices (i)–(iii), compute the matrix exponential
e
t
A
.
(You may use
eig
for the eigenvalues and eigenvectors, but please construct the matrix exponential
‘by hand’ (not with
expm
). For diagonalizable
A
=
VΛV

1
, recall the formula
e
t
A
=
V
e
t
Λ
V

1
.)
(b) Use your answers in part (a) to explain the behavior of
x
0
(
t
) =
Ax
(
t
) as
t
→ ∞
given that
x
(0) = [2
,
0]
T
(e.g., exponential growth, exponential decay, or neither) for each of the three
matrices (i)–(iii).
(c) For the matrix (ii), describe how large one can choose the time step Δ
t
so that the forward Euler
method applied to
x
0
(
t
) =
Ax
(
t
),
x
k
+1
=
x
k
+ Δ
t
Ax
k
,
will produce a solution that qualitatively matches the behavior of the true solution (i.e., the
approximations
x
k
should grow, decay, or remain of the same size as the true solution does).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Tompson
 Numerical Analysis, Runge–Kutta methods, Numerical ordinary differential equations, Backward Euler, Backward Euler method

Click to edit the document details