Euler Method: Example (3)
Second time step:
Third time step
3.01.02.0
=+=+=
ttt
2.01.01.0
12
=+=+=
ttt
Systems of First Order ODEs
Review of Matrix Algebra
Vector Algebra in n Dimensions
a
1
b
1
+
ba
11
=
ka
ka
ak …
2
1
Matrix Algebra in n Dimensions
+
+
+
+
+
+
+
+
nnjj
nnjj
nj
nj
BABABABA
BABABABA
AAAA
AAAA
......
......
......
......
222222222121
111112121111
222221
111211
nj
nj
kAkAkAkA
kAkAkAkA
……
……
222221
111211
Matrix Algebra in n Dimensions (2)
=
y
y
xAy
2
1
nj
nj
AAAA
AAAA
……
……
222221
111211
x
x
2
1
Matrix Algebra in n Dimensions (3)
xAy =
Matrix Algebra in n Dimensions (4)
= CAB
nj
nj
CCCC
CCCC
………………
……
……
222221
111211
nj
nj
AAAA
AAAA
..................
......
......
222221
111211
=
=n
k
kjikij CAB
1
nj
nj
BBBB
BBBB
………………
……
……
222221
111211
CAB =
Matrix Algebra in n Dimensions (5)
Matrix Algebra in n Dimensions (6)
A
bxA
=
If
and and are known,
b
Example: Inverting a 2 by 2 matrix
=43
52
A
Problem: Invert the matrix A
IAA = −1
Definition of inverse
Example: Inverting a 2 by 2 matrix (2)
043
152
=+
=+
ca
ca
Multiplying the matrices and equating terms component
by component
143
052
=+
=+
db
db
Solving
Example: Inverting a 2 by 2 matrix (3)
Hence: