Game Development Reference
In-Depth Information
1
i
1
.
x
=
r
L r
i
i
ik
k
L
ii
k
=
1
The solution to a linear system
Ux
=
r , where U is an nn
×
upper triangular ma-
trix, can be found by backward substitution:
1
n
.
x
=
r
Ur
i
i
ik
k
U
ii
ki
=+
1
A matrix M can be decomposed into the product LU , where L is lower triangular
and U is upper triangular, using Doolittle's method. The linear system
Mxr
=
then becomes
L Ux r , which can be solved in two stages by first using for-
ward substitution to solve
(
) =
Lyr and then backward substitution to solve
=
Ux
=
y .
Eigenvalues and Eigenvectors
The eigenvalues and eigenvectors of a 3
symmetric matrix M can be numeri-
cally calculated by applying the Jacobi method to diagonalize M . When M is
transformed by one of the rotation matrices
×
3
()
pq
given by Equation (16.39), the
R
new entries of M are given by
=
M
M
ii
ii
M
=
cM
sM
if
i
p
and
i
q
;
ip pi
,
ip
iq
MsMcM
=
+
iq qi
,
ip
iq
M
=−
MtM
pp
pp
pq
M
=+
MtM
qq
qq
pq
M
=
0,
pq qp
,
where t
=
sc
.
Ordinary Differential Equations
The first-order ordinary differential equation
()
(
)
xf
=
xy
,
can be approximat-
y
ed using Euler's method as follows.
(
)
y
=+
y f
xy
,
i
1
i
i
i
+
The improved Euler's method, also known as Heun's method, uses the step
formula