NUMERICAL ANALYSIS
1. Differentiation
( )
( )
( ) (
)
h
h
x
g
x
g
x
g
x
f
−
−
=
′
=
2. Integration
( )
( )
( )
(
)
( )
( ) (
)
[
]
h
x
g
x
g
h
x
f
h
x
f
x
f
initial
dx
x
g
x
f
+
+
+
=
+
=
∫
2
,
3. Solving non-linear equation ,
( )
0
=
x
f
3.1. Bisection Method
( ) ( )
[ ]
( ) ( )
[ ]
( ) ( )
[ ]
b
c
x
b
f
c
f
if
or
c
a
x
c
f
a
f
if
b
a
c
b
a
x
b
f
a
f
if
,
,
0
,
,
0
2
,
,
0
∈
<
∈
<
+
=
∈
<
3.2. Newton’s Method , 1 initial
( )
n
x
( )
( )
n
n
n
n
x
f
x
f
x
x
′
−
=
+1
3.3. Secant Method , 2 initial
(
)
n
n
x
x
,
1
−
( )(
)
( ) ( )
1
1
1
−
−
+
−
−
−
=
n
n
n
n
n
n
n
x
f
x
f
x
x
x
f
x
x
4. Definite integration
( )
n
a
b
h
dx
x
f
A
b
a
−
=
=
∫
4.1. Trapezoidal Method
( )
(
) ( )
(
)
( )
( )
a
b
a
b
x
f
a
b
dx
x
f
f
f
h
a
b
e
n
b
f
ih
a
f
a
f
h
A
b
a
n
i
−
′
=
−
′′
=
′′
−
′′
−
≤
=
⎥
⎦
⎤
⎢
⎣
⎡
+
+
+
=
∫
∑
−
=
12
...
3
,
2
,
1
2
2
2
1
1
4.2. Simpson’s
3
1
Method
( )
(
)
(
) ( )
(
)
( )
( )
( )
( )
( )
( )
a
b
a
b
x
f
a
b
dx
x
f
f
f
h
a
b
e
n
b
f
ih
a
f
ih
a
f
a
f
h
A
b
a
n
odd
i
i
n
even
i
i
−
=
−
=
−
−
≤
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
+
+
+
+
=
∫
∑
∑
−
=
=
−
=
=
3
4
4
4
4
1
1
2
2
180
...
6
,
4
,
2
2
4
3
4.3. Simpson’s
8
3
Method
( )
(
)
(
)
(
) ( )
(
)
( )
( )
( )
( )
( )
( )
a
b
a
b
x
f
a
b
dx
x
f
f
f
h
a
b
e
n
b
f
ih
a
f
ih
a
f
ih
a
f
a
f
h
A
b
a
n
i
i
n
i
i
n
i
i
−
=
−
=
−
−
≤
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
+
+
+
+
+
+
=
∫
∑
∑
∑
−
=
=
−
=
=
−
=
=
3
4
4
4
4
2
..
7
,
4
,
1
1
3
...
9
,
6
,
3
3
1
...
8
,
5
,
2
2
80
...
9
,
6
,
3
2
3
3
3
5. Solving differential equation
( )
( )
( )
0
0
,
,
y
x
y
initial
y
x
f
x
y
=
=
′
5.1. Euler’s Method
(
) ( )
(
i
i
i
i
y
x
hf
x
y
h
x
y
,
+
=
+
)
5.2. Runge-Kutta Method
(
) ( )
(
)
(
)
(
)
3
4
2
3
1
2
1
4
3
2
1
,
2
,
2
2
,
2
,
2
2
6
1
k
y
h
x
hf
k
k
y
h
x
hf
k
k
y
h
x
hf
k
y
x
hf
k
k
k
k
k
x
y
h
x
y
i
i
i
i
i
i
i
i
i
i
+
+
=
⎟
⎠
⎞
⎜
⎝
⎛
+
+
=
⎟
⎠
⎞
⎜
⎝
⎛
+
+
=
=
+
+
+
+
=
+
6. Interpolation polynomial
6.1. Discrete data ,
(
)
i
i
y
x ,
Minimized
( )
(
)
2
∑
−
i
i
y
x
P
For polynomial degree 2,
( )
x
a
a
x
P
1
0
+
=
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∑
∑
∑
∑
∑
i
i
i
i
i
i
y
x
y
a
a
x
x
x
n
1
0
2
For polynomial degree 3,
( )
2
2
1
0
x
a
x
a
a
x
P
+
+
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
i
i
i
i
i
i
i
i
i
i
i
i
i
y
x
y
x
y
a
a
a
x
x
x
x
x
x
x
x
n
2
2
1
0
4
3
2
3
2
2
6.2. Continues function,
( )
x
f
Minimized
( ) ( )
(
)
∫
−
−
1
1
2
dx
x
f
x
P
( )
( )
( )
( )
[ ]
( )
( )
( )
(
)
( )
(
)
( )
(
)
( )
(
)
( ) ( )
dx
x
P
x
f
k
a
x
x
x
x
P
x
x
x
P
x
x
x
P
x
x
P
x
x
P
x
P
x
x
P
a
x
P
a
x
P
a
x
P
k
k
∫
−
+
=
+
−
=
+
−
=
−
=
−
=
=
=
−
∈
+
+
+
=
1
1
3
5
5
2
4
4
3
3
2
2
1
0
1
1
1
1
0
0
2
1
2
15
70
63
8
1
3
30
35
8
1
3
5
2
1
1
3
2
1
1
1
,
1
,
...
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