Numerical Methods – Reference Formula Sheet Newton-Rahson Method / Method of Tan!ents
Bisecon Method / I ntermediate Values Theorem
f ( ( an ) . f ( ( bn ) < 0
{
an , bn , c n=
,S
ol. an , bn
an + bn 2
, f ( ( c n )
'
f μ= ' '
f μ = ' ' ' μ
[
1
h 1
h
f =
2
1
[ [
3
h
1
∆ f 0+
2
{
2
μ=
,
2
0
2
2
3
0
6
1
3
∆ f 0 +
Sol. x0 , x 1
x x n , x n+1= x n−
( 2 μ −1 ) ∆ f + 1 ( 3 μ −6 μ +2 ) ∆ f +
1
,
f ( ( x n ) f ' ( x x n )
f ( ( x 0 ) . f ( ( x 1 ) < 0
,
}
x − x 0
( 6 μ −18 μ +11 ) ∆ 2
12
1
4
( 2 μ −3 ) ∆ f + 0
4
1
a = x 0
h
3
24
f 0 +
4
0
1
( 60 μ − 240 μ +210 ) ∆
1
( 5 μ
120
( 20 μ −120 μ +210 μ−100) ∆ 3
120
2
120
2
5
f 0
2
5
f 0
4
b = x n
,
h=
,
,
x 1− x 0 y 1− y 0
)
}
b −a n
x 0
¿ f ( ( x 1+ f (¿ ( ¿) ]− E ( f ) ) ∧ E ( f ) ) =
''
3
h f ( μ μ ) 12
∧ μ ϵ [ a , b ]
x 1
∫ f ( ( x x ) dx= h2 ¿
]
]
Sol. x0 , x 1
x x 0 , x1 , y 0 , y 1 , xn = x 0− y 0 (
Num& Inte!raon – Trae'oidal Rule
( 4 μ −18 μ +22 μ −6 ) ∆ f +
,
{
μ ( μ μ −1 ) 2 μ ( μ−1 ) ( μ− 2 ) 3 μ ( μ −1 ) ( μ μ −2 ) ( μ −3) 4 μ ( μ μ −1 ) ( ∆ y 0+ ∆ y 0 + ∆ y 0 + 2! 3! 4!
∆ f 0 + ( μ −1 ) ∆ f 0 + 3
}
h= x 1− x 0
Newton#s $re!or% Forward Interolaon Method
f ( ( x ) = y 0 + μ ∆ y 0 +
f ( ( x 0 ) . f ( ( x 1 ) < 0
,
Method of False "osion / Re!ular Falsi Method
x 0
x0
{
j=n−1
f ( ( x n+ f (¿ (¿ )} +2 {
∑ = i
f ( ( xi )}
1
]
¿ μ ) (b −a ) h f '' ( μ 2
− E ( f ) ) ∧ E ( f ) ) =
12
∧ μ ϵ [ a , b ]
b
∫ f ( ( x ) dx = h2 ¿ a
(b −a ) h f ( μ ) 2
E ( f ) ) =
' '
12
'' ¿ Truncated Truncated Error Error ≡ for f ( a) →
h= x 1− x 0
Sterlin! or (entral )i*erence Formula
1
(
)
f ( ( x ) = f 0 + p δf −1 + δf +1 + 2
2
2
p=
,
2
2!
2
p δ f 0 +
Num& Inte!raon – Simson#s +/ Rule
h
p ( p −1 ) 3 p ( p − 1 3 2 2 4 δ f −1 + δ f + 1 + p ( p −1 ) δ f 0 + 2∗3 ! 4! 4 2 2 2
1
x p− x 0
(
&&& '
f p=
[
1 1
h
2
2
(δ − +δ + )+ p δ f + 1
2
1
2
0
1 12
( 3 p − 1 ) (δ − + δ + )+ 2
3
3
1
2
1
2
1 12
( 2 p − p ) δ f + 3
4
0
1 480
( 5 p −4 p −12 4
3
2
12
a = x 0
12
b = x n
,
2
|f ' ' ( a )|, for f ' ' ( b )→ (b −a ) h |f '' ,
b −a h= n
x 0
2
)
( b− a )h
¿ f ( ( x 2 + 4 f ( ( x1 ) + f (¿ (¿ ) ] − E ( f ) )∧ E ( f ) )= x 2
μ ) −h f ' ' '' ( μ
∫ f ( ( x x ) dx = h3 ¿ x 0
5
90
∧ μ ϵ [ a ,b ]
' '
f p=
[
1
h
2
2
δ f 0 +
p
3
1
2
1
3
(δ − +δ + )+ 1
2
2
5
5
( 6 p −1 ) δ f + 2
0
12
1
4
( 20 p −12 p −24 p +8 ) (δ − + δ + ) 3
480
2
5
5
1
1
2
2
]
a
60 p 1
1
2
2
(
3
)
3
2
1
4
' ''
f p =
1
h
3
480
¿
' '
f ϑ =
[
1
h 1
h
2
[
3
h
2
∇
1
'
f ' ' ϑ=
∇ f n +
[
1 2
ϑ ( ϑ + 1 ) 2!
h= x 1− x 0
y n +
ϑ=
,
ϑ ( ϑ + 1 ) ( ϑ +2 ) 3!
∇
3
y n +
( 2 ϑ + 1 ) ∇ f n + 1 ( 3 ϑ + 6 ϑ + 2 ) ∇ f n+ 2
2
3
6
3
3
2
∇
f n+ ( ϑ + 1 ) ∇ f n +
∇
f n +
1 12
1
( 6 ϑ +18 ϑ +11 ) ∇ 2
12
4
( 12 ϑ + 18 ) ∇ f n +
1 120
4
1
ϑ ( ϑ +1 ) ( ϑ + 2 ) ( ϑ + 3 ) 4!
( 2 ϑ + 9 ϑ +11 ϑ + 3 ) ∇ 1
120
2
4
4
y n +
|f '' ' ' ( a )|,forf '' '' (b ) → ( b− a )h 180
f n +
1 120
(5 ϑ +4
3
5
2
f n
]
4
5
f n
]
x 0
x i = n− 1
f (¿¿ !ultiple of 3 −i )}+ 3 {
f ( x ∑ = i
¿ ∏ = 0
¿ i% #
f ( x ) $ ( x ) ∑ =
0
f ( x 0 ) $0 ( x ) + f ( x1 ) $1 ( x ) + f ( x 2 ) $2 ( x ) + f ( x 3 ) $3 ( x ) + f ( x 4 ) $ 4 ( x ) + f ( x5 ) $5 ( x ) … + f ( x n ) $n ( x )
0
Newton#s )i1ided )i*erence Interolaon Formula
re"t of theter!"−i
1
i = n− 1
{f ( x ) + f ( x ) }+2 { ∑ =
n
n
h=
x 3
(¿ ¿ # − x i ) i
,
∫ f ( x ) dx = 38h ¿
a
$# ( x ) =
b = x n
¿
x ( x − x i ) a!ran!e Interolaon Formula
,
f ( x 3 + 3 {f ( x 1 ) + f ( x 2 ) }+ f (¿) ]
b
#
180
2
x 0
ϑ ( ϑ + 1) ( ϑ
( 20 ϑ +120 ϑ +210 ϑ +100) ∇
( 60 ϑ + 240 ϑ +210 ) ∇ 2
∇
n
∫ f ( x ) dx = 38h ¿
#
( b −a ) h
a = x 0
Num& Inte!raon – Simson#s /. Rule
h
3
12
f n +
x − x n
i
#
]
180
0
&n=
i =1
¿
f ( x ) = y n + ϑ ∇ y n+ '
1
) }+2 { ∑ f ( x een−i ) } − E ( f
(b −a ) h f ' '' ' ( μ )
¿ Truncated Error ≡ for f '' ' ' ( a ) →
Newton#s $re!or% Bacward Interolaon Method
f ϑ=
i
j=n−1
odd − i
2
δ − 1 + δ +1 + p δ f 0+ 2
j=n −1
n
0
2
E ( f ) =
(¿ ¿ 2 −24 p− 24 )( δ − + δ + ) 1
[
b
f ( x ∫ f ( x ) dx= h3 {f ( x )+ f ( x )}+ 4 { ∑ =
1
¿
)}
b −a n
2
δ ( x 0 , x1 ) =
y 1− y 0
,
x1− x0
δ ( x 0 , x1 , x2 ) =
δ ( x 0 , x 1 )− δ ( x 1 , x 2 )
δ ( x 0 , x1 , x2 , x3 ) =
,
x 2− x 0
δ ( x 3 , x 2 , x 1 )− δ ( x 2 , x 1 , x 0 ) x 3− x 0
δ ( x n−2 , x n−1 , x n ) =
,
δ ( x n−2 , x n−1 )−δ ( x n−1 , x n) x n− x n−2
f ( x ) = f ( x 0 )+ ( x − x 0 ) δ ( x 0 , x 1 ) + ( x − x 0 ) ( x − x 1 ) δ ( x 0 , x 1 , x 2 ) + ( x − x 0 ) ( x − x 1 ) ( x − x 2 ) δ ( x 0 , x1 , x 2 , x 3 ) … + ( x − x 0 ) ( x − x 1 ) … ( x − x n− 1 ) δ ( x 0 , x 1 , x 2 …x n ) 2uler Method
Modi3ed 2uler Method or 4"redictor – (orrector Method5
x n= x 0 + n h
y
[
y n +1 = y n+ h f ( x n , y n )
yn+1 = yn +
1 2
[K1 + K2]
K1 = h f (xn , yn ) K2 = h f (xn+h , yn+k1)
]
y
p n +1 c n+ 1
= y n+ h [ f ( x n , y n ) ] = y n+
yn+1 = yn +
1 6
p
y n = y n− 4+
h
[f ( x , y ) +f ( x , y )]
2
n
n
Miln – Simson Method
n
[K1 + 2 (K2 + K3) + K4]
K1 = h f (x n , yn , zn) K2 = h f (x n+h/2 , yn+(k1/2) , zn+(L1/2)) K3 = h f (x n+h/2 , yn+(k2)/2 , zn+(L2)/2) K4 = h f (x n+h , yn+k3 , zn+L3)
p n
p
y n = y n−2 + zn+1 = zn +
1 6
4h 3
h 3
[ 2 f n− + f n − +2 f n− ] 3
2
1
[ f − +4 f − + f ] n
2
n
1
p n
[L1 + 2 (L2 + L3) + L4]
L1 = h g (xn , yn , zn) L2 = h g (xn+h/2 , yn+(k1/2) , zn+(L1/2)) L3 = h g (xn+h/2 , yn+(k2)/2 , zn+(L2)/2) L4 = h g (xn+h , yn+k3 , zn+L3)
