Three-point formula with Error Bound(Forward, Central, Backward)
𝐟 ′ (𝐱 𝟎 ) ≈
−𝟑𝐟(𝐱 𝟎 ) + 𝟒𝐟(𝐱 𝟎 + 𝐡) − 𝐟(𝐱 𝟎 + 𝟐𝐡) 𝐡𝟐 (𝟑)
+ 𝐟 (𝛏),
𝟐𝐡
𝟑
𝐟 ′ (𝐱 𝟎 ) ≈
𝐟 ′ (𝐱 𝟎 ) ≈
𝐟(𝐱 𝟎 + 𝐡) − 𝐟(𝐱 𝟎 − 𝐡) 𝐡𝟐 (𝟑)
− 𝐟 (𝛏),
𝟐𝐡
𝟔
𝛏 ∈ [𝐱 𝟎 , 𝐱 𝟎 + 𝟐𝐡]
𝛏 ∈ [𝐱 𝟎 − 𝐡, 𝐱 𝟎 + 𝐡]
𝟑𝐟(𝐱 𝟎 ) − 𝟒𝐟(𝐱 𝟎 − 𝐡) + 𝐟(𝐱 𝟎 − 𝟐𝐡) 𝐡𝟐 (𝟑)
+ 𝐟 (𝛏),
𝟐𝐡
𝟑
𝛏 ∈ [𝐱 𝟎 − 𝟐𝐡, 𝐱 𝟎 ]
Five-point Formula with Error Bound(Forward, Central, Backward)
𝐟 ′ (𝐱𝟎 ) ≈
−𝟐𝟓𝐟(𝐱𝟎 ) + 𝟒𝟖𝐟(𝐱𝟎 + 𝐡) − 𝟑𝟔𝐟(𝐱𝟎 + 𝟐𝐡) + 𝟏𝟔𝐟(𝐱𝟎 + 𝟑𝐡) − 𝟑𝐟(𝐱𝟎 + 𝟒𝐡) 𝐡𝟒 (𝟓)
+ 𝐟 (𝛏),
𝟏𝟐𝐡
𝟓
𝐟 ′ (𝐱𝟎 ) ≈
𝐟 ′ (𝐱𝟎 ) ≈
𝐟(𝐱𝟎 − 𝟐𝐡) − 𝟖𝐟(𝐱𝟎 − 𝐡) + 𝟖𝐟(𝐱𝟎 + 𝐡) − 𝐟(𝐱𝟎 + 𝟐𝐡) 𝐡𝟒 (𝟓)
+
𝐟 (𝛏),
𝟏𝟐𝐡
𝟑𝟎
𝛏 ∈ [𝐱𝟎 , 𝐱𝟎 + 𝟒𝐡]
𝛏 ∈ [𝐱𝟎 − 𝟐𝐡, 𝐱𝟎 + 𝟐𝐡]
𝟐𝟓𝐟(𝐱𝟎 ) − 𝟒𝟖𝐟(𝐱𝟎 − 𝐡) + 𝟑𝟔𝐟(𝐱𝟎 − 𝟐𝐡) − 𝟏𝟔𝐟(𝐱𝟎 − 𝟑𝐡) + 𝟑𝐟(𝐱𝟎 − 𝟒𝐡) 𝐡𝟒 (𝟓)
+ 𝐟 (𝛏),
𝟏𝟐𝐡
𝟓
𝛏 ∈ [𝐱𝟎 − 𝟒𝐡, 𝐱𝟎 ]
Second Derivative Mid-point Formula with Error Bound
𝐟(𝐱𝟎 + 𝐡) − 𝟐𝐟(𝐱𝟎 ) + 𝐟(𝐱𝟎 − 𝐡) 𝐡𝟐 (𝟒)
−
𝐟 (𝛏),
𝐡𝟐
𝟏𝟐
𝐟 ′′ (𝐱𝟎 ) ≈
𝛏 ∈ [𝐱𝟎 − 𝐡, 𝐱𝟎 + 𝐡]
Richardson Extrapolation
𝐍𝐣−𝟏 (𝐡/𝟐) − 𝐍𝐣−𝟏 (𝐡)
𝐡
𝐍𝐣 (𝐡) = 𝐍𝐣−𝟏 ( ) +
𝟐
𝟒𝐣−𝟏 − 𝟏
Trapezoid Rule and Composite Trapezoid with Error
𝐛
∫ 𝐟(𝐱) 𝐝𝐱 ≈
𝐚
(𝐛 − 𝐚)𝟑 ′′
𝐛−𝐚
[𝐟(𝐚) + 𝐟(𝐛)] −
𝐟 (𝛏),
𝟐
𝟏𝟐
𝛏 ∈ [𝐚, 𝐛]
𝐧−𝟏
𝐛
∫ 𝐟(𝐱) 𝐝𝐱 ≈
𝐚
(𝐛 − 𝐚)𝐡𝟐 ′′
𝐡
[𝐟(𝐱 𝟎 ) + 𝟐 ∑ 𝐟(𝐱 𝐢 ) + 𝐟(𝐱 𝐧 )] −
𝐟 (𝛏),
𝟐
𝟏𝟐
𝐡=
𝐢=𝟏
𝐛−𝐚
, 𝛏 ∈ [𝐚, 𝐛]
𝐧
Simpson Rule and Composite Simpson with Error
𝐱𝟐
∫ 𝐟(𝐱) 𝐝𝐱 ≈
𝐱𝟎
𝐡
𝐡𝟓 (𝟒)
[𝐟(𝐱 𝟎 ) + 𝟒𝐟(𝐱 𝟏 ) + 𝐟(𝐱 𝟐 )] −
𝐟 (𝛏),
𝟑
𝟗𝟎
𝐧−𝟏
𝐱𝐧
𝐡
∫ 𝐟(𝐱) 𝐝𝐱 ≈ [𝐟(𝐱 𝟎 ) + 𝟒 ∑ 𝐟(𝐱 𝐢 ) + 𝟐
𝟑
𝐱𝟎
𝐢=𝟏, 𝐢 odd
𝛏 ∈ [𝐱 𝟎 , 𝐱 𝟐 ]
𝐧−𝟐
∑
𝐟(𝐱 𝐢 ) + 𝐟(𝐱 𝐧 )] −
𝐢=𝟐, 𝐢 even
(𝐱 𝐧 − 𝐱 𝟎 )𝐡𝟒 (𝟒)
𝐟 (𝛏)
𝟏𝟖𝟎
Simpson Three-Eight(n=3 and n=4) with Error
𝐱𝟑
∫ 𝐟(𝐱) 𝐝𝐱 ≈
𝐱𝟎
𝐱4
∫ 𝐟(𝐱) 𝐝𝐱 ≈
𝐱𝟎
𝟑𝐡
𝟑𝐡𝟓 (𝟒)
[𝐟(𝐱 𝟎 ) + 𝟑𝐟(𝐱 𝟏 ) + 𝟑𝐟(𝐱 𝟐 ) + 𝐟(𝐱 𝟑 )] −
𝐟 (𝛏)
𝟖
𝟖𝟎
(𝐛 − 𝐚)𝟓 (𝟒)
𝟑𝐡
[𝐟(𝐱𝟎 ) + 𝟑𝐟(𝐱𝟏 ) + 𝟑𝐟(𝐱𝟐 ) + 𝟑𝐟(𝐱𝟑 ) + 𝐟(𝐱𝟒 )] −
𝐟 (𝛏)
𝟖
𝟑𝟖𝟒𝟎
Midpoint(n=0,1,2,3) and Composite Midpoint with Error
𝐱𝟏
∫ 𝐟(𝐱) 𝐝𝐱 = 𝟐𝐡𝐟(𝐱 𝟎 ) +
𝐱 −𝟏
𝐱𝟐
∫ 𝐟(𝐱) 𝐝𝐱 =
𝐱 −𝟏
𝐱𝟑
∫ 𝐟(𝐱) 𝐝𝐱 =
𝐱 −𝟏
𝐱𝟒
∫ 𝐟(𝐱) 𝐝𝐱 =
𝐱−𝟑
𝐡𝟐 ′′
𝐟 (𝛏),
𝟑
where 𝐱 −𝟏 < 𝛏 < 𝐱 𝟏 .
𝟑𝐡
𝟑𝐡𝟑 ′′
[𝐟(𝐱 𝟎 ) + 𝐟(𝐱 𝟏 )] +
𝐟 (𝛏),
𝟐
𝟒
where 𝐱 −𝟏 < 𝛏 < 𝐱 𝟐 .
𝟒𝐡
𝟏𝟒𝐡𝟓 (𝟒)
[𝟐𝐟(𝐱 𝟎 ) − 𝐟(𝐱 𝟏 ) + 𝟐𝐟(𝐱 𝟐 )] +
𝐟 (𝛏),
𝟑
𝟒𝟓
where 𝐱−𝟏 < 𝛏 < 𝐱 𝟑 .
𝟓𝐡
𝟗𝟓𝐡𝟕 (𝟔)
[𝟏𝟏𝐟(𝐱 𝟎 ) + 𝐟(𝐱 𝟏 ) + 𝐟(𝐱 𝟐 ) + 𝟏𝟏𝐟(𝐱 𝟑 )] +
𝐟 (𝛏),
𝟐𝟒
𝟏𝟒𝟒
𝐧
𝟐
𝐛
∫ 𝐟(𝐱) 𝐝𝐱 = 𝟐𝐡 ∑ 𝐟(𝐱 𝟐𝐣 ) +
𝐚
𝐣=0
where 𝐱 −𝟑 < 𝛏 < 𝐱 𝟒 .
(𝐛 − 𝐚)𝒉𝟐 (𝟐)
𝐟 (𝛏)
𝟔
Romberg Integration
𝐑 𝐤,𝐣 = 𝐑 𝐤,𝐣−𝟏 +
𝟏
(𝐑
𝟒𝐣−𝟏 − 𝟏 𝐤,𝐣−𝟏
− 𝐑 𝐤−𝟏,𝐣−𝟏 ),
for 𝐤 = 𝐣, 𝐣 + 𝟏, . . . ..