辛普森的 1/3 法则和 3/8 法则的区别

在辛普森的 1/3 规则中，我们基于二次逼近来逼近多项式。在这种情况下，每个近似值实际上涵盖了两个子区间。这就是为什么我们要求子区间的数量是偶数的。有些近似看起来更像一条线而不是二次曲线，但它们确实是二次曲线。

辛普森的¹/₃ 规则公式

ₐ∫ᵇ f (x) dx = h/₃ [(y₀ + yₙ) + 4 (y₁ + y₃ + ..) + 2(y₂ + y₄ + ..)]

where,

a, b is the interval of integration

h = (b – a)/ n

y₀ means the first terms and yₙ means last terms.

(y₁ + y₃ + ..) means the sum of odd terms.

(y₂ + y₄ + …) means sum of even terms.

示例：使用辛普森的 1/3 规则找到解决方案。

x	f(x)
0.0	1.0000
0.1	0.9975
0.2	0.9900
0.3	0.9776
0.4	0.8604

解决方案：

Using Simpson’s 1/3 rule

ₐ∫ᵇ f (x) dx = h/₃ [(y₀ + yₙ) + 4 (y₁ + y₃ + …) + 2 (y₂ + y₄ + …)] 

h = 0.1

ₐ∫ᵇ f (x) dx = 0.1/3 [(1+0.8604)+4×(0.9975+0.9776)+2×(0.99)]

ₐ∫ᵇ f (x) dx = 0.1/3 [(1+0.8604)+4×(1.9751)+2×(0.99)]

ₐ∫ᵇ f (x) dx = 0.39136

Solution of Simpson’s 1/3 rule =  0.39136

在辛普森的 3/8 规则中，我们基于二次逼近来逼近多项式。然而，每个近似实际上覆盖了三个子区间而不是两个。

辛普森 3/8 法则的公式

 ₐ∫ᵇ f (x) dx = 3h/8[(y₀ + yₙ) + 3(y₁ + y₂ + y₄ + …) + 2(y₃ + y₆ +…)]

where,

a, b is the interval of integration

h = (b – a )/ n

y₀ means the first terms and yₙ means the last terms.

( y₁ + y₂ + y₄ + … ) means the sum of remaining terms.

( y₃ + y₆ +…) means the multiples of 3.

示例：使用辛普森的 1/3 规则找到解决方案。

x	f(x)
0.0	1.0000
0.1	0.9975
0.2	0.9900
0.3	0.9776
0.4	0.8604

解决方案：

Using Simpson’s 3/8 rule:  

ₐ∫ᵇ f (x) dx = 3h/8[(y₀ + yₙ) + 3(y₁ + y₂ + y₄ + …) + 2(y₃ + y₆ +…)]

h = 0.1

ₐ∫ᵇ f (x) dx = 3h/8 [(y₀+y₄)+2(y₃)+3(y₁+y₂)]

ₐ∫ᵇ f (x) dx = 3* 0.1/8 [(1+0.8604)+2×(0.9776)+3×(0.9975+0.99)]

ₐ∫ᵇ f (x) dx = 3* 0.1/8 [(1+0.8604)+2×(0.9776)+3×(1.9875)]

ₐ∫ᵇ f (x) dx = 0.36668

Solution of Simpson’s 3/8 rule = 0.36668

以下是辛普森 1/3 规则和辛普森 3/8 规则之间的差异表

Sr. No	Simpson’s 1/3 rule	Simpson’s 3/8 rule
1.	It approximates function y = f(x) by a parabola i.e. by 2nd order polynomial.	It approximates the function y = f(x) by a parabola i.e. by 3rd order polynomial.
2.	In this, the chances of error are more than Simpson’s 3/8 rule.	In this, the chances of error are less.
3.	The integral function can be calculated as = h/3 [(sum of 1st and last ordinates) + 4 (sum of odd ordinates) + 2  (sum of even ordinates)].	The integral function can be calculated as = 3h/8 [(sum of 1st and last ordinates) + 2 (sum of multiple of 3 ordinates) + 3  (sum of remaining ordinates)].
4.	Estimation in truncation error in Simpson ‘s 1/3 rule is  E< -nh5/180 y iv(x) where h = (b-a)/n	Estimation in truncation error in Simpson ‘s 3/8 rule is  E< -nh5/90 y iv(x) where h = (b-a)/n
5.	This rule is applied where N is an even number.	This rule is applied where N is a multiple of 3.