Difference between Simpson ‘s 1/3 rule and 3/8 rule

In Simpson’s 1/3 rule, we approximate the polynomial based on quadratic approximation. In this, each approximation actually covers two of the subintervals. This is why we require the number of subintervals to be even. Some of the approximations look more like a line than a quadratic, but they really are quadratics.  

Formula of Simpson’s¹/₃ rule

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 term and yₙ means last term.
• (y₁ + y₃ + ..) means the sum of odd terms.
• (y₂ + y₄ + …) means sum of even terms.

Example: Find the Solution using Simpson’s 1/3 rule.

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

Solution:

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

In Simpson’s 3/8 rule, we approximate the polynomial based on quadratic approximation. However, each approximation actually covers three of the subintervals instead of two.

Formula of Simpson’s 3/8 rule

 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 term and yₙ means the last term.
• ( y₁ + y₂ + y₄ + … ) means the sum of remaining terms.
• ( y₃ + y₆ +…) means the multiples of 3.

Example: Find a Solution using Simpson’s 1/3 rule.

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

Solution:

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

Following is a table of differences between the Simpson’s 1/3 rule and Simpson’s 3/8 rule

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< -h5/90 y4(x) where h = (b-a)/n	Estimation in truncation error in Simpson ‘s 3/8 rule is  E< -3h5/80 y4(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.