Relationship between Simpson’s rule and numerical integration

Simpson’s rules are a set of numerical integration approximations for definite integrals named after Thomas Simpson (1710–1761).

                      Simpson’s rule can be developed by 

                       using the quadratic interpolant P(x) 

                        to approximate the integrand f (x) 

Definition:

Simpson’s rule is one of the numerical methods for calculating the definite integral . To get the definite integral, we usually employ the fundamental theorem of calculus, which requires us to use antiderivative integration techniques. However, in other cases, such as in Scientific Experiments, where the function must be calculated from observed data, finding the antiderivative of an integral is difficult. In such situations, numerical approaches are utilized to approximate the integral. Other numerical methods include the trapezoidal rule, midpoint rule, and left or right approximation using Riemann sums. We’ll go through the Simpson’s rule formula, the 1/3 rule, the 3/8 rule, and some examples in this section.

Simpson’s Rule Formula:

Simpson’s rule approaches are more accurate than other numerical approximations, and the formula for n+1 evenly spaced subdivision is;

b∫a f(x)dx = Sₙ = ∆x/3 [ ƒ ( x₀ ) + 4f ( x₁ ) + 2ƒ ( x₂ ) + 4ƒ ( x₃ ) + … + 2ƒ ( xₙ₋₂ ) + 4f ( xₙ₋₁ ) + f ( xₙ ) ]

Where n is the even number, △x = (b – a)/n and xi = a + i△x

If we have f(x) = y, which is equally spaced between [a, b] and if a = x₀, x₁ = x₀ + h, x₂ = x₀ + 2h …., xₙ = x₀ + nh, where h is the difference between the terms. Or we can say that y₀ = f(x₀), y₁ = f(x₁), y₂ = f(x₂),……,yₙ = f(xₙ) are the analogous values of y with each value of x.

Simpson’s 1/3 Rule:

The trapezoidal rule is extended by Simpson’s 1/3rd rule, in which the integrand is approximated by a second-order polynomial. The Simpson rule can be determined in a variety of ways, including utilizing Newton’s divided difference polynomial, Lagrange polynomial, and the coefficient technique. The Simpson’s rule of thirds is as follows:

b∫a f(x) dx = h/3 [(y₀ + yₙ) + 4(y₁ + y₃ + y₅ + …. + yₙ₋₁) + 2(y₂ + y₄ + y₆ + ….. + yₙ₋₂)]

This guideline is known as Simpson’s One-third Rule.

Simpson’s ⅓ Rule for Integration:

We can get a quick approximation for definite integrals by dividing a small interval [a, b] into two parts. We get: as a result of splitting the interval: 

X₀= a, x₁= a + b, x₂ = b

As a result, the approximation can be written as;

b∫a f(x) dx ≈ S₂ = h/3[f(x₀) + 4f(x₁) + f(x₂)]

S₂ = h/3 [f(a) + 4 f((a+b)/2) + f(b)]

Where h = (b – a)/2

For integration, the Simpson’s 13 rule applies.

Simpson’s 3/8 Rule:

The “Simpson’s 3/8 rule” is another way of numerical integration. Rather than quadratic interpolation, it is entirely based on cubic interpolation. The Simpson 3/8 or three-eight rule is defined as follows:

b∫a f(x) dx = 3h/8 [(y₀ + yₙ) + 3(y₁ + y₂ + y₄ + y₅ + …. + yₙ₋₁) + 2(y₃ + y₆ + y₉ + ….. + yₙ₋₃)]

This rule is more accurate than the standard method since it uses an extra functional value. The 3/8 rule also has a composite Simpson’s 3/8 rule that is comparable to the generalized form. Simpson’s second rule of integration is known as the 3/8 rule.

Alternative extended Simpson’s rule:

Instead of applying Simpson’s rule to unconnected segments of the integral to be approximated, this formulation applies Simpson’s rule to overlapping segments, obtaining

b∫a f(x) dx ≈ h/48 · [ 17ƒ ( x₀ ) + 59ƒ ( x₁ ) + 43ƒ ( x₂ ) + 49ƒ ( x₃ ) + 48i=4n-4 f ( xᵢ ) + 49 f ( xₙ₋₃ ) + 43 ƒ ( xₙ₋₂ ) + 59ƒ ( xₙ₋₁ ) + 17 f ( xₙ ) 

The formula above is generated by combining the original composite Simpson’s rule with the one that uses Simpson’s 3/8 rule in the extreme subintervals and the conventional 3-point rule in the remainder. The outcome is then calculated by averaging the two formulas.

Simpson’s rules in the case of narrow peaks:

Simpson’s rules are substantially less efficient than the trapezoidal rule in estimating the whole area of narrow peak-like functions. To put it another way, composite Simpson’s 1/3 rule requires 1.8 times more points than the trapezoidal rule to obtain the same accuracy[6]. Simpson’s 3/8 rule for composites is considerably less precise. Integral by Simpson’s 1/3 rule is equal to the sum of 2/3 of integral by trapezoidal rule with step h and 1/3 of integral by rectangle rule with step 2h, with the second (2h step) term governing precision. The following rules emerge from averaging Simpson’s 1/3 rule composite sums with suitably moved frames:

b∫a f(x) dx ≈ h/24 (−ƒx₋₁+ 12ƒx₀  + 25ƒx₁ + 24 Σᵢ₌₂ⁿ⁻²ƒ ( xᵢ ) + 25 ƒ ( xₙ₋₁ ) + 12ƒ ( xₙ ) – ƒ ( xₙ₊₁ ) )

When two sites are explored outside of the integrated region, and

b∫a f(x) dx ≈ h/24 ( 9f ( x₀ ) + 28ƒ ( x₁ ) + 23ƒ ( x₂ ) + 24 Σᵢ₌₃ⁿ⁻³ƒ ( xᵢ ) + 23ƒ ( xₙ₊₂ ) + 28ƒ ( xₙ₋₁ ) + 9f ( xₙ ) )

Only points within the integration region are used in this case. When the second rule is applied to a three-point region, it yields 1/3 Simpson’s rule and 3/8 Simpson’s rule for four points.

These rules are extremely similar to Press’s expanded Simpson’s rule alternative. Within the largest part of the region that is being integrated, the coefficients are equal, and the differences exist only at the edges. These two rules are known as First order Euler–MacLaurin integration rules because they are related with the Euler–MacLaurin formula with the first derivative term. [6] The sole difference between the two rules is how the first derivative at the region end is calculated. In Euler–MacLaurin integration rules, the first derivative term accounts for the integral of the second derivative, which equals the difference of the first derivatives at the integration region’s edges. By combining a difference of 3rd, 5th, and so on derivatives with coefficients specified by the Euler–MacLaurin formula, higher order Euler–MacLaurin rules can be generated.

Use of Simpson’s rule:

Simpson’s (1/3) and (3/8) formulas estimate the values of definite integrals throughout an interval [a,b] using the values of the integrand function at a few equidistant places in the interval without integrating the function, to a remarkable degree of precision.

When the functions defining the curves have no antiderivatives in the closed form or are difficult to identify, it is particularly useful for finding areas under curves or areas limited by closed curves.

Conclusion:

Simpson’s rule is a rule of the fourth order.

We can use the composite Simpson’s 3/8 rule to split the interval x = a to x = b into a number of segments with four nodes each and sum over these segments to get the integral. The number n should be a multiple of three (with the node at x = a being counted as 0).