Comparing the Simpson 1/3 and Simpson 3/8 Rules using Newton's Progressive Formula

Introduction

end points of the two periods, where the function P 2 (x) is approximated. A parabola, which is a quadratic polynomial, is represented by (x).

Derivation of Simpson's 1/3 Rule

The method can be derived using Newton's forward formula for the insertion given in the following form

where the integration is defined on the interval [x 0 , x 2 ], we will replace the integral of f(x) with a second-order polynomial using Newton's formula for insertion with the third difference truncation:

Equation ( 1) is called the Simpson formula

For Numerical Integration It is the general formula where it gives a fixed approximate value of the integral when the function is polynomial of the third degree and below, and by generalizing equation (1) we get the following general formula for the period [x 0 , x n ]:

The last equation is called the Simpson formula 1 3

Error of Simpson's 1/3

The amputation error at Simpson's rule is equal to

That is the error order of the simple Simpson rule is O(h 5 ) The amputation error in the Simpsons vehicle base is

The amount of error for the Simpson formula is added to the value of the integration resulting from this method to get the true integration. Third: i=0,1,2…N-1,

Fifth: print a value S n and then stop. Now Using Simpson's rule, find the value of the following integration if you know that n=6

The following integration ∫ e -1 2 x 2 1 solution is Using the Simpsons Base 3 1 and through 4 intervals Since the interval count is an even number, this means using the Simpson 3 1 rule would be appropriate to solve the question:

The following order shows the integration values by applying the Simpsons rule 3 1 with the use of a different number of partitions that the true integrity value of this function is

x 2 1 = 0.4773024366 , So we can see that the value of the integration is far from the true value, so we use a value of n less.

Rule 3/8 Simpson's

The Simpson 8 3 method is considered one of the methods of numerical integration and is considered an extension of the Simpson In addition, the difference between the Simpson 3 1 method and the Simpson 8 3 method is that the amputation error rank is equal in both formulas, but the coefficient of the first is lower than the coefficient of the second, which makes the Simpson 3 1 method a desirable method in numerical integration.

Derivation of Simpson's 3/8 Rule

We derive this rule using Newton's formula for forward differences, where the period from x 0 to x 3 (i.e., for the period from [0,3] relative to n) includes four values of the function (f 0 , f 1 , f 2 , f 2 ) as well as four points where the polynomial is written as follows:

Cutting off the boundaries containing the fourth forward differences and beyond, by taking the integration period from [0,3] and using the following relations:

Equation ( 3) The previous one represents the Simpson 8 3 formula and by generalizing the equation along the period [x 0 , x n ] we get:

Whereas equation ( 4) represents the Simpson 8 3 formula.

Rule 3/8 of Simpson's Error

The amputation error of the Simpson formula is

Represents the approximate amount of integration after adding the amount of error to the integration value From the previous table, we can see that the value of the integration is closer to the true value the larger n is and especially if the value of n is even where the more divisions the faster the solution is reached, as the truth value of the integration is

Using the Simpsons 3/8 formula to find the integration for the following polynomial The value of true integration is I = ∫ f(x) dx = 1.640533 0.8 0

The true error equal to the difference between the true error and the approximate error deduced from the Simpson 3/8 formula is as follows:

E t = 1.640533 -1.519170 = 0.1213630

When the higher derivatives of the function are known, one of Newton-Coates' higher-order rules (i.e., one that contains a greater number of function values) is chosen. Accordingly, the Simpson rule 1/3 is more accurate than the Simpson rule 3/8, which has a high error order and a larger number of divisions, giving accurate results compared to the trapezoidal rule. The number of points determines which special methods are used, and while they take more time and work, they produce accurate results that are extremely near to the genuine results.

References

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