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Likewise, utilizing the third-order Lagrange interpolating polynomial (2.1) becomes.

Figure 2 Likewise, utilizing the third-order Lagrange interpolating polynomial (2.1) becomes.

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We obtain the standard 1/3 Simpson’s rule, which is one of the most recognized closed Newton- Cotes quadrature approaches in practice.
We obtain the standard 1/3 Simpson’s rule, which is one of the most recognized closed Newton- Cotes quadrature approaches in practice.
Combining and simplifying, we obtain the 3/8 Simpson’s rule
Combining and simplifying, we obtain the 3/8 Simpson’s rule
These indicates that the degree of precision of (2.7) is n+ 1 and (2.8) is n, although the interpolation polynomial is of degree at most n.
These indicates that the degree of precision of (2.7) is n+ 1 and (2.8) is n, although the interpolation polynomial is of degree at most n.
Since f € C?[a, b] then by Extreme Value Theorem, The error term in (2.14) can be derived from the summation of all errors in the application of standard trapezoidal rule on each subinterval. Thus,
Since f € C?[a, b] then by Extreme Value Theorem, The error term in (2.14) can be derived from the summation of all errors in the application of standard trapezoidal rule on each subinterval. Thus,
Additionally, the comparative analysis of the absolute errors across the three methods displayed in Figure 2 reveals that the Simpson's 3/8 rule consistently yields the minimal magnitudes of absolute errors until the desired precision is achieved. The first subinterval signifies larger disproportions on the absolute errors. The gaps are minimized when the succeeding subintervals are considered.
Additionally, the comparative analysis of the absolute errors across the three methods displayed in Figure 2 reveals that the Simpson's 3/8 rule consistently yields the minimal magnitudes of absolute errors until the desired precision is achieved. The first subinterval signifies larger disproportions on the absolute errors. The gaps are minimized when the succeeding subintervals are considered.
Table 3. Approximation of { C te~‘dt within 10-* The incomplete gamma function given by y(a,x) = f. 7 t*-1 e~'dt is commonly applied in the field of heat conduction, probability theory, and Fourier and Laplace transformations. The convergence analysis to approximate the value of the incomplete gamma function utilizing a = 2 on the interval [0,1] within 10~* is presented in Table 3. the field of heat conduction, probability theory, and Fourier and Laplace transformations. The 3.3. Incomplete Gamma Function
Table 3. Approximation of { C te~‘dt within 10-* The incomplete gamma function given by y(a,x) = f. 7 t*-1 e~'dt is commonly applied in the field of heat conduction, probability theory, and Fourier and Laplace transformations. The convergence analysis to approximate the value of the incomplete gamma function utilizing a = 2 on the interval [0,1] within 10~* is presented in Table 3. the field of heat conduction, probability theory, and Fourier and Laplace transformations. The 3.3. Incomplete Gamma Function
There is an exceptionally large discrepancy in terms of the robustness of the trapezoidal rule as compared to Simpson’s 1/8 and 3/8 rules. To arrive at the desired accuracy level, the trapezoidal rule required more than 20 times the number of subintervals of the Simpsons rules, n = 2 and n = 4 subintervals for the 1/3 rule andn = 3 andn = 6 for the 3/8 rule.
There is an exceptionally large discrepancy in terms of the robustness of the trapezoidal rule as compared to Simpson’s 1/8 and 3/8 rules. To arrive at the desired accuracy level, the trapezoidal rule required more than 20 times the number of subintervals of the Simpsons rules, n = 2 and n = 4 subintervals for the 1/3 rule andn = 3 andn = 6 for the 3/8 rule.
Table 4. Approximation of f 7 - edt within 10-* : —x2 fl . . . . The Dawson’s integral, f(x) = e~* f e¥?dy, bears important contribution in the concept of heat conduction and the theory of electrical oscillations of special vacuum tubes. Table 4 shows the results when approximating the value of the Dawson’s integral on the interval [0,1]. Comparative analysis with the results obtained in incomplete gamma function shows the same convergence behaviour using the trapezoidal rule. The data further reveals that it takes 58 subintervals to attain the targeted level of precision. The magnitude is significantly larger as compared to the outcomes of Simpson’s 1/3 and 3/8 rules. It is apparent on the results that the 3/8 rule is the most efficient method, requiring only 3 subintervals (n = 3, 6, and 9), in comparison to the 1/3 rule, which demands 4 subintervals (n = 2, 4, 6, and 8) for the approximate answer to converge with the exact answer.
Table 4. Approximation of f 7 - edt within 10-* : —x2 fl . . . . The Dawson’s integral, f(x) = e~* f e¥?dy, bears important contribution in the concept of heat conduction and the theory of electrical oscillations of special vacuum tubes. Table 4 shows the results when approximating the value of the Dawson’s integral on the interval [0,1]. Comparative analysis with the results obtained in incomplete gamma function shows the same convergence behaviour using the trapezoidal rule. The data further reveals that it takes 58 subintervals to attain the targeted level of precision. The magnitude is significantly larger as compared to the outcomes of Simpson’s 1/3 and 3/8 rules. It is apparent on the results that the 3/8 rule is the most efficient method, requiring only 3 subintervals (n = 3, 6, and 9), in comparison to the 1/3 rule, which demands 4 subintervals (n = 2, 4, 6, and 8) for the approximate answer to converge with the exact answer.
As observed from Figure 4, the absolute errors of the trapezoidal rule considering the first few intervals exhibits a wide gap as compared to Simpson’s 1/3 and 3/8 rules. Although there is a marginal difference on the errors between 1/3 and 3/8 rules, the latter bagged the lowest magnitude. When the tolerance level is increased to 10~?°, it can be observed from Table 8 that trapezoidal rule requires 9000 subintervals to achieve convergence. This is comparable to the outcomes of the incomplete gamma function and standard normal distribution function. Simpson’s 3/8 Rule has the fastest convergence which necessitates the use of 75 subintervals while the 1/3 Rule requires 92 subintervals.
As observed from Figure 4, the absolute errors of the trapezoidal rule considering the first few intervals exhibits a wide gap as compared to Simpson’s 1/3 and 3/8 rules. Although there is a marginal difference on the errors between 1/3 and 3/8 rules, the latter bagged the lowest magnitude. When the tolerance level is increased to 10~?°, it can be observed from Table 8 that trapezoidal rule requires 9000 subintervals to achieve convergence. This is comparable to the outcomes of the incomplete gamma function and standard normal distribution function. Simpson’s 3/8 Rule has the fastest convergence which necessitates the use of 75 subintervals while the 1/3 Rule requires 92 subintervals.
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