if number of steps is not multiple of 3 how to do simpsons 3/8 rule? why im getting more error in simpsons 3/8 rule than in simpsons 1/3 rule?

5 次查看（过去 30 天）
Shamila
Shamila 2024-4-4
评论： Shamila 2024-4-8
Ran in:
Here, as dx is 0.1 and 0.01, n is not multiple of 3 for 3/8 rule.tried by making n same for all 3 rules. but error in 3/8 > error in 1/3. how to solve this issue?
% Define the function
f = @(x) 2-x+log(x);
% Define integration limits
a = 1;
b = 2;
% Exact integral value
og = integral(f,a,b);
%Step sizes
h1 =(b-a)/12;
h2 =(b-a)/102;
% Trapezoidal Rule
trph1 = h1/2 * (f(a) + f(b) + 2*sum(f(a+h1:h1:b-h1)));
trph2 = h2/2 * (f(a) + f(b) + 2*sum(f(a+h2:h2:b-h2)));
trperh1 = abs((trph1-og) / og) * 100;
trperh2 = abs((trph2-og) / og) * 100;
% Simpson's 1/3 Rule
simp1h1 = h1/3 * (f(a) + f(b) + 2*sum(f(a+2*h1:2*h1:b-2*h1)) + 4*sum(f(a+h1:2*h1:b-h1)));
simp1h2 = h2/3 * (f(a) + f(b) + 2*sum(f(a+2*h2:2*h2:b-2*h2)) + 4*sum(f(a+h2:2*h2:b-h2)));
simp1erh1 = abs((simp1h1 - og) / og) * 100;
simp1erh2 = abs((simp1h2 - og) / og) * 100;
% Simpson's 3/8 Rule
simp2h1 = (3*h1/8) * (f(a) +f(b) + 3*sum(f(a+h1:3*h1:b-2*h1)) + 3*sum(f(a+2*h1:3*h1:b-h1)) + 2*sum(f(a+3*h1:3*h1:b-3*h1)));
simp2h2 = (3*h2/8) * (f(a) +f(b) + 3*sum(f(a+h2:3*h2:b-2*h2)) + 3*sum(f(a+2*h2:3*h2:b-h2)) + 2*sum(f(a+3*h2:3*h2:b-3*h2)));
simp2erh1 = abs((simp2h1 - og) / og) * 100;
simp2erh2 = abs((simp2h2 - og) / og) * 100;
% Display results
disp('for h = 0.1 :-');
for h = 0.1 :-
disp(['Trapezoidal Rule: ',num2str(trph1), ', Error: ', num2str(trperh1)]);
Trapezoidal Rule: 0.88601, Error: 0.032634
disp(['Simpson''s 1/3 Rule: ', num2str(simp1h1), ', Error: ', num2str(simp1erh1)]);
Simpson's 1/3 Rule: 0.88629, Error: 5.2332e-05
disp(['Simpson''s 3/8 Rule: ', num2str(simp2h1), ', Error: ', num2str(simp2erh1)]);
Simpson's 3/8 Rule: 0.88629, Error: 0.00011653
% disp('for h = 0.01 :-');
disp(['Trapezoidal Rule: ', num2str(trph2), ', Error: ', num2str(trperh2)]);
Trapezoidal Rule: 0.88629, Error: 0.00045186
disp(['Simpson''s 1/3 Rule: ', num2str(simp1h2), ', Error: ', num2str(simp1erh2)]);
Simpson's 1/3 Rule: 0.88629, Error: 1.0133e-08
disp(['Simpson''s 3/8 Rule: ', num2str(simp2h2), ', Error: ', num2str(simp2erh2)]);
Simpson's 3/8 Rule: 0.88629, Error: 2.2795e-08

请先登录，再回答此问题。

采纳的回答

Torsten
Torsten 2024-4-4
移动：Torsten 2024-4-4
if number of steps is not multiple of 3 how to do simpsons 3/8 rule?
The 1/3 rule can be used for the remaining subintervals without changing the order of the error term (conversely, the 3/8 rule can be used with a composite 1/3 rule for odd-numbered subintervals).
why im getting more error in simpsons 3/8 rule than in simpsons 1/3 rule?
Why do you think this should not be the case ? Both composite rules are of the same order as far as I know.

更多回答（0 个）

类别

在 Help Center 和 File Exchange 中查找有关 Numerical Integration and Differential Equations 的更多信息

标签

产品


版本

R2023b

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by