Numerical method MATLAB code

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Shoaib Salim
Shoaib Salim 2023-4-6
评论： Oguz Kaan Hancioglu 2023-4-7
I am unable to get plot of following code of numerical methods. Plot command throw error of not same vector length
% Title: Comparison of Trapezoid and Simpson's Composite Methods for T-distribution and Normal Distribution
% Name: TXXXX XXXXXXX
% Class: Scientific Computing
% Define the probability density functions for T-distribution and normal distribution
f_t = @(x, v) gamma((v+1)/2) / (sqrt(v*pi) * gamma(v/2)) * (1 + x.^2/v).^(-(v+1)/2);
f_norm = @(x) exp(-x.^2/2) / sqrt(2*pi);
% Define the integration interval and the number of subintervals for approximation
a = -5;
b = 5;
n = 1000;
% Use trapezoid method to approximate the integrals
h = (b-a)/n;
x = linspace(a, b, n+1);
t_trap = h/2 * (f_t(x(1:end-1), 10) + f_t(x(2:end), 10));
norm_trap = h/2 * (f_norm(x(1:end-1)) + f_norm(x(2:end)));
% Use Simpson's composite method to approximate the integrals
x = linspace(a, b, 2*n+1);
h = (b-a)/(2*n);
t_simp = h/3 * (f_t(x(1:2:end-2), 10) + 4*f_t(x(2:2:end-1), 10) + f_t(x(3:2:end), 10));
norm_simp = h/3 * (f_norm(x(1:2:end-2)) + 4*f_norm(x(2:2:end-1)) + f_norm(x(3:2:end)));
% Calculate the exact values of the integrals
t_exact = integral(@(x) f_t(x, 10), -Inf, Inf);
norm_exact = integral(@(x) f_norm(x), -Inf, Inf);
% Compare the approximations with the exact values
fprintf('T-distribution integral:\n');
fprintf('Trapezoid method: %f\n', t_trap);
fprintf('Simpson''s composite method: %f\n', t_simp);
fprintf('Exact value: %f\n', t_exact);
fprintf('\nNormal distribution integral:\n');
fprintf('Trapezoid method: %f\n', norm_trap);
fprintf('Simpson''s composite method: %f\n', norm_simp);
fprintf('Exact value: %f\n', norm_exact);
% Plot the approximations and the exact values
figure;
subplot(2,1,1);
plot(x, f_t(x, 10), 'k', x(1:end-1), t_trap, 'r', x(1:2:end), t_simp, 'b--');
legend('Exact', 'Trapezoid', 'Simpson''s');
title('T-distribution Probability Density Function');
subplot(2,1,2);
plot(x, f_norm(x), 'k', x(1:end-1), norm_trap, 'r', x(1:2:end), norm_simp, 'b--');
legend('Exact', 'Trapezoid', 'Simpson''s');
title('Normal Distribution Probability Density Function');
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Shoaib Salim
Shoaib Salim 2023-4-6
In plot command i get error of different vector length. How can i change it to get plot similar to exact value of 1
Torsten
Torsten 2023-4-7
编辑：Torsten 2023-4-7
Didn't you understand what I mean ?
What do you want to plot in case you determine the value of an integral over a predefined interval ?
The only thing I can think of is to plot integral_{t=a}^{t=x} f(t) dt in a given grid point x. But you don't calculate these integrals in your code and you didn't comment on this suggestion I've already done earlier.

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采纳的回答

Oguz Kaan Hancioglu
Ran in:
The length(x(1:end-1)) and length(t_trap) are not equal. Also, the length(x(1:2:end)) and length(t_simp) are not the equal. I think you can equal the length(x(1:2:end)) and length(t_simp) by changing x(3:2:end). However, the length(x(1:end-1)) is equal twice of length(t_trap).
% Title: Comparison of Trapezoid and Simpson's Composite Methods for T-distribution and Normal Distribution
% Name: TXXXX XXXXXXX
% Class: Scientific Computing
% Define the probability density functions for T-distribution and normal distribution
f_t = @(x, v) gamma((v+1)/2) / (sqrt(v*pi) * gamma(v/2)) * (1 + x.^2/v).^(-(v+1)/2);
f_norm = @(x) exp(-x.^2/2) / sqrt(2*pi);
% Define the integration interval and the number of subintervals for approximation
a = -5;
b = 5;
n = 1000;
% Use trapezoid method to approximate the integrals
h = (b-a)/n;
x = linspace(a, b, n+1);
t_trap = h/2 * (f_t(x(1:end-1), 10) + f_t(x(2:end), 10));
norm_trap = h/2 * (f_norm(x(1:end-1)) + f_norm(x(2:end)));
% Use Simpson's composite method to approximate the integrals
x = linspace(a, b, 2*n+1);
h = (b-a)/(2*n);
t_simp = h/3 * (f_t(x(1:2:end-2), 10) + 4*f_t(x(2:2:end-1), 10) + f_t(x(3:2:end), 10));
norm_simp = h/3 * (f_norm(x(1:2:end-2)) + 4*f_norm(x(2:2:end-1)) + f_norm(x(3:2:end)));
% Calculate the exact values of the integrals
t_exact = integral(@(x) f_t(x, 10), -Inf, Inf);
norm_exact = integral(@(x) f_norm(x), -Inf, Inf);
% Compare the approximations with the exact values
fprintf('T-distribution integral:\n');
T-distribution integral:
fprintf('Trapezoid method: %f\n', t_trap);
Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000008 Trapezoid method: 0.000008 Trapezoid method: 0.000008 Trapezoid method: 0.000008 Trapezoid method: 0.000008 Trapezoid method: 0.000008 Trapezoid method: 0.000008 Trapezoid method: 0.000008 Trapezoid method: 0.000009 Trapezoid method: 0.000009 Trapezoid method: 0.000009 Trapezoid method: 0.000009 Trapezoid method: 0.000009 Trapezoid method: 0.000009 Trapezoid method: 0.000010 Trapezoid method: 0.000010 Trapezoid method: 0.000010 Trapezoid method: 0.000010 Trapezoid method: 0.000010 Trapezoid method: 0.000010 Trapezoid method: 0.000010 Trapezoid method: 0.000011 Trapezoid method: 0.000011 Trapezoid method: 0.000011 Trapezoid method: 0.000011 Trapezoid method: 0.000011 Trapezoid method: 0.000012 Trapezoid method: 0.000012 Trapezoid method: 0.000012 Trapezoid method: 0.000012 Trapezoid method: 0.000012 Trapezoid method: 0.000013 Trapezoid method: 0.000013 Trapezoid method: 0.000013 Trapezoid method: 0.000013 Trapezoid method: 0.000013 Trapezoid method: 0.000014 Trapezoid method: 0.000014 Trapezoid method: 0.000014 Trapezoid method: 0.000014 Trapezoid method: 0.000015 Trapezoid method: 0.000015 Trapezoid method: 0.000015 Trapezoid method: 0.000015 Trapezoid method: 0.000016 Trapezoid method: 0.000016 Trapezoid method: 0.000016 Trapezoid method: 0.000016 Trapezoid method: 0.000017 Trapezoid method: 0.000017 Trapezoid method: 0.000017 Trapezoid method: 0.000018 Trapezoid method: 0.000018 Trapezoid method: 0.000018 Trapezoid method: 0.000019 Trapezoid method: 0.000019 Trapezoid method: 0.000019 Trapezoid method: 0.000019 Trapezoid method: 0.000020 Trapezoid method: 0.000020 Trapezoid method: 0.000020 Trapezoid method: 0.000021 Trapezoid method: 0.000021 Trapezoid method: 0.000022 Trapezoid method: 0.000022 Trapezoid method: 0.000022 Trapezoid method: 0.000023 Trapezoid method: 0.000023 Trapezoid method: 0.000023 Trapezoid method: 0.000024 Trapezoid method: 0.000024 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Trapezoid method: 0.000089 Trapezoid method: 0.000090 Trapezoid method: 0.000092 Trapezoid method: 0.000093 Trapezoid method: 0.000095 Trapezoid method: 0.000097 Trapezoid method: 0.000098 Trapezoid method: 0.000100 Trapezoid method: 0.000102 Trapezoid method: 0.000104 Trapezoid method: 0.000105 Trapezoid method: 0.000107 Trapezoid method: 0.000109 Trapezoid method: 0.000111 Trapezoid method: 0.000113 Trapezoid method: 0.000115 Trapezoid method: 0.000117 Trapezoid method: 0.000119 Trapezoid method: 0.000121 Trapezoid method: 0.000123 Trapezoid method: 0.000125 Trapezoid method: 0.000128 Trapezoid method: 0.000130 Trapezoid method: 0.000132 Trapezoid method: 0.000134 Trapezoid method: 0.000137 Trapezoid method: 0.000139 Trapezoid method: 0.000142 Trapezoid method: 0.000144 Trapezoid method: 0.000147 Trapezoid method: 0.000149 Trapezoid method: 0.000152 Trapezoid method: 0.000154 Trapezoid method: 0.000157 Trapezoid method: 0.000160 Trapezoid method: 0.000163 Trapezoid method: 0.000165 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Trapezoid method: 0.000316 Trapezoid method: 0.000321 Trapezoid method: 0.000327 Trapezoid method: 0.000332 Trapezoid method: 0.000338 Trapezoid method: 0.000344 Trapezoid method: 0.000349 Trapezoid method: 0.000355 Trapezoid method: 0.000361 Trapezoid method: 0.000367 Trapezoid method: 0.000373 Trapezoid method: 0.000380 Trapezoid method: 0.000386 Trapezoid method: 0.000392 Trapezoid method: 0.000399 Trapezoid method: 0.000406 Trapezoid method: 0.000412 Trapezoid method: 0.000419 Trapezoid method: 0.000426 Trapezoid method: 0.000433 Trapezoid method: 0.000440 Trapezoid method: 0.000447 Trapezoid method: 0.000455 Trapezoid method: 0.000462 Trapezoid method: 0.000470 Trapezoid method: 0.000477 Trapezoid method: 0.000485 Trapezoid method: 0.000493 Trapezoid method: 0.000501 Trapezoid method: 0.000509 Trapezoid method: 0.000518 Trapezoid method: 0.000526 Trapezoid method: 0.000534 Trapezoid method: 0.000543 Trapezoid method: 0.000552 Trapezoid method: 0.000561 Trapezoid method: 0.000570 Trapezoid method: 0.000579 Trapezoid method: 0.000588 Trapezoid method: 0.000597 Trapezoid method: 0.000607 Trapezoid method: 0.000616 Trapezoid method: 0.000626 Trapezoid method: 0.000636 Trapezoid method: 0.000646 Trapezoid method: 0.000656 Trapezoid method: 0.000666 Trapezoid method: 0.000677 Trapezoid method: 0.000687 Trapezoid method: 0.000698 Trapezoid method: 0.000709 Trapezoid method: 0.000720 Trapezoid method: 0.000731 Trapezoid method: 0.000742 Trapezoid method: 0.000754 Trapezoid method: 0.000765 Trapezoid method: 0.000777 Trapezoid method: 0.000789 Trapezoid method: 0.000801 Trapezoid method: 0.000813 Trapezoid method: 0.000825 Trapezoid method: 0.000837 Trapezoid method: 0.000850 Trapezoid method: 0.000863 Trapezoid method: 0.000876 Trapezoid method: 0.000889 Trapezoid method: 0.000902 Trapezoid method: 0.000915 Trapezoid method: 0.000929 Trapezoid method: 0.000942 Trapezoid method: 0.000956 Trapezoid method: 0.000970 Trapezoid method: 0.000984 Trapezoid method: 0.000999 Trapezoid method: 0.001013 Trapezoid method: 0.001028 Trapezoid method: 0.001042 Trapezoid method: 0.001057 Trapezoid method: 0.001072 Trapezoid method: 0.001088 Trapezoid method: 0.001103 Trapezoid method: 0.001119 Trapezoid method: 0.001134 Trapezoid method: 0.001150 Trapezoid method: 0.001166 Trapezoid method: 0.001182 Trapezoid method: 0.001199 Trapezoid method: 0.001215 Trapezoid method: 0.001232 Trapezoid method: 0.001249 Trapezoid method: 0.001266 Trapezoid method: 0.001283 Trapezoid method: 0.001300 Trapezoid method: 0.001318 Trapezoid method: 0.001336 Trapezoid method: 0.001353 Trapezoid method: 0.001371 Trapezoid method: 0.001389 Trapezoid method: 0.001408 Trapezoid method: 0.001426 Trapezoid method: 0.001445 Trapezoid method: 0.001463 Trapezoid method: 0.001482 Trapezoid method: 0.001501 Trapezoid method: 0.001520 Trapezoid method: 0.001540 Trapezoid method: 0.001559 Trapezoid method: 0.001579 Trapezoid method: 0.001599 Trapezoid method: 0.001618 Trapezoid method: 0.001638 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Trapezoid method: 0.002477 Trapezoid method: 0.002500 Trapezoid method: 0.002524 Trapezoid method: 0.002547 Trapezoid method: 0.002570 Trapezoid method: 0.002593 Trapezoid method: 0.002616 Trapezoid method: 0.002640 Trapezoid method: 0.002663 Trapezoid method: 0.002686 Trapezoid method: 0.002709 Trapezoid method: 0.002732 Trapezoid method: 0.002755 Trapezoid method: 0.002778 Trapezoid method: 0.002800 Trapezoid method: 0.002823 Trapezoid method: 0.002846 Trapezoid method: 0.002868 Trapezoid method: 0.002891 Trapezoid method: 0.002913 Trapezoid method: 0.002936 Trapezoid method: 0.002958 Trapezoid method: 0.002980 Trapezoid method: 0.003002 Trapezoid method: 0.003024 Trapezoid method: 0.003045 Trapezoid method: 0.003067 Trapezoid method: 0.003088 Trapezoid method: 0.003110 Trapezoid method: 0.003131 Trapezoid method: 0.003152 Trapezoid method: 0.003172 Trapezoid method: 0.003193 Trapezoid method: 0.003213 Trapezoid method: 0.003234 Trapezoid method: 0.003254 Trapezoid method: 0.003273 Trapezoid method: 0.003293 Trapezoid method: 0.003312 Trapezoid method: 0.003332 Trapezoid method: 0.003351 Trapezoid method: 0.003369 Trapezoid method: 0.003388 Trapezoid method: 0.003406 Trapezoid method: 0.003424 Trapezoid method: 0.003442 Trapezoid method: 0.003459 Trapezoid method: 0.003476 Trapezoid method: 0.003493 Trapezoid method: 0.003510 Trapezoid method: 0.003526 Trapezoid method: 0.003542 Trapezoid method: 0.003558 Trapezoid method: 0.003573 Trapezoid method: 0.003589 Trapezoid method: 0.003603 Trapezoid method: 0.003618 Trapezoid method: 0.003632 Trapezoid method: 0.003646 Trapezoid method: 0.003659 Trapezoid method: 0.003673 Trapezoid method: 0.003685 Trapezoid method: 0.003698 Trapezoid method: 0.003710 Trapezoid method: 0.003722 Trapezoid method: 0.003733 Trapezoid method: 0.003744 Trapezoid method: 0.003755 Trapezoid method: 0.003765 Trapezoid method: 0.003775 Trapezoid method: 0.003784 Trapezoid method: 0.003794 Trapezoid method: 0.003802 Trapezoid method: 0.003811 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Trapezoid method: 0.003819 Trapezoid method: 0.003811 Trapezoid method: 0.003802 Trapezoid method: 0.003794 Trapezoid method: 0.003784 Trapezoid method: 0.003775 Trapezoid method: 0.003765 Trapezoid method: 0.003755 Trapezoid method: 0.003744 Trapezoid method: 0.003733 Trapezoid method: 0.003722 Trapezoid method: 0.003710 Trapezoid method: 0.003698 Trapezoid method: 0.003685 Trapezoid method: 0.003673 Trapezoid method: 0.003659 Trapezoid method: 0.003646 Trapezoid method: 0.003632 Trapezoid method: 0.003618 Trapezoid method: 0.003603 Trapezoid method: 0.003589 Trapezoid method: 0.003573 Trapezoid method: 0.003558 Trapezoid method: 0.003542 Trapezoid method: 0.003526 Trapezoid method: 0.003510 Trapezoid method: 0.003493 Trapezoid method: 0.003476 Trapezoid method: 0.003459 Trapezoid method: 0.003442 Trapezoid method: 0.003424 Trapezoid method: 0.003406 Trapezoid method: 0.003388 Trapezoid method: 0.003369 Trapezoid method: 0.003351 Trapezoid method: 0.003332 Trapezoid method: 0.003312 Trapezoid method: 0.003293 Trapezoid method: 0.003273 Trapezoid method: 0.003254 Trapezoid method: 0.003234 Trapezoid method: 0.003213 Trapezoid method: 0.003193 Trapezoid method: 0.003172 Trapezoid method: 0.003152 Trapezoid method: 0.003131 Trapezoid method: 0.003110 Trapezoid method: 0.003088 Trapezoid method: 0.003067 Trapezoid method: 0.003045 Trapezoid method: 0.003024 Trapezoid method: 0.003002 Trapezoid method: 0.002980 Trapezoid method: 0.002958 Trapezoid method: 0.002936 Trapezoid method: 0.002913 Trapezoid method: 0.002891 Trapezoid method: 0.002868 Trapezoid method: 0.002846 Trapezoid method: 0.002823 Trapezoid method: 0.002800 Trapezoid method: 0.002778 Trapezoid method: 0.002755 Trapezoid method: 0.002732 Trapezoid method: 0.002709 Trapezoid method: 0.002686 Trapezoid method: 0.002663 Trapezoid method: 0.002640 Trapezoid method: 0.002616 Trapezoid method: 0.002593 Trapezoid method: 0.002570 Trapezoid method: 0.002547 Trapezoid method: 0.002524 Trapezoid method: 0.002500 Trapezoid method: 0.002477 Trapezoid method: 0.002454 Trapezoid method: 0.002431 Trapezoid method: 0.002408 Trapezoid method: 0.002384 Trapezoid method: 0.002361 Trapezoid method: 0.002338 Trapezoid method: 0.002315 Trapezoid method: 0.002292 Trapezoid method: 0.002269 Trapezoid method: 0.002246 Trapezoid method: 0.002223 Trapezoid method: 0.002200 Trapezoid method: 0.002178 Trapezoid method: 0.002155 Trapezoid method: 0.002132 Trapezoid method: 0.002110 Trapezoid method: 0.002087 Trapezoid method: 0.002065 Trapezoid method: 0.002043 Trapezoid method: 0.002020 Trapezoid method: 0.001998 Trapezoid method: 0.001976 Trapezoid method: 0.001954 Trapezoid method: 0.001932 Trapezoid method: 0.001911 Trapezoid method: 0.001889 Trapezoid method: 0.001867 Trapezoid method: 0.001846 Trapezoid method: 0.001825 Trapezoid method: 0.001803 Trapezoid method: 0.001782 Trapezoid method: 0.001761 Trapezoid method: 0.001741 Trapezoid method: 0.001720 Trapezoid method: 0.001699 Trapezoid method: 0.001679 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Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004
fprintf('Simpson''s composite method: %f\n', t_simp);
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Simpson's composite method: 0.001300 Simpson's composite method: 0.001318 Simpson's composite method: 0.001336 Simpson's composite method: 0.001353 Simpson's composite method: 0.001371 Simpson's composite method: 0.001389 Simpson's composite method: 0.001408 Simpson's composite method: 0.001426 Simpson's composite method: 0.001445 Simpson's composite method: 0.001463 Simpson's composite method: 0.001482 Simpson's composite method: 0.001501 Simpson's composite method: 0.001520 Simpson's composite method: 0.001540 Simpson's composite method: 0.001559 Simpson's composite method: 0.001579 Simpson's composite method: 0.001599 Simpson's composite method: 0.001618 Simpson's composite method: 0.001638 Simpson's composite method: 0.001659 Simpson's composite method: 0.001679 Simpson's composite method: 0.001699 Simpson's composite method: 0.001720 Simpson's composite method: 0.001741 Simpson's composite method: 0.001761 Simpson's composite method: 0.001782 Simpson's composite method: 0.001803 Simpson's composite method: 0.001825 Simpson's composite method: 0.001846 Simpson's composite method: 0.001867 Simpson's composite method: 0.001889 Simpson's composite method: 0.001911 Simpson's composite method: 0.001932 Simpson's composite method: 0.001954 Simpson's composite method: 0.001976 Simpson's composite method: 0.001998 Simpson's composite method: 0.002020 Simpson's composite method: 0.002043 Simpson's composite method: 0.002065 Simpson's composite method: 0.002087 Simpson's composite method: 0.002110 Simpson's composite method: 0.002132 Simpson's composite method: 0.002155 Simpson's composite method: 0.002178 Simpson's composite method: 0.002200 Simpson's composite method: 0.002223 Simpson's composite method: 0.002246 Simpson's composite method: 0.002269 Simpson's composite method: 0.002292 Simpson's composite method: 0.002315 Simpson's composite method: 0.002338 Simpson's composite method: 0.002361 Simpson's composite method: 0.002384 Simpson's composite method: 0.002408 Simpson's composite method: 0.002431 Simpson's composite method: 0.002454 Simpson's composite method: 0.002477 Simpson's composite method: 0.002500 Simpson's composite method: 0.002524 Simpson's composite method: 0.002547 Simpson's composite method: 0.002570 Simpson's composite method: 0.002593 Simpson's composite method: 0.002616 Simpson's composite method: 0.002640 Simpson's composite method: 0.002663 Simpson's composite method: 0.002686 Simpson's composite method: 0.002709 Simpson's composite method: 0.002732 Simpson's composite method: 0.002755 Simpson's composite method: 0.002778 Simpson's composite method: 0.002800 Simpson's composite method: 0.002823 Simpson's composite method: 0.002846 Simpson's composite method: 0.002868 Simpson's composite method: 0.002891 Simpson's composite method: 0.002913 Simpson's composite method: 0.002936 Simpson's composite method: 0.002958 Simpson's composite method: 0.002980 Simpson's composite method: 0.003002 Simpson's composite method: 0.003024 Simpson's composite method: 0.003045 Simpson's composite method: 0.003067 Simpson's composite method: 0.003088 Simpson's composite method: 0.003110 Simpson's composite method: 0.003131 Simpson's composite method: 0.003152 Simpson's composite method: 0.003172 Simpson's composite method: 0.003193 Simpson's composite method: 0.003213 Simpson's composite method: 0.003234 Simpson's composite method: 0.003254 Simpson's composite method: 0.003273 Simpson's composite method: 0.003293 Simpson's composite method: 0.003312 Simpson's composite method: 0.003332 Simpson's composite method: 0.003351 Simpson's composite method: 0.003369 Simpson's composite method: 0.003388 Simpson's composite method: 0.003406 Simpson's composite method: 0.003424 Simpson's composite method: 0.003442 Simpson's composite method: 0.003459 Simpson's composite method: 0.003476 Simpson's composite method: 0.003493 Simpson's composite method: 0.003510 Simpson's composite method: 0.003526 Simpson's composite method: 0.003542 Simpson's composite method: 0.003558 Simpson's composite method: 0.003573 Simpson's composite method: 0.003589 Simpson's composite method: 0.003603 Simpson's composite method: 0.003618 Simpson's composite method: 0.003632 Simpson's composite method: 0.003646 Simpson's composite method: 0.003659 Simpson's composite method: 0.003673 Simpson's composite method: 0.003685 Simpson's composite method: 0.003698 Simpson's composite method: 0.003710 Simpson's composite method: 0.003722 Simpson's composite method: 0.003733 Simpson's composite method: 0.003744 Simpson's composite method: 0.003755 Simpson's composite method: 0.003765 Simpson's composite method: 0.003775 Simpson's composite method: 0.003784 Simpson's composite method: 0.003794 Simpson's composite method: 0.003802 Simpson's composite method: 0.003811 Simpson's composite method: 0.003819 Simpson's composite method: 0.003826 Simpson's composite method: 0.003833 Simpson's composite method: 0.003840 Simpson's composite method: 0.003846 Simpson's composite method: 0.003852 Simpson's composite method: 0.003858 Simpson's composite method: 0.003863 Simpson's composite method: 0.003868 Simpson's composite method: 0.003872 Simpson's composite method: 0.003876 Simpson's composite method: 0.003879 Simpson's composite method: 0.003882 Simpson's composite method: 0.003885 Simpson's composite method: 0.003887 Simpson's composite method: 0.003888 Simpson's composite method: 0.003890 Simpson's composite method: 0.003891 Simpson's composite method: 0.003891 Simpson's composite method: 0.003891 Simpson's composite method: 0.003891 Simpson's composite method: 0.003890 Simpson's composite method: 0.003888 Simpson's composite method: 0.003887 Simpson's composite method: 0.003885 Simpson's composite method: 0.003882 Simpson's composite method: 0.003879 Simpson's composite method: 0.003876 Simpson's composite method: 0.003872 Simpson's composite method: 0.003868 Simpson's composite method: 0.003863 Simpson's composite method: 0.003858 Simpson's composite method: 0.003852 Simpson's composite method: 0.003846 Simpson's composite method: 0.003840 Simpson's composite method: 0.003833 Simpson's composite method: 0.003826 Simpson's composite method: 0.003819 Simpson's composite method: 0.003811 Simpson's composite method: 0.003802 Simpson's composite method: 0.003794 Simpson's composite method: 0.003784 Simpson's composite method: 0.003775 Simpson's composite method: 0.003765 Simpson's composite method: 0.003755 Simpson's composite method: 0.003744 Simpson's composite method: 0.003733 Simpson's composite method: 0.003722 Simpson's composite method: 0.003710 Simpson's composite method: 0.003698 Simpson's composite method: 0.003685 Simpson's composite method: 0.003673 Simpson's composite method: 0.003659 Simpson's composite method: 0.003646 Simpson's composite method: 0.003632 Simpson's composite method: 0.003618 Simpson's composite method: 0.003603 Simpson's composite method: 0.003589 Simpson's composite method: 0.003573 Simpson's composite method: 0.003558 Simpson's composite method: 0.003542 Simpson's composite method: 0.003526 Simpson's composite method: 0.003510 Simpson's composite method: 0.003493 Simpson's composite method: 0.003476 Simpson's composite method: 0.003459 Simpson's composite method: 0.003442 Simpson's composite method: 0.003424 Simpson's composite method: 0.003406 Simpson's composite method: 0.003388 Simpson's composite method: 0.003369 Simpson's composite method: 0.003351 Simpson's composite method: 0.003332 Simpson's composite method: 0.003312 Simpson's composite method: 0.003293 Simpson's composite method: 0.003273 Simpson's composite method: 0.003254 Simpson's composite method: 0.003234 Simpson's composite method: 0.003213 Simpson's composite method: 0.003193 Simpson's composite method: 0.003172 Simpson's composite method: 0.003152 Simpson's composite method: 0.003131 Simpson's composite method: 0.003110 Simpson's composite method: 0.003088 Simpson's composite method: 0.003067 Simpson's composite method: 0.003045 Simpson's composite method: 0.003024 Simpson's composite method: 0.003002 Simpson's composite method: 0.002980 Simpson's composite method: 0.002958 Simpson's composite method: 0.002936 Simpson's composite method: 0.002913 Simpson's composite method: 0.002891 Simpson's composite method: 0.002868 Simpson's composite method: 0.002846 Simpson's composite method: 0.002823 Simpson's composite method: 0.002800 Simpson's composite method: 0.002778 Simpson's composite method: 0.002755 Simpson's composite method: 0.002732 Simpson's composite method: 0.002709 Simpson's composite method: 0.002686 Simpson's composite method: 0.002663 Simpson's composite method: 0.002640 Simpson's composite method: 0.002616 Simpson's composite method: 0.002593 Simpson's composite method: 0.002570 Simpson's composite method: 0.002547 Simpson's composite method: 0.002524 Simpson's composite method: 0.002500 Simpson's composite method: 0.002477 Simpson's composite method: 0.002454 Simpson's composite method: 0.002431 Simpson's composite method: 0.002408 Simpson's composite method: 0.002384 Simpson's composite method: 0.002361 Simpson's composite method: 0.002338 Simpson's composite method: 0.002315 Simpson's composite method: 0.002292 Simpson's composite method: 0.002269 Simpson's composite method: 0.002246 Simpson's composite method: 0.002223 Simpson's composite method: 0.002200 Simpson's composite method: 0.002178 Simpson's composite method: 0.002155 Simpson's composite method: 0.002132 Simpson's composite method: 0.002110 Simpson's composite method: 0.002087 Simpson's composite method: 0.002065 Simpson's composite method: 0.002043 Simpson's composite method: 0.002020 Simpson's composite method: 0.001998 Simpson's composite method: 0.001976 Simpson's composite method: 0.001954 Simpson's composite method: 0.001932 Simpson's composite method: 0.001911 Simpson's composite method: 0.001889 Simpson's composite method: 0.001867 Simpson's composite method: 0.001846 Simpson's composite method: 0.001825 Simpson's composite method: 0.001803 Simpson's composite method: 0.001782 Simpson's composite method: 0.001761 Simpson's composite method: 0.001741 Simpson's composite method: 0.001720 Simpson's composite method: 0.001699 Simpson's composite method: 0.001679 Simpson's composite method: 0.001659 Simpson's composite method: 0.001638 Simpson's composite method: 0.001618 Simpson's composite method: 0.001599 Simpson's composite method: 0.001579 Simpson's composite method: 0.001559 Simpson's composite method: 0.001540 Simpson's composite method: 0.001520 Simpson's composite method: 0.001501 Simpson's composite method: 0.001482 Simpson's composite method: 0.001463 Simpson's composite method: 0.001445 Simpson's composite method: 0.001426 Simpson's composite method: 0.001408 Simpson's composite method: 0.001389 Simpson's composite method: 0.001371 Simpson's composite method: 0.001353 Simpson's composite method: 0.001336 Simpson's composite method: 0.001318 Simpson's composite method: 0.001300 Simpson's composite method: 0.001283 Simpson's composite method: 0.001266 Simpson's composite method: 0.001249 Simpson's composite method: 0.001232 Simpson's composite method: 0.001215 Simpson's composite method: 0.001199 Simpson's composite method: 0.001182 Simpson's composite method: 0.001166 Simpson's composite method: 0.001150 Simpson's composite method: 0.001134 Simpson's composite method: 0.001119 Simpson's composite method: 0.001103 Simpson's composite method: 0.001088 Simpson's composite method: 0.001072 Simpson's composite method: 0.001057 Simpson's composite method: 0.001042 Simpson's composite method: 0.001028 Simpson's composite method: 0.001013 Simpson's composite method: 0.000999 Simpson's composite method: 0.000984 Simpson's composite method: 0.000970 Simpson's composite method: 0.000956 Simpson's composite method: 0.000942 Simpson's composite method: 0.000929 Simpson's composite method: 0.000915 Simpson's composite method: 0.000902 Simpson's composite method: 0.000889 Simpson's composite method: 0.000876 Simpson's composite method: 0.000863 Simpson's composite method: 0.000850 Simpson's composite method: 0.000837 Simpson's composite method: 0.000825 Simpson's composite method: 0.000813 Simpson's composite method: 0.000801 Simpson's composite method: 0.000789 Simpson's composite method: 0.000777 Simpson's composite method: 0.000765 Simpson's composite method: 0.000754 Simpson's composite method: 0.000742 Simpson's composite method: 0.000731 Simpson's composite method: 0.000720 Simpson's composite method: 0.000709 Simpson's composite method: 0.000698 Simpson's composite method: 0.000687 Simpson's composite method: 0.000677 Simpson's composite method: 0.000666 Simpson's composite method: 0.000656 Simpson's composite method: 0.000646 Simpson's composite method: 0.000636 Simpson's composite method: 0.000626 Simpson's composite method: 0.000616 Simpson's composite method: 0.000607 Simpson's composite method: 0.000597 Simpson's composite method: 0.000588 Simpson's composite method: 0.000579 Simpson's composite method: 0.000570 Simpson's composite method: 0.000561 Simpson's composite method: 0.000552 Simpson's composite method: 0.000543 Simpson's composite method: 0.000534 Simpson's composite method: 0.000526 Simpson's composite method: 0.000518 Simpson's composite method: 0.000509 Simpson's composite method: 0.000501 Simpson's composite method: 0.000493 Simpson's composite method: 0.000485 Simpson's composite method: 0.000477 Simpson's composite method: 0.000470 Simpson's composite method: 0.000462 Simpson's composite method: 0.000455 Simpson's composite method: 0.000447 Simpson's composite method: 0.000440 Simpson's composite method: 0.000433 Simpson's composite method: 0.000426 Simpson's composite method: 0.000419 Simpson's composite method: 0.000412 Simpson's composite method: 0.000406 Simpson's composite method: 0.000399 Simpson's composite method: 0.000392 Simpson's composite method: 0.000386 Simpson's composite method: 0.000380 Simpson's composite method: 0.000373 Simpson's composite method: 0.000367 Simpson's composite method: 0.000361 Simpson's composite method: 0.000355 Simpson's composite method: 0.000349 Simpson's composite method: 0.000344 Simpson's composite method: 0.000338 Simpson's composite method: 0.000332 Simpson's composite method: 0.000327 Simpson's composite method: 0.000321 Simpson's composite method: 0.000316 Simpson's composite method: 0.000311 Simpson's composite method: 0.000306 Simpson's composite method: 0.000301 Simpson's composite method: 0.000296 Simpson's composite method: 0.000291 Simpson's composite method: 0.000286 Simpson's composite method: 0.000281 Simpson's composite method: 0.000276 Simpson's composite method: 0.000272 Simpson's composite method: 0.000267 Simpson's composite method: 0.000263 Simpson's composite method: 0.000258 Simpson's composite method: 0.000254 Simpson's composite method: 0.000250 Simpson's composite method: 0.000245 Simpson's composite method: 0.000241 Simpson's composite method: 0.000237 Simpson's composite method: 0.000233 Simpson's composite method: 0.000229 Simpson's composite method: 0.000225 Simpson's composite method: 0.000222 Simpson's composite method: 0.000218 Simpson's composite method: 0.000214 Simpson's composite method: 0.000210 Simpson's composite method: 0.000207 Simpson's composite method: 0.000203 Simpson's composite method: 0.000200 Simpson's composite method: 0.000197 Simpson's composite method: 0.000193 Simpson's composite method: 0.000190 Simpson's composite method: 0.000187 Simpson's composite method: 0.000183 Simpson's composite method: 0.000180 Simpson's composite method: 0.000177 Simpson's composite method: 0.000174 Simpson's composite method: 0.000171 Simpson's composite method: 0.000168 Simpson's composite method: 0.000165 Simpson's composite method: 0.000163 Simpson's composite method: 0.000160 Simpson's composite method: 0.000157 Simpson's composite method: 0.000154 Simpson's composite method: 0.000152 Simpson's composite method: 0.000149 Simpson's composite method: 0.000147 Simpson's composite method: 0.000144 Simpson's composite method: 0.000142 Simpson's composite method: 0.000139 Simpson's composite method: 0.000137 Simpson's composite method: 0.000134 Simpson's composite method: 0.000132 Simpson's composite method: 0.000130 Simpson's composite method: 0.000128 Simpson's composite method: 0.000125 Simpson's composite method: 0.000123 Simpson's composite method: 0.000121 Simpson's composite method: 0.000119 Simpson's composite method: 0.000117 Simpson's composite method: 0.000115 Simpson's composite method: 0.000113 Simpson's composite method: 0.000111 Simpson's composite method: 0.000109 Simpson's composite method: 0.000107 Simpson's composite method: 0.000105 Simpson's composite method: 0.000104 Simpson's composite method: 0.000102 Simpson's composite method: 0.000100 Simpson's composite method: 0.000098 Simpson's composite method: 0.000097 Simpson's composite method: 0.000095 Simpson's composite method: 0.000093 Simpson's composite method: 0.000092 Simpson's composite method: 0.000090 Simpson's composite method: 0.000089 Simpson's composite method: 0.000087 Simpson's composite method: 0.000086 Simpson's composite method: 0.000084 Simpson's composite method: 0.000083 Simpson's composite method: 0.000081 Simpson's composite method: 0.000080 Simpson's composite method: 0.000078 Simpson's composite method: 0.000077 Simpson's composite method: 0.000076 Simpson's composite method: 0.000074 Simpson's composite method: 0.000073 Simpson's composite method: 0.000072 Simpson's composite method: 0.000071 Simpson's composite method: 0.000069 Simpson's composite method: 0.000068 Simpson's composite method: 0.000067 Simpson's composite method: 0.000066 Simpson's composite method: 0.000065 Simpson's composite method: 0.000064 Simpson's composite method: 0.000063 Simpson's composite method: 0.000062 Simpson's composite method: 0.000060 Simpson's composite method: 0.000059 Simpson's composite method: 0.000058 Simpson's composite method: 0.000057 Simpson's composite method: 0.000056 Simpson's composite method: 0.000055 Simpson's composite method: 0.000054 Simpson's composite method: 0.000054 Simpson's composite method: 0.000053 Simpson's composite method: 0.000052 Simpson's composite method: 0.000051 Simpson's composite method: 0.000050 Simpson's composite method: 0.000049 Simpson's composite method: 0.000048 Simpson's composite method: 0.000047 Simpson's composite method: 0.000047 Simpson's composite method: 0.000046 Simpson's composite method: 0.000045 Simpson's composite method: 0.000044 Simpson's composite method: 0.000043 Simpson's composite method: 0.000043 Simpson's composite method: 0.000042 Simpson's composite method: 0.000041 Simpson's composite method: 0.000041 Simpson's composite method: 0.000040 Simpson's composite method: 0.000039 Simpson's composite method: 0.000039 Simpson's composite method: 0.000038 Simpson's composite method: 0.000037 Simpson's composite method: 0.000037 Simpson's composite method: 0.000036 Simpson's composite method: 0.000035 Simpson's composite method: 0.000035 Simpson's composite method: 0.000034 Simpson's composite method: 0.000034 Simpson's composite method: 0.000033 Simpson's composite method: 0.000032 Simpson's composite method: 0.000032 Simpson's composite method: 0.000031 Simpson's composite method: 0.000031 Simpson's composite method: 0.000030 Simpson's composite method: 0.000030 Simpson's composite method: 0.000029 Simpson's composite method: 0.000029 Simpson's composite method: 0.000028 Simpson's composite method: 0.000028 Simpson's composite method: 0.000027 Simpson's composite method: 0.000027 Simpson's composite method: 0.000026 Simpson's composite method: 0.000026 Simpson's composite method: 0.000026 Simpson's composite method: 0.000025 Simpson's composite method: 0.000025 Simpson's composite method: 0.000024 Simpson's composite method: 0.000024 Simpson's composite method: 0.000023 Simpson's composite method: 0.000023 Simpson's composite method: 0.000023 Simpson's composite method: 0.000022 Simpson's composite method: 0.000022 Simpson's composite method: 0.000022 Simpson's composite method: 0.000021 Simpson's composite method: 0.000021 Simpson's composite method: 0.000020 Simpson's composite method: 0.000020 Simpson's composite method: 0.000020 Simpson's composite method: 0.000019 Simpson's composite method: 0.000019 Simpson's composite method: 0.000019 Simpson's composite method: 0.000019 Simpson's composite method: 0.000018 Simpson's composite method: 0.000018 Simpson's composite method: 0.000018 Simpson's composite method: 0.000017 Simpson's composite method: 0.000017 Simpson's composite method: 0.000017 Simpson's composite method: 0.000016 Simpson's composite method: 0.000016 Simpson's composite method: 0.000016 Simpson's composite method: 0.000016 Simpson's composite method: 0.000015 Simpson's composite method: 0.000015 Simpson's composite method: 0.000015 Simpson's composite method: 0.000015 Simpson's composite method: 0.000014 Simpson's composite method: 0.000014 Simpson's composite method: 0.000014 Simpson's composite method: 0.000014 Simpson's composite method: 0.000013 Simpson's composite method: 0.000013 Simpson's composite method: 0.000013 Simpson's composite method: 0.000013 Simpson's composite method: 0.000013 Simpson's composite method: 0.000012 Simpson's composite method: 0.000012 Simpson's composite method: 0.000012 Simpson's composite method: 0.000012 Simpson's composite method: 0.000012 Simpson's composite method: 0.000011 Simpson's composite method: 0.000011 Simpson's composite method: 0.000011 Simpson's composite method: 0.000011 Simpson's composite method: 0.000011 Simpson's composite method: 0.000010 Simpson's composite method: 0.000010 Simpson's composite method: 0.000010 Simpson's composite method: 0.000010 Simpson's composite method: 0.000010 Simpson's composite method: 0.000010 Simpson's composite method: 0.000010 Simpson's composite method: 0.000009 Simpson's composite method: 0.000009 Simpson's composite method: 0.000009 Simpson's composite method: 0.000009 Simpson's composite method: 0.000009 Simpson's composite method: 0.000009 Simpson's composite method: 0.000008 Simpson's composite method: 0.000008 Simpson's composite method: 0.000008 Simpson's composite method: 0.000008 Simpson's composite method: 0.000008 Simpson's composite method: 0.000008 Simpson's composite method: 0.000008 Simpson's composite method: 0.000008 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004
fprintf('Exact value: %f\n', t_exact);
Exact value: 1.000000
fprintf('\nNormal distribution integral:\n');
Normal distribution integral:
fprintf('Trapezoid method: %f\n', norm_trap);
Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000000 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000001 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000002 Trapezoid method: 0.000003 Trapezoid method: 0.000003 Trapezoid method: 0.000003 Trapezoid method: 0.000003 Trapezoid method: 0.000003 Trapezoid method: 0.000003 Trapezoid method: 0.000003 Trapezoid method: 0.000003 Trapezoid method: 0.000003 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000004 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000005 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000006 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000007 Trapezoid method: 0.000008 Trapezoid method: 0.000008 Trapezoid method: 0.000008 Trapezoid method: 0.000009 Trapezoid method: 0.000009 Trapezoid method: 0.000009 Trapezoid method: 0.000010 Trapezoid method: 0.000010 Trapezoid method: 0.000010 Trapezoid method: 0.000011 Trapezoid method: 0.000011 Trapezoid method: 0.000011 Trapezoid method: 0.000012 Trapezoid method: 0.000012 Trapezoid method: 0.000013 Trapezoid method: 0.000013 Trapezoid method: 0.000013 Trapezoid method: 0.000014 Trapezoid method: 0.000014 Trapezoid method: 0.000015 Trapezoid method: 0.000015 Trapezoid method: 0.000016 Trapezoid method: 0.000016 Trapezoid method: 0.000017 Trapezoid method: 0.000018 Trapezoid method: 0.000018 Trapezoid method: 0.000019 Trapezoid method: 0.000019 Trapezoid method: 0.000020 Trapezoid method: 0.000021 Trapezoid method: 0.000021 Trapezoid method: 0.000022 Trapezoid method: 0.000023 Trapezoid method: 0.000023 Trapezoid method: 0.000024 Trapezoid method: 0.000025 Trapezoid method: 0.000026 Trapezoid method: 0.000027 Trapezoid method: 0.000028 Trapezoid method: 0.000028 Trapezoid method: 0.000029 Trapezoid method: 0.000030 Trapezoid method: 0.000031 Trapezoid method: 0.000032 Trapezoid method: 0.000033 Trapezoid method: 0.000034 Trapezoid method: 0.000035 Trapezoid method: 0.000036 Trapezoid method: 0.000038 Trapezoid method: 0.000039 Trapezoid method: 0.000040 Trapezoid method: 0.000041 Trapezoid method: 0.000042 Trapezoid method: 0.000044 Trapezoid method: 0.000045 Trapezoid method: 0.000046 Trapezoid method: 0.000048 Trapezoid method: 0.000049 Trapezoid method: 0.000051 Trapezoid method: 0.000052 Trapezoid method: 0.000054 Trapezoid method: 0.000055 Trapezoid method: 0.000057 Trapezoid method: 0.000059 Trapezoid method: 0.000060 Trapezoid method: 0.000062 Trapezoid method: 0.000064 Trapezoid method: 0.000066 Trapezoid method: 0.000068 Trapezoid method: 0.000070 Trapezoid method: 0.000072 Trapezoid method: 0.000074 Trapezoid method: 0.000076 Trapezoid method: 0.000078 Trapezoid method: 0.000080 Trapezoid method: 0.000083 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Simpson's composite method: 0.000072 Simpson's composite method: 0.000070 Simpson's composite method: 0.000068 Simpson's composite method: 0.000066 Simpson's composite method: 0.000064 Simpson's composite method: 0.000062 Simpson's composite method: 0.000060 Simpson's composite method: 0.000059 Simpson's composite method: 0.000057 Simpson's composite method: 0.000055 Simpson's composite method: 0.000054 Simpson's composite method: 0.000052 Simpson's composite method: 0.000051 Simpson's composite method: 0.000049 Simpson's composite method: 0.000048 Simpson's composite method: 0.000046 Simpson's composite method: 0.000045 Simpson's composite method: 0.000044 Simpson's composite method: 0.000042 Simpson's composite method: 0.000041 Simpson's composite method: 0.000040 Simpson's composite method: 0.000039 Simpson's composite method: 0.000038 Simpson's composite method: 0.000036 Simpson's composite method: 0.000035 Simpson's composite method: 0.000034 Simpson's composite method: 0.000033 Simpson's composite method: 0.000032 Simpson's composite method: 0.000031 Simpson's composite method: 0.000030 Simpson's composite method: 0.000029 Simpson's composite method: 0.000028 Simpson's composite method: 0.000028 Simpson's composite method: 0.000027 Simpson's composite method: 0.000026 Simpson's composite method: 0.000025 Simpson's composite method: 0.000024 Simpson's composite method: 0.000023 Simpson's composite method: 0.000023 Simpson's composite method: 0.000022 Simpson's composite method: 0.000021 Simpson's composite method: 0.000021 Simpson's composite method: 0.000020 Simpson's composite method: 0.000019 Simpson's composite method: 0.000019 Simpson's composite method: 0.000018 Simpson's composite method: 0.000018 Simpson's composite method: 0.000017 Simpson's composite method: 0.000016 Simpson's composite method: 0.000016 Simpson's composite method: 0.000015 Simpson's composite method: 0.000015 Simpson's composite method: 0.000014 Simpson's composite method: 0.000014 Simpson's composite method: 0.000013 Simpson's composite method: 0.000013 Simpson's composite method: 0.000013 Simpson's composite method: 0.000012 Simpson's composite method: 0.000012 Simpson's composite method: 0.000011 Simpson's composite method: 0.000011 Simpson's composite method: 0.000011 Simpson's composite method: 0.000010 Simpson's composite method: 0.000010 Simpson's composite method: 0.000010 Simpson's composite method: 0.000009 Simpson's composite method: 0.000009 Simpson's composite method: 0.000009 Simpson's composite method: 0.000008 Simpson's composite method: 0.000008 Simpson's composite method: 0.000008 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000007 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000006 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000005 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000004 Simpson's composite method: 0.000003 Simpson's composite method: 0.000003 Simpson's composite method: 0.000003 Simpson's composite method: 0.000003 Simpson's composite method: 0.000003 Simpson's composite method: 0.000003 Simpson's composite method: 0.000003 Simpson's composite method: 0.000003 Simpson's composite method: 0.000003 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000002 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000001 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000 Simpson's composite method: 0.000000
fprintf('Exact value: %f\n', norm_exact);
Exact value: 1.000000
% Plot the approximations and the exact values
figure;
subplot(2,1,1);
length(x)
ans = 2001
length(f_t(x, 10))
ans = 2001
length(x(1:end-1))
ans = 2000
length(t_trap)
ans = 1000
length(x(1:2:end))
ans = 1001
length(t_simp)
ans = 1000
plot(x, f_t(x, 10), 'k', x(1:end-1), t_trap, 'r', x(1:2:end), t_simp, 'b--');
Error using plot
Vectors must be the same length.
legend('Exact', 'Trapezoid', 'Simpson''s');
title('T-distribution Probability Density Function');
subplot(2,1,2);
plot(x, f_norm(x), 'k', x(1:end-1), norm_trap, 'r', x(1:2:end), norm_simp, 'b--');
legend('Exact', 'Trapezoid', 'Simpson''s');
title('Normal Distribution Probability Density Function');
  2 个评论
Shoaib Salim
Shoaib Salim 2023-4-6
How can i edit to get rid of this error as i tried above mentioned steps but not succeeded
Oguz Kaan Hancioglu
Oguz Kaan Hancioglu 2023-4-7
it is caused by you assigning the same variable name different array.
x = linspace(a, b, n+1); % length of x = 1001
t_trap = h/2 * (f_t(x(1:end-1), 10) + f_t(x(2:end), 10)); % length of t_trap 1001
norm_trap = h/2 * (f_norm(x(1:end-1)) + f_norm(x(2:end)));
% Use Simpson's composite method to approximate the integrals
x = linspace(a, b, 2*n+1); % length of x 2001
% now length of x = 2001 but t_trap = 1001
plot(x, f_t(x, 10), 'k', x(1:end-1), t_trap, 'r', x(1:2:end), t_simp, 'b--');
plesae assign different names for x arrays.

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更多回答（1 个）

Joe Vinciguerra
Joe Vinciguerra 2023-4-6
The x and y vectors for a plot need to be the same length. You were close, but looks like you had a typo. Try this:
% Plot the approximations and the exact values
figure;
subplot(2,1,1);
% plot(x, f_t(x, 10), 'k', x(1:end-1), t_trap, 'r', x(1:2:end), t_simp, 'b--');
plot(x, f_t(x, 10), 'k', x(1:2:end-1), t_trap, 'r', x(1:2:end-1), t_simp, 'b--');
legend('Exact', 'Trapezoid', 'Simpson''s');
title('T-distribution Probability Density Function');
subplot(2,1,2);
% plot(x, f_norm(x), 'k', x(1:end-1), norm_trap, 'r', x(1:2:end), norm_simp, 'b--');
plot(x, f_norm(x), 'k', x(1:2:end-1), norm_trap, 'r', x(1:2:end-1), norm_simp, 'b--');
legend('Exact', 'Trapezoid', 'Simpson''s');
title('Normal Distribution Probability Density Function');
For more details: 2-D line plot - MATLAB plot (mathworks.com)

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