Azzera filtri
Azzera filtri

Numerical method MATLAB code

2 visualizzazioni (ultimi 30 giorni)
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');
  7 Commenti
Shoaib Salim
Shoaib Salim il 6 Apr 2023
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 il 7 Apr 2023
Modificato: Torsten il 7 Apr 2023
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.

Accedi per commentare.

Risposta accettata

Oguz Kaan Hancioglu
Oguz Kaan Hancioglu il 6 Apr 2023
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: 0.003293 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 Commenti
Shoaib Salim
Shoaib Salim il 6 Apr 2023
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 il 7 Apr 2023
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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Più risposte (1)

Joe Vinciguerra
Joe Vinciguerra il 6 Apr 2023
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');
  1 Commento
Shoaib Salim
Shoaib Salim il 6 Apr 2023
I tried this earlier but it does not make sense in plot

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