Different limits of function sequence

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Andrei 2023-10-25

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https://ww2.mathworks.cn/matlabcentral/answers/2038511-different-limits-of-function-sequence

评论： Andrei 2023-10-25

Write a function convergenceFunc(fn, f, a, b, n, convType) that takes as input arguments: a function fn(n,x), fn(n,x) = f_n(x) - and a function f considered as the limit of the sequence f_n(x) on [a,b] in the sense given by the string convType: it can be pointwise convergence, uniform convergence, mean-square convergence.

And at this point I am a bit puzzled, since I want to enter only fn(n,x) and write matlab function, whichtake fn and find limits in different senses according to convType. But I dont know what to chage in limit to obtain different funcions for different types of convergeance.

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Andrei 2023-10-25

For some functions there is only one type of convergeance - then convType can return error in case when expression doesnt convege in sense of convType. Also inside function I will evaluate errors which depend on convType

Andrei 2023-10-25

在 MATLAB Online 中打开

I have written my realization, but it is no automized

clc
clear
close all
a = 0;
b = 1;
n = 10;
fn = @(n, x) (2.*x.*n + (-1).^n.*x^2)./n;
f = @(x) 2.*x;
% Uniform Convergence
convergenceFunc(@(n, x) (2.*x.*n + ((-1).^n).*x.^2)./n , @(x) 2.*x, a, b, n, 'uniform');
% Mean Squared Convergence
convergenceFunc(@(n, x) (2.*x.*n + ((-1).^n).*x.^2)./n , @(x) 2.*x, a, b, n, 'mean_squared');
% Pointwise Convergence
convergenceFunc(@(n, x) (2.*x.*n + ((-1).^n).*x.^2)./n , @(x) 2.*x, a, b, n, 'pointwise');
function convergenceFunc(fn, f, a, b, n, convType)
    
    x = linspace(a, b, 1000);  
    y_fn = zeros(1000, n);    
    y_f = f(x);               
    figure
    for i = 1:n
        y_fn(:, i) = fn(i, x);  
        
        if strcmp(convType, 'uniform')
            diff_metric = max(abs(y_fn(:, i) - y_f));
            title_str = 'Uniform Convergence';
        elseif strcmp(convType, 'mean_squared')
            diff_metric = sqrt(mean((y_fn(:, i) - y_f).^2));
            title_str = 'Mean Squared Convergence';
        elseif strcmp(convType, 'pointwise')
            diff_metric = NaN;
            title_str = 'Pointwise Convergence';
        end
        
        
        plot(x, y_fn(:, i), 'b', x, y_f, 'r');
        title(title_str);
        legend(['f_', num2str(i)], 'f');
        xlabel('x');
        ylabel('y');
        title(['Metric: ', num2str(diff_metric)]);
        drawnow;
        hold on
        pause(0.5);
    end
end