design the function function out = x_m(x,m) with input data matrix x of size 𝑁 × 𝑑 hat produces the data matrix of size 𝑁 × (𝑀 + 1) in matlab

To design the function x_m(x,m), follow these steps:

1. Initialize the output matrix out with size N x (M+1)
2. Copy the input data matrix x into the first d columns of out. This can be done using indexing and slicing, as shown in the code below.
3. Loop over the remaining columns of out, from column d+1 to column M+1.
4. For each column, raise the input data matrix x to the power of its column index minus d (or simply use the built-in power function in MATLAB) and set the corresponding column of out to the result. This can also be done using indexing and slicing.

Here's the code implementation in MATLAB:

main.m
function out = x_m(x,m)
    % Get the dimensions of the input data matrix x
    [N,d] = size(x);
    
    % Initialize the output matrix out
    out = zeros(N, M+1);
    
    % Copy the first d columns of x into the first d columns of out
    out(:,1:d) = x;
    
    % Loop over the remaining columns of out
    for i = d+1:M+1
        % Raise x to the power of (i-d) and set the corresponding column of out to the result
        out(:,i) = power(x,i-d);
    end
end
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You can now call this function with any input data matrix x and any value of M to produce the desired output matrix of size N x (M+1).