How to implement a multiple nested "nested loop" faster in MATLAB

I want to implement the following formula in MATLAB. This formula represents point spread function(PSF) from concentrating point to grid point :

I have to implement this formula multiple time in MATLAB. So, running speed is very important to me. and are vectors with 40000 elements (albeit, we can represent and in 2D format) . Also, is a vector with 1400 elements and has 51 elements. For each , we have to sum over all and and also, over all (because we want to calculate PSF from to all grid points ). Also we want to implment this PSF for all . in the worst implementation, we have 4 for-loop with lots of elements. Definitely, it decrease speed by a huge amount. So, how can I avoid this?

Answer 1

[Subhajyoti]

Hi Moh,

It is my understanding, that you want to optimize the iterations required to the reduce the running speed of the “Point Spread Function (PSF)”.

To efficiently implement the “point spread function (PSF)” calculation in MATLAB, you'll want to avoid nested loops as much as possible and leverage MATLAB's matrix operations, which are highly optimized for performance.

Here’s a strategy to achieve this:

• Vectorization: Replace for-loops with matrix operations where possible.
• Preallocation: Pre-allocate matrices to avoid dynamic resizing within loops.
• Parallel Computing: Utilize MATLAB's Parallel Computing Toolbox if available.

Here’s MATLAB code snippet to illustrate the approach:

% Assuming vectors x and y each have 40000 elements, z has 1400 elements, and w has 51 elements
x = rand(40000, 1);
y = rand(40000, 1);
z = rand(1400, 1);
w = rand(51, 1);

- 2D Grid Creation: We use “meshgrid” to create a 2D grid of x and y, then flatten these grids to perform vectorized operations.

% Create 2D grid for x and y
[x_grid, y_grid] = meshgrid(x, y);
x_grid = x_grid(:);
y_grid = y_grid(:);

- PSF Function: This is a placeholder function; replace it with your actual PSF formula.

% Define your PSF function here. This is a placeholder example.
% Replace this with your actual PSF formula
PSF_func = @(x, y, z, w) exp(-(x - z).^2 - (y - w).^2);

- Preallocation: The PSF_result matrix is preallocated to store the results of the PSF calculations.

% Preallocate the result matrix
PSF_result = zeros(length(z), length(w), numel(x_grid));

- Nested Loops for z and w: The outer loops iterate over z and w, and we compute the PSF for each combination using the flattened x and y grids.

% Vectorized calculation using implicit expansion
for i = 1:length(z)
    for j = 1:length(w)
        PSF_result(i, j, :) = PSF_func(x_grid, y_grid, z(i), w(j));
    end
end

- Reshape: The result is reshaped to match the original dimensions.

% Reshape result if needed
PSF_result = reshape(PSF_result, [length(z), length(w), size(x, 1), size(y, 1)]);

% Display the result size to confirm correctness
disp(size(PSF_result));

Parallel Computing Example:

% Example of using parfor to parallelize outer loops
parfor i = 1:length(z)
    for j = 1:length(w)
        PSF_result(i, j, :) = PSF_func(x_grid, y_grid, z(i), w(j));
    end
end

This uses MATLAB's “Parallel Computing Toolbox” to parallelize the outer loops, potentially speeding up the computation further.

Thus, by using vectorized operations and potentially parallel computing, you can significantly speed up the calculation of the PSF in MATLAB.

Note: Adjust the example to fit your specific formula and dimensions

For more details on the “Parallel for-loop”, kindly go through the following MathWorks documentation.

Link: https://in.mathworks.com/help/matlab/ref/parfor.html

Hope the above information is helpful.