When working with large matrices in Julia, it is not uncommon to encounter performance issues. In this case, the input matrix of size 41924192 cannot be squared and the computation keeps looping for more than 7 minutes. In this article, we will explore three different approaches to solve this problem and determine which one is the most efficient.

## Approach 1: Optimize Matrix Multiplication

One possible reason for the slow computation is inefficient matrix multiplication. Julia provides several built-in functions for matrix multiplication, such as the dot product operator `*` and the `mul!` function. However, these functions may not be optimized for large matrices.

```
# Julia code
using LinearAlgebra
function square_matrix(matrix)
return matrix * matrix
end
# Call the function with the input matrix
input_matrix = rand(41924192, 41924192)
result = square_matrix(input_matrix)
```

This approach uses the built-in matrix multiplication operator `*` to square the input matrix. However, for large matrices, this operation may be slow. Therefore, we need to explore alternative approaches.

## Approach 2: Utilize Parallel Computing

Parallel computing can significantly improve the performance of matrix operations by distributing the computation across multiple cores or processors. Julia provides the `Distributed` module for parallel computing.

```
# Julia code
using Distributed
function square_matrix_parallel(matrix)
@distributed for i = 1:size(matrix, 1)
matrix[i, :] = matrix[i, :] * matrix
end
return matrix
end
# Call the function with the input matrix
input_matrix = rand(41924192, 41924192)
result = square_matrix_parallel(input_matrix)
```

This approach utilizes parallel computing to distribute the matrix multiplication across multiple cores or processors. By doing so, we can significantly reduce the computation time for large matrices.

## Approach 3: Optimize Memory Usage

Another possible reason for the slow computation is excessive memory usage. When working with large matrices, it is crucial to optimize memory allocation and deallocation to avoid performance issues.

```
# Julia code
function square_matrix_optimized(matrix)
result = similar(matrix)
mul!(result, matrix, matrix)
return result
end
# Call the function with the input matrix
input_matrix = rand(41924192, 41924192)
result = square_matrix_optimized(input_matrix)
```

This approach optimizes memory usage by pre-allocating the result matrix using the `similar` function. It then uses the `mul!` function to perform the matrix multiplication in-place, reducing memory overhead and improving performance.

After evaluating the three approaches, it is evident that Approach 3, which optimizes memory usage, is the most efficient solution for squaring a large matrix in Julia. By pre-allocating the result matrix and performing the multiplication in-place, we can significantly reduce memory overhead and computation time.