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# How to speed up large double sums in a table?

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I am calculating a 3-by-3 matrix whose elements are given as follows:

$$M_{mn} = \frac{1}{N}\sum_{i=1}^N \sum_{j=1}^N (r^i_m - r^j_m)(r^i_n - r^j_n) \tag{1}$$

where $$N$$ is the total number of particles, and $$r^i_m$$ denotes the m-th component of the i-th particle's vector. The sums can be computationally expensive as often $$N$$ is around $$10000.$$

Below is my implementation of the matrix $$M$$ in Mathematica:

matrixM = Table[(1./Nnpart)*
Sum[Sum[(Part[vecs, i, m] - Part[vecs, j, m])*(Part[vecs, i, n] -
Part[vecs, j, n]), {j, 1, Nnpart}], {i, 1, Nnpart}], {m, 3}, {n, 3}];


and here vecs is the array of all particle vectors (so one line per particle and each line of the array is a vector of 3 components).

• How could I speed up computations of this nature? Is there possibly a bottleneck in my efficiency due to the way I generate the matrix using Table or the fact that I access the components stored in a big array using Part? Any advice would be very helpful.

I am calculating a 3-by-3 matrix whose elements are given as follows:

$$M_{mn} = \frac{1}{N}\sum_{i=1}^N \sum_{j=1}^N (r^i_m - r^j_m)(r^i_n - r^j_n) \tag{1}$$

where $$N$$ is the total number of particles, and $$r^i_m$$ denotes the m-th component of the i-th particle's vector. The sums can be computationally expensive as often $$N$$ is around $$10000.$$

Below is my implementation of the matrix $$M$$ in Mathematica:

matrixM = Table[(1./N)*
Sum[Sum[(Part[vecs, i, m] - Part[vecs, j, m])*(Part[vecs, i, n] -
Part[vecs, j, n]), {j, 1, N}], {i, 1, N}], {m, 3}, {n, 3}];


and here vecs is the array of all particle vectors (so one line per particle and each line of the array is a vector of 3 components).

• How could I speed up computations of this nature? Is there possibly a bottleneck in my efficiency due to the way I generate the matrix using Table or the fact that I access the components stored in a big array using Part? Any advice would be very helpful.

I am calculating a 3-by-3 matrix whose elements are given as follows:

$$M_{mn} = \frac{1}{N}\sum_{i=1}^N \sum_{j=1}^N (r^i_m - r^j_m)(r^i_n - r^j_n) \tag{1}$$

where $$N$$ is the total number of particles, and $$r^i_m$$ denotes the m-th component of the i-th particle's vector. The sums can be computationally expensive as often $$N$$ is around $$10000.$$

Below is my implementation of the matrix $$M$$ in Mathematica:

matrixM = Table[(1./npart)*
Sum[Sum[(Part[vecs, i, m] - Part[vecs, j, m])*(Part[vecs, i, n] -
Part[vecs, j, n]), {j, 1, npart}], {i, 1, npart}], {m, 3}, {n, 3}];


and here vecs is the array of all particle vectors (so one line per particle and each line of the array is a vector of 3 components).

• How could I speed up computations of this nature? Is there possibly a bottleneck in my efficiency due to the way I generate the matrix using Table or the fact that I access the components stored in a big array using Part? Any advice would be very helpful.
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# How to speed up large double sums in a table

I am calculating a 3-by-3 matrix whose elements are given as follows:

$$M_{mn} = \frac{1}{N}\sum_{i=1}^N \sum_{j=1}^N (r^i_m - r^j_m)(r^i_n - r^j_n) \tag{1}$$

where $$N$$ is the total number of particles, and $$r^i_m$$ denotes the m-th component of the i-th particle's vector. The sums can be computationally expensive as often $$N$$ is around $$10000.$$

Below is my implementation of the matrix $$M$$ in Mathematica:

matrixM = Table[(1./N)*
Sum[Sum[(Part[vecs, i, m] - Part[vecs, j, m])*(Part[vecs, i, n] -
Part[vecs, j, n]), {j, 1, N}], {i, 1, N}], {m, 3}, {n, 3}];


and here vecs is the array of all particle vectors (so one line per particle and each line of the array is a vector of 3 components).

• How could I speed up computations of this nature? Is there possibly a bottleneck in my efficiency due to the way I generate the matrix using Table or the fact that I access the components stored in a big array using Part? Any advice would be very helpful.