Why shouldn't Matrix class's interface look like an array-of-array?

A: Some people build a Matrix class that has an operator[] that returns a reference to an Array object (or possibly to a raw array, shudder), & that Array object contain an operator[] which returns an element of the Matrix (for example a reference to a double). Therefore they access elements of the matrix via syntax such as m[i][j] instead of syntax like m(i,j).

The array-of-array solution clearly works, however it is less flexible than the operator () approach. Exclusively, there are easy performance tuning tricks which can be done along with the operator () approach which are more complicated in the [][] approach, and thus the [][] approach is more likely to lead to bad performance, at least in some of the cases.

For instance, the easiest way to implement the [][] approach is to employ a physical layout of the matrix as a dense matrix which is stored in row-major form. By contrast, the operator () approach entirely hides the physical layout of the matrix, and which can lead to better performance in some of the cases.

Put it this way: the operator() approach is never bad than, and sometimes better than, the [][] approach.

The operator() approach is never bad since it is simple to implement the dense, row-major physical layout utilizing the operator() approach, so while that configuration happens to be the optimal layout through a performance standpoint, the operator() approach is only as simple as the [][] approach (possibly the operator() approach is a tiny bit simpler, The operator() approach is at times better since whenever the optimal layout for a given application happens to be something other than row-major, dense the implementation is frequently significantly easier by the operator() approach compared to the [][] approach.

As an instance of when a physical layout makes a important difference, a recent project happened to access matrix elements in columns (i.e., the algorithm accesses all the elements in one column, then the elements in another, etc.), & if the physical layout is row- major, the accesses may "stride the cache". For instance, if the rows happen to be approximately as big as the processor's cache size, machine might end up along with "cache miss" for almost all element access. In this specific project, we got a 20% development in performance by altering the mapping from the logical layout (row, column) to the physical layout (row , column).

Certainly there are several examples of this kind of thing from numerical methods, & sparse matrices are an entire other dimension on this issue. As it is, generally, easier to implement a sparse matrix or swap row/column ordering via the operator() approach, the operator() approach nothing lose and might achieve something it has no down-side & a potential up-side. Use the operator() approach.