Element-wise power of sparse matrix: scalar vs matrix exponent

[tcunis]

Dear all,

I found what is, in my opinion, an inconsistent behaviour in computing the element-wise power of a sparse matrix with scalar exponent compared to the same power with matrix exponent:

• Taking a sparse zero scalar to a scalar power of one (e.g., DM(1,1).^1) returns a dense zero.
• Taking a sparse zero matrix element-wise to a scalar power of one (e.g., DM(2,2).^1) returns a sparse zero matrix.
• Taking a sparse zero matrix element-wise to a matrix power of ones (e.g., DM(2,2).^DM.ones(2,2)) returns a dense zero matrix.

I would think that all three expressions should return the same result (namely -- but that might be a separate discussion -- a sparse zero matrix).

Is this behaviour intended?

I am using CasADi v3.7.2 with MATLAB R2024a on an Apple M3 with MacOS Tahoe 26.5.1.

Thanks!

[mohammadrezwankhan]

The numerical results are consistent; the difference is in CasADi's stored sparsity pattern.

Binary operations currently dispatch as follows: scalar left operand to scalar_matrix, scalar right operand to matrix_scalar, otherwise to matrix_matrix (source).

• DM(2,2).^1 uses matrix_scalar. The result starts with the base matrix's sparsity, and 0^1 = 0, so it remains sparse.
• DM(2,2).^DM.ones(2,2) uses matrix_matrix, which combines both operand patterns. Because the exponent is structurally dense, the result becomes dense. This is conservative but valid for general exponents because 0^0 = 1 and zero raised to a negative exponent is not zero.
• DM(1,1).^1 takes the scalar-left path because a 1-by-1 matrix is scalar. That path initializes the result using the exponent's dense 1-by-1 sparsity (source).

Therefore, the 1-by-1 result looks like a sparsity consistency or optimization gap rather than an incorrect numerical result. For numeric DM results, sparsify(result) removes stored numerical zeros. When the exponent is exactly one, reusing the original matrix also preserves its pattern.