[Type] Introduce function to compute the determinant of any square matrix

[alxbilger]

• tests

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[bakpaul]

Wouldn't it be more efficient to only create a vector of col and row indices that is reduced of one element for each determinant development ? So in the end your 'submatrix' is like a view on the original matrix, instead of copying the same values multiple times ?

[alxbilger]

I did not try to reach high performances because I don't have the need. Currently, I use it only in init().

[hugtalbot]

would your changes be easily introduced @bakpaul ?
if yes, we can add it inside this PR
else, it could be done in a later one

[bakpaul]

Yes I think so, I just need to take time to do it. Give me one week ?

[alxbilger]

[bakpaul]

See alxbilger#9

[alxbilger]

@bakpaul I finally changed the algorithm for a Gaussian elimination. Neither your code or mine worked on a 12x12 matrix. It hanged infinitely.

I also implemented benchmarks for this function: alxbilger/SofaBenchmark#44. Here is a possible result:

Gaussian elimination:

------------------------------------------------------------------------------------------
Benchmark                                                Time             CPU   Iterations
------------------------------------------------------------------------------------------
BM_Matrix_typemat_determinant<float, 3>/512           18.5 us         16.3 us        34462
BM_Matrix_typemat_determinant<float, 3>/1024          31.6 us         30.5 us        23579
BM_Matrix_typemat_determinant<float, 3>/2048          62.3 us         55.8 us        11200
BM_Matrix_typemat_determinant<double, 3>/512          15.9 us         14.3 us        44800
BM_Matrix_typemat_determinant<double, 3>/1024         31.5 us         30.7 us        22400
BM_Matrix_typemat_determinant<double, 3>/2048         62.2 us         58.6 us        11200
BM_Matrix_typemat_determinant<float, 4>/512           73.3 us         71.5 us         8960
BM_Matrix_typemat_determinant<float, 4>/1024           148 us          148 us         4978
BM_Matrix_typemat_determinant<float, 4>/2048           296 us          276 us         2489
BM_Matrix_typemat_determinant<double, 4>/512          72.5 us         67.0 us         7467
BM_Matrix_typemat_determinant<double, 4>/1024          146 us          144 us         4978
BM_Matrix_typemat_determinant<double, 4>/2048          311 us          301 us         2489
BM_Matrix_typemat_determinant<float, 12>/512          1485 us         1343 us          640
BM_Matrix_typemat_determinant<float, 12>/1024         2887 us         2790 us          224
BM_Matrix_typemat_determinant<float, 12>/2048         5387 us         5000 us          100
BM_Matrix_typemat_determinant<double, 12>/512         1425 us         1255 us          498
BM_Matrix_typemat_determinant<double, 12>/1024        2658 us         2511 us          280
BM_Matrix_typemat_determinant<double, 12>/2048        5455 us         5000 us          100

BM_Matrix_eigenmat_determinant<float, 3>/512          2.11 us         2.03 us       407273
BM_Matrix_eigenmat_determinant<float, 3>/1024         3.75 us         3.54 us       172308
BM_Matrix_eigenmat_determinant<float, 3>/2048         8.20 us         7.67 us       112000
BM_Matrix_eigenmat_determinant<double, 3>/512         1.90 us         1.76 us       320000
BM_Matrix_eigenmat_determinant<double, 3>/1024        4.21 us         3.84 us       179200
BM_Matrix_eigenmat_determinant<double, 3>/2048        7.59 us         7.50 us        89600
BM_Matrix_eigenmat_determinant<float, 4>/512          5.61 us         5.02 us       112000
BM_Matrix_eigenmat_determinant<float, 4>/1024         9.94 us         9.42 us        89600
BM_Matrix_eigenmat_determinant<float, 4>/2048         20.6 us         19.9 us        34462
BM_Matrix_eigenmat_determinant<double, 4>/512         4.89 us         4.43 us       144516
BM_Matrix_eigenmat_determinant<double, 4>/1024        10.5 us         10.5 us        64000
BM_Matrix_eigenmat_determinant<double, 4>/2048        21.3 us         18.4 us        37333
BM_Matrix_eigenmat_determinant<float, 12>/512         1755 us         1674 us          560
BM_Matrix_eigenmat_determinant<float, 12>/1024        3266 us         2860 us          224
BM_Matrix_eigenmat_determinant<float, 12>/2048        6458 us         5781 us          100
BM_Matrix_eigenmat_determinant<double, 12>/512        1126 us         1001 us          640
BM_Matrix_eigenmat_determinant<double, 12>/1024       2691 us         2613 us          299
BM_Matrix_eigenmat_determinant<double, 12>/2048       5246 us         5022 us          112

For small matrices, even 3x3 we are way slower than Eigen.

[bakpaul]

Well it makes sense, thinking about it now, it was in n! complexity