[ilayn]

> Actually the column-major order of Fortran is more efficient for some linear algebra operations than the order of C, which has been inherited by many modern languages that do not care about high performance in scientific computations.

This is a plausible assumption to make but unfortunately it is not true at large. Especially when the traditional sizes are exceeded say n >= 2000 certain operations such as LU can be improved in terms of performance with C-major arrays. However the correct statement is you lose at some place you win at other. There are certainly linalg operations that F-major can give you more performance. However it is also true for C-major layout.

In your example matrix vector product or any BLAS2 or BLAS3 level operations you can also swap out the for loop order to convert things around (row*col buffer multiplication vs sum of weighted column sum interpretation). In particular matrix norm operations are the only exceptions (abs column sum, row abs sum etc.) that certain norms prefer certain orders. In fact if you go into the Goto method deep enough you'll see that internal order is a bit like Morton ordering to fit things into L1 Cache.

The reason why column-major is preferred is historical and requires more surgery to get it running with C-major ordering. Trust me I tried but it's too much work to gain not so much. Maybe someday when I retire I can attempt it. Hence I kept it column major in my retranslation of LAPACK https://github.com/ilayn/semicolon-lapack

Instead I implemented a "high"-performance AVX2 matrix transpose operation so that swapping the memory layout is trivial compared to the linalg cost.