Finding optimal rotation and translation between corresponding 3D points Last update: The corresponding points have the same colour, R is the rotation and t is the translation. We want to find the best rotation and translation that will align the points in dataset A to dataset B.
This kind of question is recurring and should be answered more clearly than "Matlab uses highly optimized libraries" or "Matlab uses the MKL" for once on Stackoverflow.
Matrix multiplication together with Matrix-vector, vector-vector multiplication and many of the matrix decompositions is are the most important problems in linear algrebra.
Engineers have been solving these problems with computers since the early days. I'm not an expert on the history, but apparently back then, everybody just rewrote his Fortran version with simple loops. Some standardization then came along, with the identification of "kernels" basic routines that most linear algebra problems needed in order to be solved.
These basic operations were then standardized in a specification called: Engineers could then call these standard, well-tested BLAS routines in their code, making their work much easier. BLAS evolved from level 1 the first version which defined scalar-vector and vector-vector operations to level 2 vector-matrix operations to level 3 matrix-matrix operationsand provided more and more "kernels" so standardized more and more of the fundamental linear algebra operations.
The original Fortran 77 implementations are still available on Netlib's website. So over the years notably between the BLAS level 1 and level 2 releases: These evolutions made it possible to increase the performance of the BLAS subroutines substantially.
Different vendors then came along with their implementation of BLAS routines which were more and more efficient. I don't know all the historical implementations I was not born or a kid back thenbut two of the most notable ones came out in the early s: Technical details on Matrix multiplication: So why is Matlab the MKL so fast at dgemm double-precision general matrix-matrix multiplication?
In more complex terms: Since I suspect you created an array of pointers to row arrays, your accesses in your inner loop to the k-th column of "matice2": Indeed, when you access matice2[k], you must get the k-th element of the array 0 of your matrix.
Then in the next iteration, you must access matice2[k], which is the k-th element of another array the array 1. Then in the next iteration you access yet another array, and so on If you just transposed the matrix, so that accesses would be in contiguous memory addresses, your code would already run much faster because now the compiler can load entire rows in the cache at the same time.
Just try this modified version: Now real dgemm implementations exploit that to a very extensive level: They perform the multiplication on blocks of the matrix defined by the size of the TLB Translation lookaside buffer, long story short: Finally, people claiming that it's because of Strassen's or Coppersmith—Winograd algorithm are wrong, both these algorithms are not implementable in practice, because of hardware considerations mentioned above.In the world of matrix switches, this one is very affordable, especially for a 4x4 - any 4 inputs to any 4 outputs.
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