Support dpotrs, dgetrs, dormqr, dtrtrs, and eigen decomposition for asymmetric matrix

[haifengl]

They are needed to solve least square problems with the QR decomposition. ND4J supports QR decomposition already. Thanks.

Aha! Link: https://skymindai.aha.io/features/ND4J-145

[agibsonccc]

@saudet maybe @rcorbish could look at these batch of usses? @haifengl thanks for reporting!

[saudet]

Let's merge issues #6762, #6763, and #6765 here too if it's alright @haifengl?

[haifengl]

Yes, it is okay to merge them. Thanks.

[haifengl]

The current support of SVD can be improved too. Typically, we use the input matrix to hold the U after the decomposition. But Nd4j's gesvd() method requires U be a m x m matrix, where m is the number of rows. In most case, m be very large (thousand to millions). We cannot hold such big matrix in memory and it is also meaningless to compute m left singular vectors.

[haifengl]

Looking into the nd4j implementations, I would suggest the following improvements:

• In geqrf (for QR), the input R is required be a matrix of size n x n. In fact, it could be just a vector of size n. Let's call it tau, which just holds the diagonal of R. The R itself is stored in A.

• Please support dorgqr that generates an M-by-N real matrix Q with orthonormal columns of QR.

• In general, we need call a function to determine the workspace size before the real decomposition (e.g. QR, SVD, eigen). It is not clear to me how do you decide the workspace size.

Thanks!

[saudet]

@haifengl Sounds good! If you have some time to implement this, feel free to send a pull request. This is not a priority for us, so unless @rcorbish has some time to work on this, we are not likely to spend time on this anytime soon, but feel free to ask questions if you need guidance.