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LAPACK has been on the cutting edge for just about three decades, and probably still is for its niche. However, given given recent developments in libraries for the simpler BLAS-type matrix operations that LAPACK traditionally builds upon, it is perhaps conceivable that we could see the emergence of serious competitors to the traditional FORTRAN-based LAPACK ...
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The reference LAPACK library prioritizes portability over performance, so it's missing many optimizations, such as vectorization and threading. (Additionally, it relies on BLAS for much of it's performance, so using the reference BLAS will significantly limit performance.)
However, most LAPACK users don't use the reference library, but various optimized ...
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Another approach, which might be of interest to you is randomized sampling. This is of particular interest if you can quickly compute matrix-vector products $x\rightarrow Ax$ and $x\rightarrow A^* x$. The core idea is to form a small sampling matrix $S = A\Omega$, where $\Omega$ is a Gaussian random matrix. If the sampling matrix is large enough, $S$ will ...
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