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PRIMME, pronounced as prime, computes a few eigenvalues and their corresponding eigenvectors of a real symmetric or complex Hermitian matrix. It can also compute singular values and vectors of a square or rectangular matrix. It can find largest, smallest, or interior singular/eigenvalues and can use preconditioning to accelerate convergence. It is especially optimized for large, difficult problems, and can be a useful tool for both non-experts and experts. PRIMME is written in C99, but complete interfaces are provided for Fortran 77, MATLAB, Python, and R.