A parallel generator of nonxe2x80x90Hermitian matrices computed from given spectra Iterative linear algebra methods to solve linear systems and eigenvalue problems with nonHermitian matrices are important for both the simulation arising from diverse scientific fields and the applications related to big data, machine learning and artificial intelligence. The spectral property of these matrices has impacts on the convergence of these solvers. Moreover, with the increase of the size of applications, iterative methods are implemented in parallel on clusters. Analysis of their behaviors with nonHermitian matrices on supercomputers is so complex that we need to generate largescale matrices with different given spectra for benchmarking. These test matrices should be nonHermitian and nontrivial, with high dimension. This paper highlights a scalable matrix generator that constructs large sparse matrices using the userdefined spectrum, and the eigenvalues of generated matrices are ensured to be the same as the predefined spectrum. This generator is implemented on CPUs and multiGPUs platforms, with good strong and weak scaling performance on several supercomputers. We also propose a method to verify its ability to guarantee the given spectra. Finally, we give an example to evaluate the numerical properties and parallel performance of iterative methods using this matrix generator.

Parallelization and scalability analysis of inverse factorization using the Chunks and Tasks programming model We present three methods for distributed memory parallel inverse factorization of blocksparse Hermitian positive definite matrices. The three methods are a recursive variant of the AINV inverse Cholesky algorithm, iterative refinement, and localized inverse factorization, respectively. All three methods are implemented using the Chunks and Tasks programming model, building on the distributed sparse quadtree matrix representation and parallel matrixmatrix multiplication in the publicly available Chunks and Tasks Matrix Library (CHTML). Although the algorithms are generally applicable, this work was mainly motivated by the need for efficient and scalable inverse factorization of the basis set overlap matrix in large scale electronic structure calculations. We perform various computational tests on overlap matrices for quasilinear Glutamic AcidAlanine molecules and threedimensional water clusters discretized using the standard Gaussian basis set STO3G with up to more than 10 million basis functions. We show that for such matrices the computational cost increases only linearly with system size for all the three methods. We show both theoretically and in numerical experiments that the methods based on iterative refinement and localized inverse factorization outperform previous parallel implementations in weak scaling tests where the system size is increased in direct proportion to the number of processes. We show also that compared to the method based on pure iterative refinement the localized inverse factorization requires much less communication.