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Recently, several task-parallel programming models have emerged to address the high synchronization and load imbalance issues as well as data movement overheads in modern shared memory architectures. OpenMP, the most commonly used shared memory parallel programming model, has added task execution support with dataflow dependencies. HPX and Regent are two more recent runtime systems that also support the dataflow execution model and extend it to distributed memory environments. We focus on parallelization of sparse matrix computations on shared memory architectures. We evaluate the OpenMP, HPX and Regent runtime systems in terms of performance and ease of implementation, and compare them against the traditional BSP model for two popular eigensolvers, Lanczos and LOBPCG. We give a general outline in regards to achieving parallelism using these runtime systems, and present a heuristic for tuning their performance to balance tasking overheads with the degree of parallelism that can be exposed. We then demonstrate their merits on two architectures, Intel Broadwell (a multicore processor) and AMD EPYC (a modern manycore processor). We observe that these frameworks achieve up to 13.7 × fewer cache misses over an efficient BSP implementation across L1, L2 and L3 cache layers. They also obtain up to 9.9 × improvement in execution time over the same BSP implementation.