{'fraction_non_alphanumeric': 0.011986301369863013, 'fraction_numerical': 0.0, 'mean_word_length': 5.423076923076923, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A computational technique for parallel solution of diagonally dominant banded linear systems

In this work, we present a WZ factorization for a nonsingular diagonally dominant banded matrix. With little modifications in the structures of W and Z, we construct a stable parallel algorithm suitable for solving narrow banded nonsingular diagonally dominant linear systems using divide and conquer technique. Partition the coefficient matrix along the main diagonal; also partition the unknown vector and the right hand side vector accordingly. The coefficient matrix of the ‘reduced system’ which is obtained by collecting the first ß and last ß equations from each partition, where ß is semibadwidth of the given banded linear system, is proved to be nonsingular diagonally dominant. The backward error analysis of the algorithm is presented and the algorithm is proved to be numerically stable. Numerical experiments are conducted to check the performance of the parallel algorithm and to compare the present parallel algorithm with the corresponding subroutines of ScaLAPACK. The performance of the parallel algorithm is evaluated in terms of speedup and scalability.