Method and apparatus for solving dense systems of linear equations with an iterative method that employs partial multiplications using rank compressed SVD basis matrices of the partitioned submatrices of the coefficient matrix

A parallel processing computer system comprising:

• a) storage means for storing program means and data in electromagnetic/optical representations;
• and
• b) processor means coupled to the storage means for executing first program means stored on the storage means, the first program means having
• b.1) first function means for partitioning a coefficient matrix of a dense system of linear equations stored on the storage means, into a plurality of submatrices, and storing the submatrices onto storage means in electromagnetic/optical representations,
• b.2) second function means for generating in parallel a plurality of sets of singular value decomposed (SVD) basis matrices for the submatrices, one set of rank compressed SVD basis matrices for each submatrix, rank compressing in parallel the generated sets of SVD basis matrices in accordance to a calculation error tolerance limit, and storing in parallel the rank compressed SVD basis matrices onto the storage means in electromagnetic/optical representations, and
• b.3) third function means for iteratively solving the dense system of linear equations, performing all matrix multiplications involving the coefficient matrix indirectly by performing in parallel partial matrix multiplications using the generated sets of rank compressed SVD basis matrices of the partitioned submatrices, and accumulating the partial matrix multiplication results.

A parallel processing computer system comprising:

• a) storage means, including N high performance volatile memory units and at least one non-volatile mass storage, for storing program means and data in electromagnetic/optical representations;
• and
• b) processor means coupled to the storage means, including at least N processors, for executing first program means stored on the storage means, the first program means having
• b.1) first function means for partitioning a coefficient matrix of a dense system of linear equations stored on the storage means, into N submatrices, and storing the submatrices in the N high performance volatile memory units in electromagnetic/optical representations,
• b.2) N second function means for generating in parallel N of sets of singular value decomposed (SVD) basis matrices for the N submatrices at the same time, one set of rank compressed SVD basis matrices for each submatrix, rank compressing in parallel the N generated sets of SVD basis matrices in accordance to a calculation error tolerance limit, and storing in parallel the rank compressed SVD basis matrices in the N high performance volatile memory units in electromagnetic/optical representations, and
• b.3) N third function means for iteratively solving the dense system of linear equations, performing all matrix multiplications involving the coefficient matrix indirectly by performing in parallel partial matrix multiplications using the N sets of rank compressed SVD basis matrices of the partitioned submatrices at the same time, and accumulating the partial matrix multiplication results.