Blocking and parallelization of the Hari–Zimmermann variant of the Falk–Langemeyer algorithm for the generalized SVD

• New parallel Jacobi-type algorithm for the generalized singular value problem.
• The algorithm is 15 times faster than DGGSVD from LAPACK.
• It is also more accurate.

The paper describes how to modify the two-sided Hari–Zimmermann algorithm for computation of the generalized eigenvalues of a matrix pair (A, B), where B is positive definite, to an implicit algorithm that computes the generalized singular values of a pair (F, G). In addition, we present blocking and parallelization techniques for speedup of the computation.For triangular matrix pairs of a moderate size, numerical tests show that the double precision sequential pointwise algorithm is several times faster than the Lapack DTGSJA algorithm, while the accuracy is slightly better, especially for small generalized singular values.Cache-aware algorithms, implemented either as the block-oriented, or as the full block algorithm, are several times faster than the pointwise algorithm. The algorithm is almost perfectly parallelizable, so parallel shared memory versions of the algorithm are perfectly scalable, and their speedup almost solely depends on the number of cores used. A hybrid shared/distributed memory algorithm is intended for huge matrices that do not fit into the shared memory.