JacobiSVD is a Julia package wrapping the LAPACK subroutines for computing the singular value decomposition (SVD) of general dense matrices using Jacobi algorithms. The advantage of the Jacobi scheme is the higher accuracy of the computed singular values (compared to QR and divide-and-conquer algorithms). The disadvantage is that they are relatively inefficient unless one combines them with elaborate preconditioning. LAPACK includes routines which organize such preconditioning.
Caveat: extra accuracy is guaranteed only for matrices which can be
expressed as A = B * D where B is well-conditioned and D is a diagonal
scaling. It also obtains for forms A = D1 * C * D2 where C is
well-conditioned and D1, D2 are diagonal scalings, but theory is lacking.
For general matrices, results may be no better than standard methods.
This package provides a LinearAlgebra-style function jsvd!.
S = jsvd!(A)which returns an SVD object, like svd!(A), but perhaps more accurate.
It also exports low-level wrappers gesvj! and gejsv!; see the docstrings
and LAPACK documentation for details.
Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I. SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342. LAPACK Working note 169
Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II. SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362. LAPACK Working note 170