1,988 research outputs found
Simultaneous Diagonalization of Incomplete Matrices and Applications
We consider the problem of recovering the entries of diagonal matrices
for from multiple "incomplete" samples
of the form , where and are unknown matrices of low rank. We
devise practical algorithms for this problem depending on the ranks of and
. This problem finds its motivation in cryptanalysis: we show how to
significantly improve previous algorithms for solving the approximate common
divisor problem and breaking CLT13 cryptographic multilinear maps.Comment: 16 page
An Accelerated Conjugate Gradient Algorithm to Compute Low-Lying Eigenvalues --- a Study for the Dirac Operator in SU(2) Lattice QCD
The low-lying eigenvalues of a (sparse) hermitian matrix can be computed with
controlled numerical errors by a conjugate gradient (CG) method. This CG
algorithm is accelerated by alternating it with exact diagonalisations in the
subspace spanned by the numerically computed eigenvectors. We study this
combined algorithm in case of the Dirac operator with (dynamical) Wilson
fermions in four-dimensional \SUtwo gauge fields. The algorithm is
numerically very stable and can be parallelized in an efficient way. On
lattices of sizes an acceleration of the pure CG method by a factor
of~ is found.Comment: 25 pages, uuencoded tar-compressed .ps-fil
Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations
The iterative diagonalization of a sequence of large ill-conditioned
generalized eigenvalue problems is a computational bottleneck in quantum
mechanical methods employing a nonorthogonal basis for {\em ab initio}
electronic structure calculations. We propose a hybrid preconditioning scheme
to effectively combine global and locally accelerated preconditioners for rapid
iterative diagonalization of such eigenvalue problems. In partition-of-unity
finite-element (PUFE) pseudopotential density-functional calculations,
employing a nonorthogonal basis, we show that the hybrid preconditioned block
steepest descent method is a cost-effective eigensolver, outperforming current
state-of-the-art global preconditioning schemes, and comparably efficient for
the ill-conditioned generalized eigenvalue problems produced by PUFE as the
locally optimal block preconditioned conjugate-gradient method for the
well-conditioned standard eigenvalue problems produced by planewave methods
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