73,337 research outputs found
Efficient numerical diagonalization of hermitian 3x3 matrices
A very common problem in science is the numerical diagonalization of
symmetric or hermitian 3x3 matrices. Since standard "black box" packages may be
too inefficient if the number of matrices is large, we study several
alternatives. We consider optimized implementations of the Jacobi, QL, and
Cuppen algorithms and compare them with an analytical method relying on
Cardano's formula for the eigenvalues and on vector cross products for the
eigenvectors. Jacobi is the most accurate, but also the slowest method, while
QL and Cuppen are good general purpose algorithms. The analytical algorithm
outperforms the others by more than a factor of 2, but becomes inaccurate or
may even fail completely if the matrix entries differ greatly in magnitude.
This can mostly be circumvented by using a hybrid method, which falls back to
QL if conditions are such that the analytical calculation might become too
inaccurate. For all algorithms, we give an overview of the underlying
mathematical ideas, and present detailed benchmark results. C and Fortran
implementations of our code are available for download from
http://www.mpi-hd.mpg.de/~globes/3x3/ .Comment: 13 pages, no figures, new hybrid algorithm added, matches published
version, typo in Eq. (39) corrected; software library available at
http://www.mpi-hd.mpg.de/~globes/3x3
Accelerating the CM method
Given a prime q and a negative discriminant D, the CM method constructs an
elliptic curve E/\Fq by obtaining a root of the Hilbert class polynomial H_D(X)
modulo q. We consider an approach based on a decomposition of the ring class
field defined by H_D, which we adapt to a CRT setting. This yields two
algorithms, each of which obtains a root of H_D mod q without necessarily
computing any of its coefficients. Heuristically, our approach uses
asymptotically less time and space than the standard CM method for almost all
D. Under the GRH, and reasonable assumptions about the size of log q relative
to |D|, we achieve a space complexity of O((m+n)log q) bits, where mn=h(D),
which may be as small as O(|D|^(1/4)log q). The practical efficiency of the
algorithms is demonstrated using |D| > 10^16 and q ~ 2^256, and also |D| >
10^15 and q ~ 2^33220. These examples are both an order of magnitude larger
than the best previous results obtained with the CM method.Comment: 36 pages, minor edits, to appear in the LMS Journal of Computation
and Mathematic
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence
We introduce a framework for quasi-Newton forward--backward splitting
algorithms (proximal quasi-Newton methods) with a metric induced by diagonal
rank- symmetric positive definite matrices. This special type of
metric allows for a highly efficient evaluation of the proximal mapping. The
key to this efficiency is a general proximal calculus in the new metric. By
using duality, formulas are derived that relate the proximal mapping in a
rank- modified metric to the original metric. We also describe efficient
implementations of the proximity calculation for a large class of functions;
the implementations exploit the piece-wise linear nature of the dual problem.
Then, we apply these results to acceleration of composite convex minimization
problems, which leads to elegant quasi-Newton methods for which we prove
convergence. The algorithm is tested on several numerical examples and compared
to a comprehensive list of alternatives in the literature. Our quasi-Newton
splitting algorithm with the prescribed metric compares favorably against
state-of-the-art. The algorithm has extensive applications including signal
processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115
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