21,401 research outputs found
Numerical Computation of the Complex Eigenvalues of a Matrix by solving a Square System of Equations
It is well known that if the largest or smallest eigenvalue of a matrix has been computed by some numerical algorithms and one is interested in computing the corresponding eigenvector, one method that is known to give such good approximations to the eigenvector is inverse iteration with a shift. For complex eigenpairs, instead of using Ruhe’s normalization, we show that the natural two norm normalization for the matrix pencil, yields an underdetermined system of equation and by adding an extra equation, the augmented system becomes square which can be solved by LU factorization at a cheaper rate and quadratic convergence is guaranteed. While the underdetermined system of equations can be solved using QR factorization as shown in an earlier work by the same authors, converting it to a square system of equations has the added advantage that besides using LU factorization, it can be solved by several approaches including iterative methods. We show both theoretically and numerically that both algorithms are equivalent in the absence of roundoff errors
LU factorization with panel rank revealing pivoting and its communication avoiding version
We present the LU decomposition with panel rank revealing pivoting (LU_PRRP),
an LU factorization algorithm based on strong rank revealing QR panel
factorization. LU_PRRP is more stable than Gaussian elimination with partial
pivoting (GEPP). Our extensive numerical experiments show that the new
factorization scheme is as numerically stable as GEPP in practice, but it is
more resistant to pathological cases and easily solves the Wilkinson matrix and
the Foster matrix. We also present CALU_PRRP, a communication avoiding version
of LU_PRRP that minimizes communication. CALU_PRRP is based on tournament
pivoting, with the selection of the pivots at each step of the tournament being
performed via strong rank revealing QR factorization. CALU_PRRP is more stable
than CALU, the communication avoiding version of GEPP. CALU_PRRP is also more
stable in practice and is resistant to pathological cases on which GEPP and
CALU fail.Comment: No. RR-7867 (2012
The double charm decays of Meson in the Perturbative QCD Approach
We make a systematic investigation on the double charm decays of meson,
by employing the perturbative QCD approach based on factorization. It is
found that the non-factorizable emission diagrams are not negligible in these
channels. We predict the branching ratios of these decays and also the
transverse polarization fractions of decays, % where V denote the vector meson.
We find that the magnitudes of the branching ratios of the decays
and are very close to
each other, which are well suited to extract the Cabibbo-Kobayashi-Maskawa
angle through the amplitude relations. In addition, a large transverse
polarization contribution that can reach is predicted in some of
the meson decay to two vector charmed mesons.Comment: 22 pages, 5 tables, to appear at PRD. arXiv admin note: text overlap
with arXiv:1112.125
Improving approximate matrix factorizations for implicit time integration in air pollution modelling
For a long time operator splitting was the only computationally feasible way of implicit time integration in large scale Air Pollution Models. A recently proposed attractive alternative is Rosenbrock schemes combined with Approximate Matrix Factorization (AMF). With AMF, linear systems arising in implicit time stepping are solved approximately in such a way that the overall computational costs per time step are not higher than those of splitting methods. We propose and discuss two new variants of AMF. The first one is aimed at yet a further reduction of costs as compared with conventional AMF. The second variant of AMF provides in certain circumstances a better approximation to the inverse of the linear system matrix than standard AMF and requires the same computational work
DMFSGD: A Decentralized Matrix Factorization Algorithm for Network Distance Prediction
The knowledge of end-to-end network distances is essential to many Internet
applications. As active probing of all pairwise distances is infeasible in
large-scale networks, a natural idea is to measure a few pairs and to predict
the other ones without actually measuring them. This paper formulates the
distance prediction problem as matrix completion where unknown entries of an
incomplete matrix of pairwise distances are to be predicted. The problem is
solvable because strong correlations among network distances exist and cause
the constructed distance matrix to be low rank. The new formulation circumvents
the well-known drawbacks of existing approaches based on Euclidean embedding.
A new algorithm, so-called Decentralized Matrix Factorization by Stochastic
Gradient Descent (DMFSGD), is proposed to solve the network distance prediction
problem. By letting network nodes exchange messages with each other, the
algorithm is fully decentralized and only requires each node to collect and to
process local measurements, with neither explicit matrix constructions nor
special nodes such as landmarks and central servers. In addition, we compared
comprehensively matrix factorization and Euclidean embedding to demonstrate the
suitability of the former on network distance prediction. We further studied
the incorporation of a robust loss function and of non-negativity constraints.
Extensive experiments on various publicly-available datasets of network delays
show not only the scalability and the accuracy of our approach but also its
usability in real Internet applications.Comment: submitted to IEEE/ACM Transactions on Networking on Nov. 201
Hadronic decays of in the perturbative QCD approach
We calculate the branching ratios and polarization fractions of the decays in the perturbative QCD(pQCD) approach at leading order, where
() stands for the axial-vector state. By
combining the phenomenological analyses with the perturbative calculations, we
find the following results: (a) the large decay rates around to
of the decays dominated by the longitudinal
polarization(except for the mode) are predicted and
basically consistent with those in the QCD factorization(QCDF) within errors,
which are expected to be tested by the Large Hadron Collider and Belle-II
experiments. The large branching ratio could provide
hints to help explore the mechanism of the color-suppressed decays. (b) the
rather different QCD behaviors between the and mesons result in the
destructive(constructive) contributions in the nonfactorizable spectator
diagrams with emission. Therefore, an interesting pattern of the
branching ratios appears for the color-suppressed and modes in the pQCD approach, , which is different
from in the QCDF and would be verified at future experiments. (c) the
large naive factorization breaking effects are observed in these decays. Specifically, the large nonfactorizable spectator(weak
annihilation) amplitudes contribute to the mode(s), which demand confirmations
via the precise measurements.Comment: 13 pages, 1 figure, 5 tables, revtex fil
The Bordering Algorithm and Path Following Near Singular Points of Higher Nullity
We study the behavior of the bordering algorithm (a form of block elimination) for solving nonsingular linear systems with coefficient matrices in the partitioned form (A & B \\ C^* & D) when N(A)≧1. Systems with this structure naturally occur in path following procedures. We show that under appropriate assumptions, the algorithm, which is based on solving systems with coefficient matrix A, works as A varies along a path and goes through singular points. The required assumptions are justified for a large class of problems coming from discretizations of boundary value problems for differential equations
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