545,530 research outputs found
Application of the matrix exponential kernel
A point matrix kernel for radiation transport, developed by the transmission matrix method, has been used to develop buildup factors and energy spectra through slab layers of different materials for a point isotropic source. Combinations of lead-water slabs were chosen for examples because of the extreme differences in shielding properties of these two materials
Exponential parameterization of the neutrino mixing matrix - comparative analysis with different data sets and CP violation
The exponential parameterization of Pontecorvo-Maki-Nakagawa-Sakata mixing
matrix for neutrino is used for comparative analysis of different neutrino
mixing data. The UPMNS matrix is considered as the element of the SU(3) group
and the second order matrix polynomial is constructed for it. The inverse
problem of constructing the logarithm of the mixing matrix is addressed. In
this way the standard parameterization is related to the exponential
parameterization exactly. The exponential form allows easy factorization and
separate analysis of the rotation and the CP violation. With the most recent
experimental data on the neutrino mixing (May 2016), we calculate the values of
the exponential parameterization matrix for neutrinos with account for the CP
violation. The complementarity hypothesis for quarks and neutrinos is
demonstrated to hold, despite significant change in the neutrino mixing data.
The values of the entries of the exponential mixing matrix are evaluated with
account for the actual degree of the CP violation in neutrino mixing and
without it. Various factorizations of the CP violating term are investigated in
the framework of the exponential parameterization
Residual, restarting and Richardson iteration for the matrix exponential
A well-known problem in computing some matrix functions iteratively is a lack of a clear, commonly accepted residual notion. An important matrix function for which this is the case is the matrix exponential. Assume, the matrix exponential of a given matrix times a given vector has to be computed. We interpret the sought after vector as a value of a vector function satisfying the linear system of ordinary differential equations (ODE), whose coefficients form the given matrix. The residual is then defined with respect to the initial-value problem for this ODE system. The residual introduced in this way can be seen as a backward error. We show how the residual can efficiently be computed within several iterative methods for the matrix exponential. This completely resolves the question of reliable stopping criteria for these methods. Furthermore, we show that the residual concept can be used to construct new residual-based iterative methods. In particular, a variant of the Richardson method for the new residual appears to provide an efficient way to restart Krylov subspace methods for evaluating the matrix exponential.\u
Fast computation of the matrix exponential for a Toeplitz matrix
The computation of the matrix exponential is a ubiquitous operation in
numerical mathematics, and for a general, unstructured matrix it
can be computed in operations. An interesting problem arises
if the input matrix is a Toeplitz matrix, for example as the result of
discretizing integral equations with a time invariant kernel. In this case it
is not obvious how to take advantage of the Toeplitz structure, as the
exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself.
The main contribution of this work are fast algorithms for the computation of
the Toeplitz matrix exponential. The algorithms have provable quadratic
complexity if the spectrum is real, or sectorial, or more generally, if the
imaginary parts of the rightmost eigenvalues do not vary too much. They may be
efficient even outside these spectral constraints. They are based on the
scaling and squaring framework, and their analysis connects classical results
from rational approximation theory to matrices of low displacement rank. As an
example, the developed methods are applied to Merton's jump-diffusion model for
option pricing
Information Geometry Approach to Parameter Estimation in Markov Chains
We consider the parameter estimation of Markov chain when the unknown
transition matrix belongs to an exponential family of transition matrices.
Then, we show that the sample mean of the generator of the exponential family
is an asymptotically efficient estimator. Further, we also define a curved
exponential family of transition matrices. Using a transition matrix version of
the Pythagorean theorem, we give an asymptotically efficient estimator for a
curved exponential family.Comment: Appendix D is adde
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