7,294 research outputs found
Efficient Rank Reduction of Correlation Matrices
Geometric optimisation algorithms are developed that efficiently find the
nearest low-rank correlation matrix. We show, in numerical tests, that our
methods compare favourably to the existing methods in the literature. The
connection with the Lagrange multiplier method is established, along with an
identification of whether a local minimum is a global minimum. An additional
benefit of the geometric approach is that any weighted norm can be applied. The
problem of finding the nearest low-rank correlation matrix occurs as part of
the calibration of multi-factor interest rate market models to correlation.Comment: First version: 20 pages, 4 figures Second version [changed content]:
21 pages, 6 figure
Spin gauge symmetry in the action principle for classical relativistic particles
We suggest that the physically irrelevant choice of a representative
worldline of a relativistic spinning particle should correspond to a gauge
symmetry in an action approach. Using a canonical formalism in special
relativity, we identify a (first-class) spin gauge constraint, which generates
a shift of the worldline together with the corresponding transformation of the
spin on phase space. An action principle is formulated for which a minimal
coupling to fields is straightforward. The electromagnetic interaction of a
monopole-dipole particle is constructed explicitly.Comment: 9 pages, 1 figur
Ehrenfest theorem, Galilean invariance and nonlinear Schr\"odinger equations
Galilean invariant Schr\"odinger equations possessing nonlinear terms
coupling the amplitude and the phase of the wave function can violate the
Ehrenfest theorem. An example of this kind is provided. The example leads to
the proof of the theorem: A Galilean invariant Schr\"odinger equation derived
from a lagrangian density obeys the Ehrenfest theorem. The theorem holds for
any linear or nonlinear lagrangian.Comment: Latex format, no figures, submitted to journal of physics
Developing the MTO Formalism
We review the simple linear muffin-tin orbital method in the atomic-spheres
approximation and a tight-binding representation (TB-LMTO-ASA method), and show
how it can be generalized to an accurate and robust Nth order muffin-tin
orbital (NMTO) method without increasing the size of the basis set and without
complicating the formalism. On the contrary, downfolding is now more efficient
and the formalism is simpler and closer to that of screened multiple-scattering
theory. The NMTO method allows one to solve the single-electron Schroedinger
equation for a MT-potential -in which the MT-wells may overlap- using basis
sets which are arbitrarily minimal. The substantial increase in accuracy over
the LMTO-ASA method is achieved by substitution of the energy-dependent partial
waves by so-called kinked partial waves, which have tails attached to them, and
by using these kinked partial waves at N+1 arbitrary energies to construct the
set of NMTOs. For N=1 and the two energies chosen infinitesimally close, the
NMTOs are simply the 3rd-generation LMTOs. Increasing N, widens the energy
window, inside which accurate results are obtained, and increases the range of
the orbitals, but it does not increase the size of the basis set and therefore
does not change the number of bands obtained. The price for reducing the size
of the basis set through downfolding, is a reduction in the number of bands
accounted for and -unless N is increased- a narrowing of the energy window
inside which these bands are accurate. A method for obtaining orthonormal NMTO
sets is given and several applications are presented.Comment: 85 pages, Latex2e, Springer style, to be published in: Lecture notes
in Physics, edited by H. Dreysse, (Springer Verlag
Projected Newton Method for noise constrained Tikhonov regularization
Tikhonov regularization is a popular approach to obtain a meaningful solution
for ill-conditioned linear least squares problems. A relatively simple way of
choosing a good regularization parameter is given by Morozov's discrepancy
principle. However, most approaches require the solution of the Tikhonov
problem for many different values of the regularization parameter, which is
computationally demanding for large scale problems. We propose a new and
efficient algorithm which simultaneously solves the Tikhonov problem and finds
the corresponding regularization parameter such that the discrepancy principle
is satisfied. We achieve this by formulating the problem as a nonlinear system
of equations and solving this system using a line search method. We obtain a
good search direction by projecting the problem onto a low dimensional Krylov
subspace and computing the Newton direction for the projected problem. This
projected Newton direction, which is significantly less computationally
expensive to calculate than the true Newton direction, is then combined with a
backtracking line search to obtain a globally convergent algorithm, which we
refer to as the Projected Newton method. We prove convergence of the algorithm
and illustrate the improved performance over current state-of-the-art solvers
with some numerical experiments
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
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