760 research outputs found
Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem
A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A1 is presented. From the eigenpairs of some real matrix A0, the eigenpairs of
A(t) ≡ (1 − t)A0 + tA1
are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix:
• bifurcation,
• ill conditioning due to nonorthogonal eigenvectors,
• jumping of eigenpaths.
These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented
Spectral methods in general relativistic astrophysics
We present spectral methods developed in our group to solve three-dimensional
partial differential equations. The emphasis is put on equations arising from
astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures,
submitted to Journal of Computational & Applied Mathematic
Short-recurrence Krylov subspace methods for the overlap Dirac operator at nonzero chemical potential
The overlap operator in lattice QCD requires the computation of the sign
function of a matrix, which is non-Hermitian in the presence of a quark
chemical potential. In previous work we introduced an Arnoldi-based Krylov
subspace approximation, which uses long recurrences. Even after the deflation
of critical eigenvalues, the low efficiency of the method restricts its
application to small lattices. Here we propose new short-recurrence methods
which strongly enhance the efficiency of the computational method. Using
rational approximations to the sign function we introduce two variants, based
on the restarted Arnoldi process and on the two-sided Lanczos method,
respectively, which become very efficient when combined with multishift
solvers. Alternatively, in the variant based on the two-sided Lanczos method
the sign function can be evaluated directly. We present numerical results which
compare the efficiencies of a restarted Arnoldi-based method and the direct
two-sided Lanczos approximation for various lattice sizes. We also show that
our new methods gain substantially when combined with deflation.Comment: 14 pages, 4 figures; as published in Comput. Phys. Commun., modified
data in Figs. 2,3 and 4 for improved implementation of FOM algorithm,
extended discussion of the algorithmic cos
Computational aspects of helicopter trim analysis and damping levels from Floquet theory
Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated
Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems
In these lecture notes, we present a pedagogical review of a number of
related {\it numerically exact} approaches to quantum many-body problems. In
particular, we focus on methods based on the exact diagonalization of the
Hamiltonian matrix and on methods extending exact diagonalization using
renormalization group ideas, i.e., Wilson's Numerical Renormalization Group
(NRG) and White's Density Matrix Renormalization Group (DMRG). These methods
are standard tools for the investigation of a variety of interacting quantum
systems, especially low-dimensional quantum lattice models. We also survey
extensions to the methods to calculate properties such as dynamical quantities
and behavior at finite temperature, and discuss generalizations of the DMRG
method to a wider variety of systems, such as classical models and quantum
chemical problems. Finally, we briefly review some recent developments for
obtaining a more general formulation of the DMRG in the context of matrix
product states as well as recent progress in calculating the time evolution of
quantum systems using the DMRG and the relationship of the foundations of the
method with quantum information theory.Comment: 51 pages; lecture notes on numerically exact methods. Pedagogical
review appearing in the proceedings of the "IX. Training Course in the
Physics of Correlated Electron Systems and High-Tc Superconductors", Vietri
sul Mare (Salerno, Italy, October 2004
A Singular Value Thresholding Algorithm for Matrix Completion
This paper introduces a novel algorithm to approximate the matrix with minimum
nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood
as the convex relaxation of a rank minimization problem and arises in many important
applications as in the task of recovering a large matrix from a small subset of its entries (the famous
Netflix problem). Off-the-shelf algorithms such as interior point methods are not directly amenable
to large problems of this kind with over a million unknown entries. This paper develops a simple
first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in
which the optimal solution has low rank. The algorithm is iterative, produces a sequence of matrices
{X^k,Y^k}, and at each step mainly performs a soft-thresholding operation on the singular values
of the matrix Y^k. There are two remarkable features making this attractive for low-rank matrix
completion problems. The first is that the soft-thresholding operation is applied to a sparse matrix;
the second is that the rank of the iterates {X^k} is empirically nondecreasing. Both these facts allow
the algorithm to make use of very minimal storage space and keep the computational cost of each
iteration low. On the theoretical side, we provide a convergence analysis showing that the sequence
of iterates converges. On the practical side, we provide numerical examples in which 1,000 × 1,000
matrices are recovered in less than a minute on a modest desktop computer. We also demonstrate
that our approach is amenable to very large scale problems by recovering matrices of rank about
10 with nearly a billion unknowns from just about 0.4% of their sampled entries. Our methods are
connected with the recent literature on linearized Bregman iterations for ℓ_1 minimization, and we
develop a framework in which one can understand these algorithms in terms of well-known Lagrange
multiplier algorithms
Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Poschl-Teller-Ginocchio potential wave functions
The fast computation of the Gauss hypergeometric function 2F1 with all its
parameters complex is a difficult task. Although the 2F1 function verifies
numerous analytical properties involving power series expansions whose
implementation is apparently immediate, their use is thwarted by instabilities
induced by cancellations between very large terms. Furthermore, small areas of
the complex plane are inaccessible using only 2F1 power series formulas, thus
rendering 2F1 evaluations impossible on a purely analytical basis. In order to
solve these problems, a generalization of R.C. Forrey's transformation theory
has been developed. The latter has been successful in treating the 2F1 function
with real parameters. As in real case transformation theory, the large
canceling terms occurring in 2F1 analytical formulas are rigorously dealt with,
but by way of a new method, directly applicable to the complex plane. Taylor
series expansions are employed to enter complex areas outside the domain of
validity of power series analytical formulas. The proposed algorithm, however,
becomes unstable in general when |a|,|b|,|c| are moderate or large. As a
physical application, the calculation of the wave functions of the analytical
Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.Comment: 29 pages; accepted in Computer Physics Communication
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