8 research outputs found
High degree interpolation polynomial in Newton form
Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4
Polynomial approximation of functions of matrices and its application to the solution of a general system of linear equations
During the process of solving a mathematical model numerically, there is often a need to operate on a vector v by an operator which can be expressed as f(A) while A is NxN matrix (ex: exp(A), sin(A), A sup -1). Except for very simple matrices, it is impractical to construct the matrix f(A) explicitly. Usually an approximation to it is used. In the present research, an algorithm is developed which uses a polynomial approximation to f(A). It is reduced to a problem of approximating f(z) by a polynomial in z while z belongs to the domain D in the complex plane which includes all the eigenvalues of A. This problem of approximation is approached by interpolating the function f(z) in a certain set of points which is known to have some maximal properties. The approximation thus achieved is almost best. Implementing the algorithm to some practical problem is described. Since a solution to a linear system Ax = b is x= A sup -1 b, an iterative solution to it can be regarded as a polynomial approximation to f(A) = A sup -1. Implementing the algorithm in this case is also described
A Chebychev propagator for inhomogeneous Schr\"odinger equations
We present a propagation scheme for time-dependent inhomogeneous
Schr\"odinger equations which occur for example in optimal control theory or in
reactive scattering calculations. A formal solution based on a polynomial
expansion of the inhomogeneous term is derived. It is subjected to an
approximation in terms of Chebychev polynomials. Different variants for the
inhomogeneous propagator are demonstrated and applied to two examples from
optimal control theory. Convergence behavior and numerical efficiency are
analyzed.Comment: explicit description of algorithm and two appendices added version
accepted by J Chem Phy
New, Highly Accurate Propagator for the Linear and Nonlinear Schr\"odinger Equation
A propagation method for the time dependent Schr\"odinger equation was
studied leading to a general scheme of solving ode type equations. Standard
space discretization of time-dependent pde's usually results in system of ode's
of the form u_t -Gu = s where G is a operator (matrix) and u is a
time-dependent solution vector. Highly accurate methods, based on polynomial
approximation of a modified exponential evolution operator, had been developed
already for this type of problems where G is a linear, time independent matrix
and s is a constant vector. In this paper we will describe a new algorithm for
the more general case where s is a time-dependent r.h.s vector. An iterative
version of the new algorithm can be applied to the general case where G depends
on t or u. Numerical results for Schr\"odinger equation with time-dependent
potential and to non-linear Schr\"odinger equation will be presented.Comment: 14 page
A Chebychev propagator with iterative time ordering for explicitly time-dependent Hamiltonians
A propagation method for time-dependent Schr\"odinger equations with an
explicitly time-dependent Hamiltonian is developed where time ordering is
achieved iteratively. The explicit time-dependence of the time-dependent
Schr\"odinger equation is rewritten as an inhomogeneous term. At each step of
the iteration, the resulting inhomogeneous Schr\"odinger equation is solved
with the Chebychev propagation scheme presented in J. Chem. Phys. 130, 124108
(2009). The iteratively time-ordering Chebychev propagator is shown to be
robust, efficient and accurate and compares very favorably to all other
available propagation schemes
Parameterization-free Projection for Geometry Reconstruction
We introduce a Locally Optimal Projection operator (LOP) for surface approximation from point-set data. The operator is parameterization free, in the sense that it does not rely on estimating a local normal, fitting a local plane, or using any other local parametric representation. Therefore, it can deal with noisy data which clutters the orientation of the points. The method performs well in cases of ambiguous orientation, e.g., if two folds of a surface lie near each other, and other cases of complex geometry in which methods based upon local plane fitting may fail. Although defined by a global minimization problem, the method is effectively local, and it provides a second order approximation to smooth surfaces. Hence allowing good surface approximation without using any explicit or implicit approximation space. Furthermore, we show that LOP is highly robust to noise and outliers and demonstrate its effectiveness by applying it to raw scanned data of complex shapes