Using a predictor-corrector tactic, this paper derives new iteration schemes for unconstrained optimization. It yields a point (predictor) by some line search from the current point; then with the two points it constructs a quadratic interpolation curve to approximate some ODE trajectory; it finally determines a new point (corrector) by searching along the quadratic curve. In particular, this p...

This study focused mainly on the derivation of the 2 and 3-point block methods with constant coefficients for solving special second order ordinary differential equations directly based on Newton-Gregory backward interpolation formula. The performance of the new methods was compared with the conventional 1-point method using a standard set of test problems. Numerical results were presented to i...

We propose a new method to approximate a given set of ordered data points by a spatial circular spline curve. At first an initial circular spline curve is generated by biarc interpolation. Then an evolution process based on a least-squares approximation is applied to the curve. During the evolution process, the circular spline curve converges dynamically to a stable shape. Our method does not n...

We propose a novel multivariate Monte Carlo method as an efficient and flexible approach to analyzing extended X-ray sources with the Reflection Grating Spectrometer (RGS) on XMM Newton. A multi-dimensional interpolation method is used to efficiently calculate the response function for the RGS in conjunction with an arbitrary spatially-varying spectral model. Several methods of event comparison...

T -adic exponential sums associated to a Laurent polynomial f are introduced. They interpolate all classical p-power order exponential sums associated to f . The Hodge bound for the Newton polygon of L-functions of T -adic exponential sums is established. This bound enables us to determine, for all m, the Newton polygons of Lfunctions of p-power order exponential sums associated to an f which i...

We present here work which is, in part, expository with proofs, exercises (2.4, 2.6, 4.3, and 7.6), and, in part, contains new results (3.1, 5.4, amd 7.7) so we ought to begin with some background: Dynamics travels a line of history from as far back as Newton, as a notion for his laws of motion, especially, as concerns the law of gravity, for which he developed the Calculus. Prior to the twenti...

2. Write a Matlab script of no more than 10 lines to calculate the Newton-Cotes quadrature weights wj , j = 1, 2, . . . , n, n + 1; go to the end of this handout for more details about wj . For uniformity, we require that the script starts with the four lines n=2; h=1/n; nodes=(0:h:1)’; f=eye(n+1); where the j-th column of the matrix f are needed for the Lagrange interpolation in order to find ...

Principal lattices in the plane are distributions of points particularly simple to use Lagrange, Newton or Aitken-Neville interpolation formulas. Principal lattices were generalized by Lee and Phillips, introducing three-pencil lattices, that is, points which are the intersection of three lines, each one belonging to a different pencil. In this contribution, a semicubical parabola is used to co...

An algorithm for the computation of error bounds for the zeros of a polynomial is described. This algorithm is derived by applying Rouch6's theorem to a Newton-like interpolation formula for the polynomial, and so it is suitable in the case where the approximations to the zeros of the polynomial are computed successively using deflation. Confluent and clustered approximations are handled easily...

Weighted essentially non-oscillatory (WENO) schemes have proved useful in a variety of physical applications. They capture sharp gradients without smearing, and feature high order of accuracy along with nonlinear stability. The high order of accuracy, robustness, and smooth numerical uxes of the WENO schemes make them ideal for use with Jacobian based iterative solvers, to directly simulate the...