The trapezoidal rule for the numerical integration of first-order ordinary differential equations is shown to possess, for a certain type of problem, an undesirable property. The removal of this difficulty is shown to be straightforward, resulting in a modified trapezoidal rule. Whilst this latent difficulty is slight (and probably rare in practice), the fact that the proposed modification invo...

The numerical computation of P{X > x} can be accomplished in a variety of ways. An appealing class of methods may be derived from the contour integral connecting P{X > x} and its Fourier representation. Statisticians have largely focused on deriving saddlepoint approximations for this contour integral. The accuracy of such approximations is generally understood in vague terms only and, perhaps ...

An important step in the automation of electronic design is the assignment of the physical components on the target semiconductor die. The major aim of floorplanning is to distribute the modules of a circuit onto a chip to optimize its area, wire length and timing. As the density of very large scale integrated (VLSI) circuits enhance, the need for faster floorplanning algorithms also grows. The...

1 Motivation We want to numerically approximate coefficients in a Fourier series. The first step is to see how the trapezoidal rule applies when numerically computing the integral (2π) −1 2π 0 F (t)dt, where F (t) is a continuous, 2π-periodic function. Applying the trapezoidal rule with the stepsize taken to be h = 2π/n for some integer n ≥ 1 results in (2π) −1 2π 0 F (t)dt ≈ 1 n n−1 j=0 Y j , ...

An important step in the automation of electronic design is the assignment of the physical components on the target semiconductor die. The major aim of floorplanning is to distribute the modules of a circuit onto a chip to optimize its area, wire length and timing. As the density of very large scale integrated (VLSI) circuits enhance, the need for faster floorplanning algorithms also grows. The...

Intuitionistic Fuzzy numbers each of which is characterized by the degree of membership and the degree of non-membership of an element are a very useful means to depict the decision information in the process of decision making. The aim of this article is to investigate the approach to multiple attribute group decision making with intuitionistic trapezoidal fuzzy numbers, some operational laws ...

Recent advances in computation technology for decision/simulation and uncertainty analyses have revived interest in the the triangular distribution and its use to describe uncertainty of bounded input phenomena. The trapezoidal distribution, explicitly suggested by Pouliquen (1970) in the framework of risk and uncertainty analysis, is a generalization of the triangular distribution that allows ...

The objective of this investigation was to determine the quasi-static indentation response and failure mode in three-dimensional (3D) printed trapezoidal core structures, and to characterize the energy absorbed by the structures. In this work, the trapezoidal sandwich structure was designed in the following two ways. Firstly, the trapezoidal core along with its facesheet was 3D printed as a sin...

ne of the greatest challenges for software developers is forecasting the development effort for a software system for the last decades. The capability to provide a good estimation on software development efforts is necessitated by the project managers. Software effort estimation models divided into two main categories: algorithmic and non-algorithmic. Developers should be able to achieve practi...