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for k = n, n 1, . . . , m + 2, m + 1, where En = 0. At each stage of this algorithm the initial error is reduced by factor 1/k rather than being magni ed as it was in the procedure of (9.4). Other than potential numerical stability problems, which may or may not be easy to solve, it is apparent that not all integration problems may be cast into a recursive form. It is also possible that f (x) is not known for all x R. We might know f (x) only on a nite subset of R, or perhaps a countably in nite subset of R. This situation might arise in the context of obtaining f (x) experimentally. Such a scenario would rule out the previous suggestions. Thus, there is much room to consider alternative methodologies. In this chapter we consider what may be collectively called quadrature methods. For the most part, these are based on applying some of the interpolation ideas considered in 6. But Gaussian quadrature (see Section 9.4) also employs orthogonal polynomials (material from 5). This is dedicated mainly to the subject of numerical integration by quadratures. But the nal section considers numerical approximations to derivatives (i.e., numerical differentiation). Numerical differentiation is relevant to the numerical solution of differential equations (to be considered in later chapters), and we have mentioned that it is relevant to spline interpolation (Section 6.5). In fact, it can also nd a role in re ned methods of numerical integration (Section 9.5). 9.2 TRAPEZOIDAL RULE A simple approach to numerical integration is the following. In this book we implicitly assume all functions are Riemann integrable. From elementary calculus such integrals are obtained by the limiting process I=

package com.wrox.beginspring.pix.dao.examples; import import import import java.sql.PreparedStatement; java.sql.ResultSet; java.sql.SQLException; java.util.List;

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FIGURE 14.43. Sampling and aliasing in one dimension: (a) the input frequency is less than the Nyquist frequency and (b) the input frequency exceeds the Nyquist frequency.

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Consider these three styles of web applications and then think about the page flow for each one. Style one probably has no need to define a formal page flow; styles two and three definitely do. But who s to say which applications need a formal page flow and which do not There s no single test to determine this, so it s like anything else a judgement call. The best way to decide is to understand the problems solved by a formal page flow mechanism to see if it will benefit your projects. This is exactly what this chapter helps you do. By the end of it you will understand the following: Web flow background The reasons for using a formal page flow mechanism, specifically Spring Web Flow The concepts that make up a web flow The configuration of Spring Web Flow Spring Web Flow components, such as views and actions How to construct an application using Spring Web Flow How to test your SWF application

the beam-beam tune shift per collision is then All=-. (6.21)

The preceding illustrates the classic three-tier architecture typical of Java EE systems. In Figure B-1, the web applications are components that are deployed and run in the web tier. Enterprise JavaBeans (EJBs) are components that run in the business tier within an EJB container. Servlets are software components that run in the web tier within a servlet container (such as Jetty or Tomcat), also called a web (tier) container. The important items to notice in this architecture are: Application servers must be installed and maintained professionally in a production environment Software components are created by the application developer and deployed to this server environment Components must be written specifically for the Java EE implementations newer editions may not maintain backwards compatibility Because components are coded to the Java EE family of specifications, it must support the implementation of specific interfaces and mandate a certain inheritance hierarchy among components Components coded to the Java EE family of specifications must be tested while being run inside the EJB container; because the container is tied to external and possibly legacy resources and servers, testing can be difficult to set up The cycle of build, unit-test, deploy, integration-test can be very time-consuming within this environment. This is because of the tight coupling between the components to the container, and the dependencies on external connections and external resources involved during this cycle. Because application servers are usually full of features (and often called heavyweight for this reason), the application components can draw on a rich set of container services available in a typical Java EE environment In large environments where multiple application servers are deployed, Java EE can provide better support for server-side application scaling (usually via proprietary means) and distributed API standards such as distributed transaction (XA)

9