انت هنا الان : شبكة جامعة بابل > موقع الكلية > نظام التعليم الالكتروني > مشاهدة المحاضرة

Numerical Integration

الكلية كلية العلوم للبنات القسم قسم فيزياء الليزر المرحلة 3
أستاذ المادة نزار سالم شنان الزبيدي 4/27/2011 7:31:00 AM

Numerical Integration Chapter 6

Nizar Salim 1 lecture 1

Numerical Integration

6.1 INTRODUCTION

A set of tabular data is illustrated in Figure 6.1 in the form of a set of [x, f(x)] pairs. The

function f(x) is known at a finite set of discrete values of x. Interpolation within a set of

discrete data is presented in Chapter 3. Differentiation within a set of tabular data is

presented in Chapter 4. Integration of a set of tabular data is presented in this chapter.

The discrete data in Figure 6.1 are actually values of the function f(x) = 1/x, which is

used as the example problem in this chapter.

The evaluation of integrals, a process known as integration or quadrature, is

required in many problems in engineering and science.

The function f(x), which is to be integrated, may be a known function or a set of discrete

data. Some known functions have an exact integral, in which case Eq. (6.1) can be

evaluated exactly in closed form. Many known functions, however, do not have an exact

integral, and an approximate numerical procedure is required to evaluate Eq. (6.1). In

many cases, the function f(x) is known only at a set of discrete points, in which case an

approximate numerical procedure is again required to evaluate Eq.(6.1). The evaluation

of integrals by approximate numerical procedures is the subject of this chapter.

Numerical Integration Chapter 6

Nizar Salim 2 lecture 1

Numerical integration (quadrature) formulas can be developed by fitting

approximating functions (e.g., polynomials) to discrete data and integrating the

approximating function:

Several types of problems arise. The function to be integrated may be known only at a

finite set of discrete points. In that case, an approximating polynomial is fit to the discrete

points, or several subsets of the discrete points, and the resulting polynomial, or

polynomials, is integrated.

When a known function is to be integrated, several parameters are under our

control.The total number of discrete points can be chosen arbitrarily. The degree of the

approximating polynomial chosen to represent the discrete data can be chosen. The

locations of the points at which the known function is discretized can also be chosen to

enhance the accuracy of the procedure.

Procedures are presented in this chapter for all of the situations discussed above.

Direct fit polynomials are applied to prespecified unequally spaced data. Integration

formulas based on Newton forward-difference polynomials, which are called Newton-

Cotes formulas

The numerical evaluation of multiple integrals is discussed briefly.

The simple function

Numerical Integration Chapter 6

Nizar Salim 3 lecture 1

6.2 DIRECT FIT POLYNOMIALS

A straightforward numerical integration procedure that can be used for both

unequally spaced data and equally spaced data is based on fitting the data by a direct fit

polynomial and integrating that polynomial. Thus,

After the approximating polynomial has been fit, the integral becomes

Numerical Integration Chapter 6

Nizar Salim 4 lecture 1

6.3 NEWTON-COTES FORMULAS

The direct fit polynomial procedure presented in Section 6.2 requires a significant

amount of effort in the evaluation of the polynomial coefficients. When the function to be

integrated is known at equally spaced points, the Newton forward-difference polynomial

presented in Section 4.6.2 can be fit to the discrete data with much less effort, thus

significantly decreasing the amount of effort required. The resulting formulas are called

Newton-Cotes formulas. Thus,

Numerical Integration Chapter 6

Nizar Salim 5 lecture 1

Equation (6.13) requires that the approximating polynomial be an explicit

function of x, whereas Eq. (6.14) is implicit in x. Either Eq. (6.14) must be made explicit

in x by introducing Eq. (6.16) into Eq. (6.14), or the second integral in Eq. (6.13) must be

transformed into an explicit function of s, so that Eq. (6.14) can be used directly. The first

approach leads to a complicated result, so the second approach is taken. Thus,

Each choice of the degree n of the interpolating polynomial yields a different

Newton-Cotes formula. Table 6.1 lists the more common formulas. Higher-order

formulas have been developed, but those presented in Table 6.1 are sufficient for most

problems in engineering and science. The rectangle rule has

Numerical Integration Chapter 6

Nizar Salim 6 lecture 1

poor accuracy, so it is not considered further. The other three roles are developed in this

section.

Some terminology must be defined before proceeding with the development of

the Newton-Cotes formulas. Figure 6.4 illustrates the region of integration. The distance

between the lower and upper limits of integration is called the range of integration. The

distance between any two data points is called an increment. A linear polynomial requires

one increment and two data points to obtain a fit. A quadratic polynomial requires two

increments and three data points to obtain a fit. And so on for higher-degree polynomials.

The group of increments required to fit a polynomial is called an interval. A

linear polynomial requires an interval consisting of only one increment. A quadratic

polynomial requires an interval containing two increments. And so on. The total range of

integration can consist of one or more intervals. Each interval consists of one or more

increments, depending on the degree of the approximating polynomial.

6.3.1 The Trapezoid Rule

The trapezoid rule for a single interval is obtained by fitting a first-degree polynomial to

two discrete points, as illustrated in Figure 6.5. The upper limit of integration x1

corresponds to s = 1. Thus, Eq.(6.19) gives

Numerical Integration Chapter 6

Nizar Salim 7 lecture 1

Example 6.2. The trapezoid rule

Let s solve the example problem presented in Section 6.1 by the trapezoid rule. Recall

that f(x) = l/x. Solving the problem for the range of integration consisting of only one

interval of h = 0.8 gives

Numerical Integration Chapter 6

Nizar Salim 8 lecture 1

This document was created with Win2PDF available at http://www.daneprairie.com.

The unregistered version of Win2PDF is for evaluation or non-commercial use only.

المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .