20.7.2012

Numerical Analysis

Module IX

Numerical Integration

The objectives attained by this module are:

  1. StudyNumerical Integration
  2. Solving problems of numerical integration
  3. Familiarizes the applicability of integration.

9.1.Introduction:

In this module,using numerical methods we find the definite integral of the function y=f(x) in the range [a,b] . The function f(x) need not be given explicitly. Usually we are given with tabular values of the function y=f(x) with. The process of finding definite integral of a function f(x) from a given set of tabulated values of f(x) is known as numerical integration or numerical quadrature.

9.2.Newton-Cote’s QuadratureFormula

Consider the integral. Let are the points dividing the interval [a,b] in to n subintervals of equal width h, where . Assume the values of the function f(x) are given for the variable values as.

Let us consider. This gives.

Now, .

In the integral, let us change the variable to u,where .

This gives . Also the range of integration changes as:

When and when .

Hence,

For some set of tabular values , we have Newton’s Forward difference Formula as,

where, .

Now,

This formula is known as Newton-Cote’s quadrature formula.

By giving different values for n, different quadrature formulae are derived.

9.3.Trapezoidal Rule

When n=1, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Trapezoidal rule.

When n=1, all the differences except the first order differences are zero.

Hence,

Similarly,

.

Proceeding like this,

Adding all these integrals, we get,

That is,

9.4.Simpson’s one third rule

When n=2, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Simpson’s one third rule.

When n=2, all the differences of order threeor higher becomes zero.

Hence,

Hence,

Similarly,

Proceeding like this and under the assumption n is even, we get

Hence the summation of the above integrals gives,

That is,

This is known as Simpson’s one third rule.

9.5.Simpson’s three eight rule

When n=3, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Simpson’s three eight rule.

When n=3, all the differences of order fouror higher becomes zero.

Hence,

Similarly,

Finally, under the assumption that n is a multiple of three,

Adding these integrals, we get,

That is,

This equation is the Simpson’s three eight rule.

9.6.Boole’s rule

When n=4, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Boole’s rule. This can be derived as,

9.7.Weddle’s rule

When n=6, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Weddle’s rule. This can be derived as,

9.8.Problems

Problem1: Using Trapezoidal rule solve the integral, with four subintervals.

Solution:

For n subintervals, the trapezoidal rule for the integral of a function in the range [a,b] is,

Here to consider n=4.

Now,

In our integral, , the range of integral [0,1] is divided into four equal subinterval of width h=0.25, by the points, 0.00,0.25,0.50,0.75 and 1.

Considering them as the x values, corresponding values of the integranddenoted by are 0.10, 0.08649, 0.07547, 0.06639 and 0.05882 respectively.

Hence,

= 0.07694.

Problem2: Find using Simpson’s one third rule.

Solution:

By Simpson’s one third rule,

In our integral,, let the range [0,10] is subdivided into 10 equal interval of width h=1, by the x values 0,1,2,3,4,5,6,7,8,9 and 10. Corresponding y values of the function are listed below:

x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
y / 1 / 0.5 / 0.2 / 0.1 / 0.0588 / 0.0385 / 0.0270 / 0.02 / 0.0154 / 0.0122 / 0.0099

Thus,

.

Problem3: Evaluate using Simpson’s three eight rule.

Solution:

By Simpson’s three eight rule,

Let the limit of integral [0,6] be divided into six equal parts with interval h=1, using the x values 0,1,2,3,4,5 and 6. Corresponding y values of the given integrand function are,

x / 0 / 1 / 2 / 3 / 4 / 5 / 6
y / 0.333 / 0.25 / 0.1429 / 0.1 / 0.0526 / 0.0357 / 0.0256

Thus,

For n=6,

.

9.9.Gaussian Quadrature Formula

The quadrature formulae so far considered are in connection with equi-spaced pre determined x values. Gauss quadrature formulae are formulae for finding integral based on the x values chosen wisely in between the interval of integration without the restrication of equally spaced x values.

(i)Two point Gaussian quadrature formula

Consider the integral.

Let us express theintegral in the form of a linear combination as, .

Here the four unknowns are solved under the assumption that f(x) satisfies the equation for any polynomial of degree less than four. That is the equation is true for.

Solving these equations, we get, .

Hence,

This expression is known as Two point Gaussian quadrature formula. This expression helps to solve the integral of any polynomial of degree less than or equal to three. If the limit of integral is from a to b, by suitable change of variable the limit can be changed to -1 to 1 and the integral is solved by the above formula.

Consider the integral. Change the variable in the integration x to t, where, . This gives, and as x=a, t=-1 and as x=b, t=+1.

Now, . This is solved using (5).

(ii)Three point Gaussian quadrature formula

The integral is expressed as . The six unknowns are solved under the assumption that f(x) satisfies the equation for any polynomial of degree less than or equal to five. That is the equation is assumed true for.

Solving the equations obtained under these assumptions, we get,

.

Problem: Evaluate using two point Gaussian quadrature formula.

Solution:

Hence,

By two point Gaussian quadrature formula, we have,

=200.222.

9.10.Assignments

  1. Use trapezoidal rule to solve the integral, with 4 subintervals.
  2. Evaluate using Simpson’s one third rule.
  3. Solve with Simpson’s three eight rule.
  4. Evaluate using two point Gaussian quadrature formula.

9.11.Quiz

  1. In trapezoidal rule, is

(i) (ii) (iii) (iv)

  1. Name of the rule when n=4, in Newton-Cote’s quadrature formula is

(i) Trapezoidal rule (ii) Simpson’s one third rule (iii) Weddle’s rule

(iv) Boole’s rule.

  1. By two point Gaussian quadrature formula

(i) (ii)

(iii) (iv) None of these

9.12.Glossary

9.12.1.Numerical quadrature:The process of finding definite integral of a function f(x) from a given set of tabulated values of f(x) is known as numerical integration or numerical quadrature.

9.12.2.Trapezoidal rule:When n=1, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Trapezoidal rule.

9.12.3.Simpson’s one third rule: When n=2, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Simpson’s one third rule.

9.12.4.Simpson’s three eight rule:When n=3, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Simpson’s three eight rule.

9.12.5.Boole’s rule: When n=4, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Boole’s rule.

9.12.6.Weddle’s rule: When n=6, in Newton-Cote’s quadrature formula, we get a quadrature formula, known as Weddle’s rule.

9.13.FAQs

1. What is meant by numerical quadrature?

2. How the numerical quadrature is performed?

3. What are the finite differences in common use?

9.14.References:

  • C.E.Froberg, Introduction to Numerical Analysis (Second Edn.), Addison-Wesley, 1979.
  • James B Scarborough,Numerical Mathematical Analysis, (Oxford and IBH Publishing Co. Pvt.Ltd.), 1966.
  • M.K.Jain,S.R.K.Iyengar,R.K.Jain,Numerical methods: Problems and solutions (New Age International (P) Ltd.), 1996.
  • M.K.Jain,S.R.K.Iyengar,R.K.Jain, Numerical methods for Scientific and Engineering Computation (New Age International (P) Ltd.), 1999.
  • S.S. Sastry, Introductory methods of Numerical Analysis (Third Edn.), PHI India Pvt Ltd, 1998.

9.15.Summary:

This module is to describe the numerical methods in integration of a function. Various methods are discussed. First we considered the integration of a function where the equally spaced variable values and corresponding function values are given. The integration is performed numerically by any of the appropriate method derived out ofNewton-Cote’s Quadrature Formula. To find integral based on the x values chosen wisely in between the limit of integration without the restriction of equally spaced x values the Gauss quadrature formulae are described. Problems are solved for illustration.

*************************

ANSWERS TO QUIZ AND FAQs

Answers to Quiz:

(1) (i)

(2) (iv) Boole’s rule.

(3) (iii)

Answers to FAQs:

  1. What is meant by numerical quadrature?

Ans:The process of finding definite integral of a function f(x) from a given set of tabulated values of f(x) is known as numerical integration or numerical quadrature.

  1. How the numerical quadrature is performed?

Ans:From the tabular values of the function for various values of the variable, apply the expressions for numerical integration derived. If the values given are equally spaced use any of the suitable expression derived out of Newton-Cote’s Quadrature Formula. On other hand numerical integration also can be performed using the function values of arbitrarily selected variable values using Gaussian Quadrature Formula.

*******************************

Prepared by :Dr.Aneesh Kumar.K

Content edited by:Dr. Bijumon.R