Part 2. Numerical Integration

TABLE 2.1 Important integration formulas

Method	Formula	Error
Left Sum Approximation		~O(h)
Right Sum Approximation		~O(h)
Midpoint Approximation		~O(h2)
Trapezoidal Approximation		~O(h2)
Simpson’s (1/3) Approximation		~O(h4)
Simple Monte Carlo Approximation		~

^* ; ^** .

Example 1

Suppose we want to evaluate the integral . The definite integral is the area under the curve , above the x-axis and between the lines x = 0 and x = 10. The actual value can be easily obtained as follows:

.

Solution. n = 4(h = 2.5)

a)Use it in conjunction with Left sum approximation to integrate the same function for four subintervals:

.

The numerical error and the true percent relative error are:

.

b)Using Right sum approximation:

.

.

c)Using Trapezoidal Rule (four subintervals):

.

.

d)Using Simpson's 1/3 Rule (four subintervals):

.

.

Example 2

Suppose we want to evaluate the following double integral:

.

For the numerical evaluations use both the trapezoidal and Simpson's 1/3 rules. For both cases, use the multiple-application version, with n = 4. Compute percent relative errors for the numerical results.

Solution.

a) Analytically the actual value can be easily obtained as follows:

b)Numerical integration with n = 4:(h_x = 1.0; h_y = 2.0)

c)Use it in conjunction with Trapezoidal Rule to integrate the same function for four subintervals:

.

d)Using Simpson's 1/3 Rule:

.

Note:

· Trapezoidal rule gives the exact value of the integral for linear polynomials;

· The Simpson's rule gives the exact value of the integral for polynomials of degree three or less;

· An even number subintervals must be utilized to the Simpson's rule;

· As the number of subintervals increases, the approximations get better.

Problems

1. Integrate the following function both analytically and numerically.

.

For the numerical evaluations use: (a) left sum approximation,
(b)right sum approximation, and (c)midpoint approximation. For all cases use version with n = 4. Compute percent relative errors for the numerical results.

2. Integrate the following function:

.

Note that the true value is I = 0.602297. Use both the trapezoidal and Simpson's 1/3 rules to numerically integrate the function with n = 4.

3. Evaluate the triple integral

(a) analytically, and (b)using Simpson's 1/3 rule with n = 4.
For (b)compute the percent relative error ( ).

4. Integrate the following function both analytically and numerically.

.

For the numerical evaluations use: (a) midpoint approximation,
(b)Monte Carlo integration.

A random sample ( ) of size 20 is given below:

i		i		i		i
	0.0105		0.2171		0.2747		0,4150
	0.0039		0.5369		0.4442		0.1651
	0.3351		0.1957		0.1089		0.8154
	0.0332		0.7003		0.6982		0.6855
	0.3557		0.9498		0.5643		0.7643

For all cases use version with n = 10. Compute percent relative errors for the numerical results. Discuss your results.

5. The function

can be used to generate the following table of unequally spaced data:

Evaluate the integral from a = 0 to b = 0.54 using: (a)analytical means, (b)left sum approximation, (c)right sum approximation, and (d)the trapezoidal rule. For (b), (c)and (d),compute the percent relative error ( ).

Date: 2016-01-14; view: 1067