Numerical integration

Numerical methods are a collection of techniques for solving mathematical problems which lend themselves to implementation as computer programs. They exist to solve all sorts of problems, for example differentiation, minimisation, sorting data and solving differential equations. Numerical methods are particularly useful when it is either impossible or impractical to solve a problem analytically.

The trapezium rule for numerical integration of a function f (x) is

$$\int_{{x_1}}^{{x_N}}$$f (x)dx = h[$${\frac{{1}}{{2}}}$$f₁ + f₂ + f₃ + ... + f_N-1 + $${\frac{{1}}{{2}}}$$f_N] + O($${\frac{{(b-a)^3 f''}}{{N^2}}}$$)

= h[$${\frac{{1}}{{2}}}$$(f₁ + f_N) + $$\sum_{{i=2}}^{{N-1}}$$f_i]

where N - 1 is the number of trapezia, x₁ and x_N are the lower limit and upper limits of integration respectively, and h is the trapezium width.

Here you will use the trapezium rule to numerically integrate a function. This exercise will allow you to put into practice many aspects of Java programming that you have learnt so far. A reminder/primer on the trapezium rule can be found in Appendix C.

Exercise 7.1

(20 marks)

Write a program that will make use of the trapezium rule to numerically integrate a mathematical function.

The following steps are given as a guide on how to proceed with the program. It is recommended that you follow these steps closely to get all the available marks.

1. Write a separate method to return the value of the function you want to integrate. For example, this is a definition of f (x) = 2x:

       public static double f(double x)
       {
           return 2*x;
       }
   

2. Write a method which implements the trapezium rule. It should take as arguments the range over which to integrate (x₁ and x_N) and the number of steps (N). It should use these parameters to compute the integration using the trapezium rule and return the result. It will call the method f you defined in the previous step.

3. Write a main method to prompt the user to type in values of x₁, x_N, and N, call the above method and print the result. It should check the value of N supplied is valid before proceeding to any calculation, but it is sufficient just to exit with a auitable error message if the value is invalid.

4. Test your program, using a function linear in x as shown above, first with N = 1 (which should fail the validity check) then N = 2 and a few values where N > 2. Calculate the answer by integrating the function by hand. You should get exactly the right answer regardless of the value of N (why?)

5. Next, test your program with a second order polynomial for several values of N. Again, compare with the analytical (hand calculated) result.

6. Study what happens to the accuracy of the answer as N increases. Show how the convergence relates to N. Is this what you expect from the formula for the trapezium rule?

7. Based on your findings, devise a method to estimate the accuracy of a calculation when the correct answer is unknown.

At this point you should have established enough confidence in your program to use it on a function which cannot be integrated analytically.

When you have written your program, show a demonstrator that it works. You will then be allocated a function to integrate from those below. It will help to sketch the function before you try to integrate it.

(i)

The function x⁴log(x + $\sqrt{{x^2+1}}$) over the interval x = 0, 1.

(ii)

The function log(3x²) over the interval x = 0.7, 2.

(iii)

The function cos(x²) over the interval x = - 1, 1.

(iv)

The function cos(x³) over the interval x = - 1, 1.

(v)

The function sin(x²) over the interval x = 0, 1.7.

(vi)

The function sin(x³) over the interval x = 0, 1.4.

(vii)

The function -log(sin(x)) over the interval x = 0.1, 1.

(viii)

The function cos(tan(x)) over the interval x = - 1, 1.

Make sure you document the design, code and testing of your program as well as the final result. Most of the marks for this exercise will be awarded for these three things. As well as giving the answer, estimate the accuracy of this number using the method you devised in step 7 above. An accuracy of around 3 significant figures is sufficient.