Homework: Some Integration

Due Wednesday, October 24, 2001 in class

Your first task is to write a subroutine that performs numerical integration. The boys in the class will use the Extended trapeziodal rule and the girls will use the Extended Simpson rule.

• The subroutine should have the following arguments: the name of the function to be integrated, the limits of integration, and n, the number of points where the function is to be evaluated.
• The integration subroutine should go in a separate file.
• Use comment statements to explain the important variables used in the subroutine.
• Try out your integration routine on a simple function like x² to see if it works.

Students, as denoted by their initials, will look at different functions:

• RM, JF:
  The Bessel function of the first kind J₀(z); alternatively, you can try J₁(z). One can write this function in terms of an integral from 0 to pi. Bessel functions are used all over the place in science and engineering. Suggested domain: [0,10].
  J₀(10)=-0.245935764451348335...
  J₁(10)=0.0434727461688614367...
• EW, JT:
  The modified Bessel function of the first kind I₀(z). One can write this function in terms of an integral from 0 to pi. Bessel functions are used all over the place in science and engineering. Suggested domain: [0,3].
  I₀(3)=4.88079258586502409...
• SH, NG:
  The Frensel Integral C(z). Frensel Integrals are used in diffraction theory (optics). Suggested domain: [0,3].
  C(3)=0.605720789297685630...
• JO:
  The error function erf(z). This function is very important in the world of probability and statistics. Suggested domain: [0,3].
  erf(3)=0.999977909503001415...
• MN:
  The sine integral Si(z). Suggested domain: [0,10].
  Si(10)=1.658347594218874049...

1. Find a definition of your function in terms of an integral. The Geneva College library has:
   • M. Abramowitz and I. Stegun, Handbook of mathematical functions
   • I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products
2. Look up the behavior of your function in the limits z goes to zero and z goes to infinity. Print out or photocopy a graph of the function.
3. Write a subroutine (in the same file as your main program), for the function to be integrated. For some of the functions, you will need to use a global variable to specify the value of z in your subroutine; in FORTRAN, you would use a COMMON block statement.
4. Evaluate the integral for various values of n at the example point given above. How large must n be to obtain an error of 1.0E-2? How large must n be to obtain an error of 1.0E-5? How large must n be to obtain an error of 1.0E-8?
5. We have discussed the error associated with both integration techniques. Do your results agree with this? Explain.
6. Graph your function versus z; a suggested domain is given above. Use at least n=50 when calculating the necessary integrals.