Math 171 Spring 2005, Lab 8

Due Saturday Saturday, February 26, 11:59 P.M.

1. Overview
2. Assignment
3. Hints

Overview

The purpose of this lab is to introduce you to using DO loops in Fortran. Additionally, you will gain practice implementing two fundamental algorithms of numerical mathematics (series approximation and bisection).

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Assignment

Write a fortran program called pie_approx that calculates either pi or e (according to the user's specification) using the maximum possible precision. Your program must satisfy the following requirements:

• You should error check the input provided by the user. If invalid input is provided, you should print an error message and terminate program execution.
• All calculations should be performed using 16-byte reals, which can be specified using a declaration of the form

  
  REAL(KIND=16) :: x, y
  
  

  Note that in general, you can specify an n-byte real (for n=4, 8, or 16) using the declaration

  
  REAL(KIND=n) :: x, y
  
  

  Warning: this behavior is specific to the Intel Fortran compiler; the correct value of n may vary from one compiler to another. For example, see the table on page 452 of the text for the correct value of n used to sepcify a 4/8/16 byte (i.e. 32/64/128 bit) real variable on various compilers.
• Your code must use only loops, branches, and the elementary arithmetic operations (i.e. +,-,/,*, and **). Note that this is actually redundant with the previous requirement since the standard trigonometric and exponential functions only perform calculations using 8-byte reals.
• You should calculate the number e using the usual power series representation of the exponential function centered at 0. This expansion can be found in your Calculus textbook or here.
• You should calculate pi using bisection on the function sin(x) on an interval containing only one zero.
• You should implement the calculation of sin(x) yourself using the usual power series expansion at 0 found in Calculus textbooks, or here.
• In each case, you should test for convergence by checking that two successive approximations are equal (i.e. the actual difference between the two approximations is a smaller number than the machine can represent at that magnitude).
• In addition to the approximation, your program should report how many terms of the power series were needed to approximate e or how many iterations of bisection were needed to approximate pi, according to which approximation the user has specified.

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Hints

1. To start out, just write a skeletal program that prompts the user for which calculation they want. Put in an IF-THEN-ELSE statement that prints a message saying something like "the calculation of e will go here". In general, any code that serves as a placeholder is called a stub.
2. The calculation of e is easier. So, your next step should be to implement a DO loop (inside the e clause of the IF statement) for the power series expansion of the exponential function. Make sure this calculation is correct to the maximum number of digits of accuracy.
3. Now get your power series for sin(x) working. Don't worry about the bisection method yet, just try your calculation for several different values of x and make sure they come back correct.
4. Finish by adding an additional DO loop for the bisection method. This loop should surround your power series approximation to sin(x). If you set up the calculation correctly, you shouldn't have to change anything inside the inner loop (except the indentation level).
5. Test your program by checking against the various approximations to pi and e that can be found on the web.
6. Finally, go back and add the WRITE statements for the requested stats on your program's performance if you haven't already done so.

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