Physics 6810: Assignment #1a

Update to area.cpp

• This is just some programming practice. Use other Activities 1 and Activities 2 programs for examples.
• Also, use a search engine (e.g., Google) to find examples. E.g., search for "c++ predefined pi" or "c++ inline square function" and look at several of the answers --- not only the first one! You'll often get pointed to stackoverflow.com, which is a great online resource!

Summing up vs. summing down

• You should find it useful to think of how the 1 + a + a + ... vs. a + a + ... + 1 problem from Activities 2 relates to this one. (That might also be a good code to start from in writing one for this problem.)
• If you only consider relatively small N, summing up and summing down should both have an error about equal to the machine precision. (Why?) So what should your plot show in this region? Be sure to increase N until you see a different trend (I used up to 10⁸).
• If you are looking for the effects of errors over a logarithmic range (e.g., N from 10² to 10⁸), you should vary N by multiplying by a factor each time through a loop and not by adding 1 (or some other number). That is, don't use:

   for (N = 100; N <= 100000000; N++) 
   {
      
   }
   

  which will take an enormous amount of time calculating lots of N values you don't need. Instead, use something like:

   for (N = 100; N <= 100000000; N *= 10) 
   {
      
   }
   

  which just calculates for 100, 1000, 100000, etc.
• Using 1000000000 is hard to read (you can't tell at once what power of 10 this is). What could you replace it with that is much clearer?
• When you output relative errors to a file for plotting, the "scientific" format is usually most useful, again because you are looking at logarithmic intervals. (See the codes we've used in class for examples.)
• It is only the absolute value of the relative error that is of interest. So use "fabs" (you need to "input "). It is particularly important to have only positive numbers if you are going to plot on logarithmic scales!
• Be careful that you don't confuse the machine precision with the smallest floating point number. For example, in single precision, the machine precision is about 10⁻⁷, but that does not mean that 10⁻¹⁰ is set to zero. It is only when you add two numbers that differ by a factor of more than 10⁻⁷ that the smaller number is effectively zero.
• A common bug when you have an outer loop stepping through N_max and an inner loop that does the sum of 1/n up to N_max is forgetting to reset your sum to zero when you start the inner loop. So if your summation variable is sum_up, make sure that sum_up = 0.; appears within the outer loop and not just at the beginning of the program.
• When interpreting a linear region I'm not looking for a quantitative interpretation of the slope. [Note: If you print enough points, the "linear" region just means the envelope of the largest errors.] You need only provide a qualitative interpretation of what you are seeing (e.g., what does linear mean on a log-log plot?).