Stat 5810, Applied Spatial Statistics

Homework Assignment 5 (10/9/00)

15 + 10 Points - Due Wed 10/18/00 in class or as a Web page

You may work in groups of 2 people on this assignment and turn in a joint solution.

You have to use S-Plus and S+SpatialStats to answer all of the following questions. Your solutions should consist of the S-Plus commands you have typed, any S-Plus functions you have written, and the output (text and postscript graphics) produced in S-Plus, i.e., exactly the same things as for Homework Assignment 3.

1) We know that the sum of integers from 1 through n is n * (n + 1) / 2, e.g., 1 + 2 + 3 + 4 + 5 + 6 = 6 * 7 / 2 = 21. In this question, you should not use this fact. Instead, you should write 3 different S-Plus functions that sum up integers from 1 through a value n which is provided by the user as the input to each of these functions.

1 of your functions should use a for loop, 1 function a while loop, and 1 function a repeat loop. Test your function for n = 1, 2, 3, 6, 10.

Try to make one of your 3 functions fool proof and print error messages when the input is not a valid integer, e.g., n = 0, -5, or 12.5. For this check, the functions floor or trunc might be helpful. (5 Points)
2) If we sum 1 / 1^2 + 1 / 2^2 + 1 / 3^2 + 1 / 4^2 + 1 / 5^2 + ... + 1 / n^2 + ... and continue indefinitely, the result is pi^2 / 6 (let's call this pi6). For practical purposes, we want to know which value of n brings us close to pi6, i.e., for which n do we observe that (1 / 1^2 + 1 / 2^2 + 1 / 3^2 + 1 / 4^2 + 1 / 5^2 + ... + 1 / n^2) > (pi^2 / 6 - delta) for the first time.

Write an S-Plus function that takes delta as its input and returns n. You may assume that only valid values for delta will be used as input to this function. Test your function with
```
  delta1 _ pi - 3
  delta2 _ 0.01
  delta3 _ 0.001
  delta4 _ 0.0001
  delta5 _ 0.00001
```
(5 Points)
3) Write an S-Plus function that prints all prime numbers from 1 through a user input n. A prime number is an integer that is only dividable by 1 and itself, e.g., 2, 3, 5, 7, 11, etc. are prime numbers. 1 is not a prime number. You probably need the modulo operator "%%" which gives the remainder when its first operand is divided by its second operand.

First develop a basic function and see what you get for n = 1, 2, 3, 4, 10, and 100.

Then add a few if statements that help to speed up your function. For example, notice that 2 is the only even prime number. Also, when checking whether an integer x is dividable by another integer y with remainder 0, the largest value for y you have to consider is sqrt (x), e.g., for x = 9 it is ymax = 3 and for x = 81 it is ymax = 9 (obviously, 81 = 9 * 9 = 27 * 3 -- but the pair (27, 3) has already been encountered before when we learned that 81 = 3 * 27). Make sure that your enhanced function still returns the same result as your initial function. (5 Points)
4) Also turn in questions 1) and 3) from Homework Assignment 4 at this time (questions 2), 4), and 5) from Homework Assignment 4 will be due later this month). For question 3), you should restrict yourself to material related to Section 2.3 from our lecture. There will be a similar question in the future that is related to Section 2.4 from our lecture. (5 + 5 Points)