DATA101 Assignment 5 Solution

1. The secant method for finding the solution x of an equation of the form
f(x) = 0
is

where two initial guesses x1 and x2 must be specified beforehand. With these guesses, we can use the formula to calculate x3, and with x2 and x3, we calculate x4, and so on. Typically, the solution is found to a few digits of accuracy in fewer than 5 steps, i.e. x6 should be a good approximation to the solution.
(a) (8 points) Write a function named secant() with three inputs including x1, x2 and a function f which returns the approximate solution of f(x) = 0 based on 3 steps of the formula given above. (An example of f could be cos for which x = 1.57 is a good approximation to a solution of cos(x) = 0.)
(b) (2 points) Use the secant function just created to verify that a solution to cos(x) = 0 is x = 1.57. Use starting values x1 = 1.56 and x2 = 1.58.
(c) (4 points) Write a function f() which takes a single argument x and returns the value of the function
f(x) = x3 − 2x + 3.
Apply the secant() function to find the solution of
x3 − 2x + 3 = 0.
Use −2 and −1.8 as your starting guesses.
2. (a) (8 points) Finish writing the function below which should be called WHunif and which computes n uniform pseudorandom numbers on the interval [0,1] using a random number generator (called the Wichman-Hill generator):
For j = 1,2,...,n,
xj = 171 xj−1 mod 30269 yj = 172 xj−1 mod 30307 zj = 170 xj−1 mod 30323 vj = xj/30269 + yj/30307 + zj/30323. uj = vj − [vj]
where x0, y0, and z0 are all initial seeds, and [v] is the integer part of v, or floor of v.
Your function should take n, and the seeds x, y, z as arguments, and return the vector u as output.
(This is a real generator which is actually used in the R function runif()).
? <- function(?, x, y, z) { u <- numeric(n) for (i in 1:n) {
?
?
?
?
? <- v - floor(v)
} ?
}
(b) (2 points) Obtain 20 uniform numbers using the above function with seeds 1, 2, and
3.
3. In this exercise, we will see how you can use uniform random numbers to simulate the tossing of a coin - where 0 represents a head, and 1 represents a tail.
(a) (5 points) Write a second function called WHcointoss which will generate n random 0’s and 10s with parameter p, based on uniform numbers generated by WHunif function, using the seeds 1, 2 and 3. That is, 1’s are generated if the corresponding uniform number is less than p, and 0’s are generated otherwise.
Input to the WHcointoss function should be n and p, and the output should be the vector of n coin toss outcomes.
(b) (2 points) Obtain 20 coin tosses with parameter p = .5 using your WHcointoss function.
4. (4 points) Write a function f that, with input x, returns the value of f(x) = log(x) + x where log(x) is the natural logarithm function. Use the secant function created in the demonstration to find the solution in the interval [0.5,1] to the equation f(x) = 0.
5. (a) (10 points) Write a function called myrandom which takes n x0 as input and that returns n values from the following random number generator. For j = 1,2,...,n,
xj = 171 xj−1 mod 30269 uj = xj/30269.
where x0 is the initial seed.
(b) (2 points) Evaluate 10 random numbers using myrandom and an initial seed of 25.
6. (a) (6 points) If a student guesses on a multiple choice test with 4 possible answers for each question, the student will be correct 25% of the time. Write a function called myguesses which takes the number of questions n as input and returns simulated outcomes (1, for correct and 0, for incorrect) for each of the n guesses. Use the function myrandom with initial seed 325.
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(b) (2 points) Try out the myguesses function on a test with 20 questions. How many answers were correct?
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