#########################################################
################ BTRY/STSCI 4520 ########################
################## Homework 1 ###########################
############# Due: Feb 9, 2018 #########################
#########################################################
# Instructions: save this file in the format <NetID>_HW1.R.
# Complete each question using code below the question number.
# You need only upload this file to CMS.
# Note, we assume your working directory contains any files
# that accompany this one.
# Further note: 10% will be deducted if your file produces
# an error when run. If your code produces an error and you
# cannot find it, comment out that portion of the code and we
# will give partial credit for it.
# Do not use either the function set.seed() or rm(list=ls())
# in your code.
################################
# Question 1: (from JMR 3.9.1) #
################################
# The function f(x) takes the values
#
# -x^3 if x < 0
# x^2 if 0 < x < 1
# sqrt(x) if x > 1
#
# Write a function to compute
# For the values
x.values = seq(-2,2,by=0.1)
# create a vector y.values that give the corresponding
# values of f(x). Plot y.values against x.values.
#################################################
# Question 2: (from JMR 3.9.2-3.9.4 and 5.7.2) #
#################################################
# For this problem we will consider calculating a
# sum of powers defining the function
#
# h(x,n) = 1 + x + x^2 + x^3 + … + x^n
#
# a) Write a function to calculate h(x,n) for any x
# and n using a for loop.
# b) Re-write the same function to use a while loop
# instead
# c) Re-write the function again to use no loops and
# only employ vector operations
hn.vec = function(x,n)
{
return (sum(x^(0:n)))
}
# (Note that JMR gives some specific values to check this,
# these will be included in the checking script for the
# homework.
# d) As a variation, we are interested in calculating only
# the sum of the even powers
#
# h2(x,n) = 1 + x^2 + x^4 + …. + x^(2*floor(n/2))
#
# where floor(n/2) is the largest whole number smaller than n/2,
# that 2*floor(n/2) is the largest even number smaller than n.
#
# Modify the code used by any of your functions above to
# calculate this in
hn.skip = function(x,n,skip=2)
{
return (sum(x^(seq(0, skip*floor(n/skip), skip))))
}
# BONUS use the skip argument in hn.skip to sum up
#
# 1 + x^s + x^(2s) + …
#
# for any integer s.
hn.bonus = function(x,s)
{
return (hn.skip(x^s, 1000, skip = 1))
}
##################################################
# Question 3: Matrix Multiplication the Slow Way #
##################################################
# Recall that for matrices A and B where A has the same
# number columns as B has rows, we can write the matrix
# product
#
# C = AB
#
# by saying that
#
# C_{ij} = sum_k A_{ik} B_{kj}
# Write a function that takes in to matrices A and B and
# returns C and which only ever sums two numbers together
# or multiplies two numbers together. That is, you are
# restricted to using only for loops and the mathematical
# operations + and *.
# Consider the two matrices
A = matrix(runif(200*200),200,200)
B = matrix(runif(200*200),200,200)
# How does the timing of
matprod(A,B)
# Compare to the built-in function
A%*%B
# it is much slower than built-in function
########################################
# Question 4: Tree Heights (JMR 6.5.3) #
########################################
# b) Write a function that extracts a particular
# habitat and produces the corresponding plot
habitatplot = function(num,data)
{
td = data[data$habitat == num, ]
plot(td[, “height.ft”], td[, “age”], xlab=”height”, ylab=”age”, main=paste(“habitat”, num))
}
# a) Create the plot requested in this question. The data in
# treegrowth.txt are available on CMS. You may do part b first
# and then provide the answer for part a.
###########################################
# Question 5: The Game of Life: JMR 5.7.6 #
###########################################
# You will need to write the function
neighbours = function (A,i,j,n)
# You can find the program to run the game of life
# in life.r; the glider gun initialization is in
# glidergun.r
###################################
# Bonus: JMR 5.4 #
###################################
# The function below gives misleading outputs
random.sum <- function(n) {
# For example, try
show(random.sum(10))
show(random.sum(5))
# Fix this function so that it does what it says it does. Note in comments below
# which lines you fixed and why they would not work. You may provide your answers
# with this function commented out.
# random.sum <- function(n) {
# # sum of n random numbers
# x[1:n] <- ceiling(10*runif(n)) # this line is changed to x <- ceiling(10*runif(n))
# cat(“x:”, x[1:n], “\n”)
# return(sum(x))
# }
# x[1:n] would change the n elements of global variable x
# the sum will sum the n elements plus remaining elements in x