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Stats 102A HW4

 Stats 102A HW4 Due March 12, 2021

General Notes
• You will submit a minimum of three files, the core files must conform to the following
naming conventions (including capitalization and underscores). 123456789 is a placeholder,
please replace these nine digits with your nine-digit Bruin ID. The files you must submit are:
1. 123456789_stats102a_hw1.R: An R script file containing all of the functions and
constructors you wrote for the homework. The first line of your .Rmd file, after
loading libraries, should be sourcing this script file.
2. 123456789_stats102a_hw1.Rmd: Your markdown file which generates the output file
of your submission.
3. 123456789_stats102a_hw1.html/pdf : Your output file, either a PDF or an HTML file
depending on the output you choose to generate.
4. included image files: You may name these what you choose, but you must include all of
the image files you generated for your structured flowcharts, otherwise your file will not
knit.
If you fail to submit any of the required core files you will receive ZERO points for the
assignment. If you submit any files which do not conform to the specified naming
convention, you will receive (at most) half credit for the assignment.
• Your .Rmd file must knit. If your .Rmd file does not knit you will receive (at most) half
credit for the assignment.
The two most common reason files fail to knit are because of workspace/directory structure
issues and because of missing include files. To remedy the first, ensure all of the file paths in
your document are relative paths pointing at the current working directory. To remedy the
second, simply make sure you upload any and all files you source or include in your .Rmd
file.
• Your coding should adhere to the tidyverse style guide: https://style.tidyverse.org/. • All pseudocode or flowcharts should be done on separate sheets of paper, but be included,
inline as images, in your final markdown document.
• Any functions/classes you write should have the corresponding comments as the following
format.
my_function <- function(x, y, ...){
#A short description of the function
#Args:
#x: Variable type and dimension
#y: Variable type and dimension
#Return:
#Variable type and dimension
Your codes begin here
} 1
Stats 102A HW4 Due March 12, 2021
NOTE: Everything you need to do this assignment is here, in your class notes, or was covered in
discussion or lecture.
• DO NOT look for solutions online.
• DO NOT collaborate with anyone inside (or outside) of this class.
• Work INDEPENDENTLY on this assignment.
• EVERYTHING you submit MUST be 100% your, original, work product. Any student
suspected of plagiarizing, in whole or in part, any portion of this assignment, will be
immediately referred to the Dean of Student’s office without warning.
1: Dealing with Large Numbers
To calculate with large floating point numbers we define objects called (p, q) number with a list as
its base type. The list should have four components. The first one is either the integer +1 or the
integer −1. It gives the sign of the number. The second and third are p and q. And the fourth
component is a vector of p + q + 1 integers between zero and nine. For example,
x <- structure(list(sign = 1, p = 3, q = 4, nums = 1:8), class = "pqnumber")
(a) Write a constructor function, an appropriate predicate function, appropriate coercion
functions, and a useful print() method.
• The constructor takes the four arguments: sign, p, q, and nums. Then check if the
arguments satisfy all requirements for a (p, q) number. If not, stop with an error
message. If yes, return a (p, q) number object.
• A predicate is a function that returns a single TRUE or FALSE, like is.data.frame(), or
is.factor(). Your predicate function should be is_pqnumber() and should behave as
expected.
• A useful print() method should allow users to print a (p, q) number object with the
decimal representation defined as follows:
x = Xq s= p xs × 10s
Thus p = 3 and q = 4 with nums = 1:8 has the decimal value
0.001 + 0.002 + 0.3 + 4 + 50 + 600 + 7000 + 80000 = 87654.321
• Please create two (p, q) number objects for the following numbers to
demonstrateis_pqnumber(), print():
(a) sign = 1, p = 3, q = 4, nums = 1:8
(b) sign = 1, p = 6, q = 0, nums = c(3,9,5,1,4,1,3)
(c) sign = -1, p = 5, q = 1, nums = c(2,8,2,8,1,7,2)
2
Stats 102A HW4 Due March 12, 2021
• A coercion function forces an object to belong to a class, such as as.factor() or
as_tibble(). You will create a generic coercion function as_pqnumber() which will
accept a numeric or integer argument x and return the appropriate (p, q) number object.
For example, given the decimal number 3.14 with p = 3 and q = 4, the function will
return a (p, q) number with num = c(0, 4, 1, 3, 0, 0, 0, 0). You should also create a
generic as_numeric() function and a method to handle a (p, q) number. You may use
the provided example to showcase your functions.
(b) Addition and Subtraction
Write an addition function and a subtraction function. Suppose we have two positive (p, q)
number objects x and y. Write a function to calculate the sum of x and y. Clearly its decimal
representation is
x + y = Xq s= p(xs + ys) × 10s
but this cannot be used directly because we can have xs + ys > 9.
So we need a carry-over algorithm that moves the extra digits in the appropriate way. Same
as one would do when adding two large numbers on paper. Also we need a special provision
for overflow, because the sum of two (p, q) number objects can be too big to be a (p, q)
number.
A subtraction algorithm should have a carry-over algorithm that borrows 10 in the same way
as you would do a subtraction with pencil-and-paper.
Your functions should work for both positive and negative (p, q) numbers.
(c) Multiplication
Write a multiplication function which can multiply two pqnumber objects. Think about how
you would multiply two large numbers by hand and implement that algorithm in R for two
pqnumber objects.
You may demonstrate your three functions using the examples provided in (a).
Note: Please attach the flowcharts or pseudo code for your addition function, subtraction
function, and multiplication function. Also, the cases provided here are only for your test. We
will use different arguments and objects to try your functions while grading. Therefore, try your
best to make your functions robust.
2: Root Finding
Use Newton’s method to compute the square root by using only simple arithmetic operations.
(a) Please write a function that computes the square root of an arbitrary positive number. The
function should take four arguments, the last three have default values:
1. a a non-negative number to compute the square root of.
2. tol determines when you consider two iterates to be equal.
3. iter_max gives the maximum number of iterations.
4. verbose determines if you want to display intermediate results.
3
Stats 102A HW4 Due March 12, 2021
(b) Give an example of your function by computing the square root of 5.
(c) How would the program change to compute an arbitrary, positive integer, root? Please use
calculus to determine an appropriate iterative equation for approximating this arbitrary root.
(d) Write an R function like the one for square root which calculates the arbitrary root of a
number.
(e) Give an example of your function by computing the fifth root of 7.
(f) Let ek = |xk √5 7|, where k indicates the kth iteration. Please print from e1 to e4, and specify
your findings.
 
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