Suppose I have a function gcf(x,y) that returns the greatest common factor of x and y. So, for example,
gcf(75,85) = 5
Now, I'm trying to create a function lcm(v) that takes a vector of integers and returns the least common multiple.
I know mathematically it will be
lcd(a,b) = a*b/((gcf(a,b))
But I have a vector as the argument. How do I start writing the code?
Also, how do I make sure that the vector has at least two integers and no more than 100?
No need to reinvent the wheel. You can try the package library(pracma) and the vectorized functions gcd & Lcm like
a1 = c(75, 30)
b1 = c(85, 10)
pracma::gcd(a1,b1)
[1] 5 10
pracma::Lcm(a1,b1)
[1] 1275 30
Check View(pracma::gcd) or hit F2 in RStudio to learn from the source code. The second question is a typical if statement:
foo <- function(x, y){
require(pracma)
if(length(x) < 2 | length(x) > 100 ) return("Length of x must be at least 2 and not more than 100")
if(length(y) < 2 | length(y) > 100 ) return("Length of y must be at least 2 and not more than 100")
list(gcd=pracma::gcd(x,y),
lcm=pracma::Lcm(x,y))
}
foo(a1, b1)
$gcd
[1] 5 10
$lcm
[1] 1275 30
Edit
As you commented you want to find the least common factor of three or more numbers. Thus, you can start and play with the following lines of code e.g. for a vector of length of six. The idea is to test all combinations of length 2 and finally reduce them from left to right.
a =c(21, 45, 3 , 90, 72, 99)
combn(a, 2, simplify = F) %>%
map(~gcd(.[1], .[2])) %>%
Reduce(function(x,y) gcd(x, y),.)
But without any guarantee for correctness of the result.
Package numbers contains number-theoretic functions, and function mLCM() therein does what you want (I think):
> numbers::mLCM(c(20,50,75))
[1] 300
If you want to write your own function, you may still want to take a look into this function -- the logic behind is simple.
Related
I am sure this is a really dumb question with a simple answer, but I have been banging my head against the desk for an hour now. The goal is to write a simple function that returns a vector of n length consisting of integers spaces as evenly as possible, from 1 to k. So:
place_in_groups <- function(n, k){
rate = (n - 1) / (k - 1)
vect <- round(seq(from = 1, to = n, by = rate), 0)
return(vect)
}
When I run the lines inside the function on the outside of the function, it does what I want it to do: creates a vector with the appropriate values. But when I run it inside the vector, I get the actual values, not the vector:
place_in_groups(4,5)
[1] 1 2 2 3 4
As I said, I'm sure it is something obvious I'm doing wrong, but it is also something I'm obviously in need of learning.
I'm not shure I undestand the question correctly. Try str() on your results. What you are getting is a vector.
vect <- place_in_groups(4,5)
str(vect)
num [1:5] 1 2 2 3 4
What do you want to do with the vector, or what is your challenge?
I am new to programming and R and would like to compute the following sum
I used the pochMpfr from the Rmpfr package for the rising factorial and a for loop in order compute the sum.
B=rep(1,k+1)
for (i in 0:k) {
B[(i+1)]= (-1)^i *choose(k,i)*pochMpfr((-i)*sigma, n)
}
sum(B)
Doing so, I get the results as list (including always: mpfr) and thus cannot compute the sum.
Is there a possibility to get the results immediately as a Matrix or to convert the list to vector including only the relevant Elements?
The solution is probably quite easy but I haven't found it while looking through the forums.
There is no need to use a for-loop, this should work:
library(Rmpfr)
# You do not define these in your question,
# so I just take some arbitrary values
k <- 10
n <- 3
sigma <- 0.3
i <- 0:k
B <- (-1)^i *choose(k,i)*pochMpfr((-i)*sigma, n)
sum(B)
## 1 'mpfr' number of precision 159 bits
## [1] 6.2977401071861993597462780570563107354142915151e-14
Closed. This question needs debugging details. It is not currently accepting answers.
Edit the question to include desired behavior, a specific problem or error, and the shortest code necessary to reproduce the problem. This will help others answer the question.
Closed 7 years ago.
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my question is simple... every reference I find in books and on the internet for learning R programming is presented in a very linear way with no context. When I try and learn things like functions, I see the code and my brain just freezes because it's looking for something to relate these R terms to and I have no frame of reference. I have a PhD and did a lot of statistics for my dissertation but that was years ago when we were using different programming languages and when it comes to R, I don't know why I can't get this into my head. Is there someone who can explain in plain english an example of this simple code? So for example:
above <- function(x, n){
use <- x > n
x[use]
}
x <- 1:20
above(x, 12)
## [1] 13 14 15 16 17 18 19 20
I'm trying to understand what's going on in this code but simply don't. As a result, I could never just write this code on my own because I don't have the language in my head that explains what is happening with this. I get stuck at the first line:
above <- function(x, n) {
Can someone just explain this code sample in plain English so I have some kind of context for understanding what I'm looking at and why I'm doing what I'm doing in this code? And what I mean by plain English is, walking through the code, step by step and not just repeating the official terms from R like vector and function and array and all these other things, but telling me, in a common sense way, what this means.
Since your background ( phd in statsitics) the best way to understand this
is in mathematics words.
Mathematically speaking , you are defining a parametric function named above that extracts all element from a vector x above a certain value n. You are just filtering the set or the vector x.
In sets notation you can write something like :
above:{x,n} --> {y in x ; y>n}
Now, Going through the code and paraphrasing it (in the left the Math side , in the right its equivalent in R):
Math R
---------------- ---------------------
above: (x,n) <---> above <- function(x, n)
{y in x ; y>n} <---> x[x > n]
So to wrap all the statments together within a function you should respect a syntax :
function_name <- function(arg1,arg2) { statements}
Applying the above to this example (we have one statement here) :
above <- function(x,n) { x[x>n]}
Finally calling this function is exactly the same thing as calling a mathematical function.
above(x,2)
ok I will try, if this is too detailed let me know, but I tried to go really slowly:
above <- function(x, n)
this defines a function, which is just some procedure which produces some output given some input, the <- means assign what is on the right hand side to what is on the left hand side, or in other words put everything on the right into the object on the left, so for example container <- 1 puts 1 into the container, in this case we put a function inside the object above,
function(x, n) everything in the paranthesis specifys what inputs the function takes, so this one takes two variables x and n,
now we come to the body of the function which defines what it does with the inputs x and n, the body of the function is everything inside the curley braces:
{
use <- x > n
x[use]
}
so let's explain that piece by piece:
use <- x > n
this part again puts whats on the right side into the object on the left, and what is happening on the right hand side? a comparison returning TRUE if x is bigger than n and FALSE if x is equal to or smaller then n, so if x is 5 and n is 3 the result will be TRUE, and this value will get stored inside use, so use contains TRUE now, now if we have more than one value inside x than every value inside x will get compared to n, so for example if x = [1, 2, 3] and n = 2
than we have
1 > 2 FALSE
2 > 2 FALSE
3 > 2 TRUE
, so use will contain FALSE, FALSE, TRUE
x[use]
now we are taking a part of x, the square brackets specify which parts of x we want, so in my example case x has 3 elements and use has 3 elements if we combine them we have:
x use
1 FALSE
2 FALSE
3 TRUE
so now we say I dont want 1,2 but i want 3 and the result is 3
so now we have defined the function, now we call it, or in normal words we use it:
x <- 1:20
above(x, 12)
first we assign the numbers 1 through 20 to x, and then we tell the function above to execute (do everything inside its curley braces with the inputs x = 1:20 and n = 12, so in other words we do the following:
above(x, 12)
execute the function above with the inputs x = 1:20 and n = 12
use <- 1:20 > 12
compare 12 to every number from 1:20 and return for each comparison TRUE if the number is in fact bigger than 12 and FALSE if otherwise, than store all the results inside use
x[use]
now give me the corresponding elements of x for which the vector use contains TRUE
so:
x use
1 FALSE
2 FALSE
3 FALSE
4 FALSE
5 FALSE
6 FALSE
7 FALSE
8 FALSE
9 FALSE
10 FALSE
11 FALSE
12 FALSE
13 TRUE
14 TRUE
15 TRUE
16 TRUE
17 TRUE
18 TRUE
19 TRUE
20 TRUE
so we get the numbers 13:20 back as a result
I'll give it a crack too. A few basic points that should get you going in the right direction.
1) The idea of a function. Basically, a function is reusable code. Say I know that in my analysis for some bizarre reason I will often want to add two numbers, multiply them by a third, and divide them by a fourth. (Just suspend disbelief here.) So one way I could do that would just be to write the operation over and over, as follows:
(75 + 93)*4/18
(847 + 3)*3.1415/2.7182
(999 + 380302)*-6901834529/2.5
But that's tedious and error-prone. (What happens if I forget a parenthesis?) Alternatively, I can just define a function that takes whatever numbers I feed into it and carries out the operation. In R:
stupidMath <- function(a, b, c, d){
result <- (a + b)*c/d
}
That code says "I'd like to store this series of commands and attach them to the name "stupidMath." That's called defining a function, and when you define a function, the series of commands is just stored in memory---it doesn't actually do anything until you "call" it. "Calling" it is just ordering it to run, and when you do so, you give it "arguments" ---the stuff in the parentheses in the first line are the arguments it expects, i.e., in my example, it wants four distinct pieces of data, which will be called 'a', 'b', 'c', and 'd'.
Then it'll do the things it's supposed to do with whatever you give it. "The things it's supposed to do" is the stuff in the curly brackets {} --- that's the "body" of the function, which describes what to do with the arguments you give it. So now, whenever you want to carry that mathematical operation you can just "call" the function. To do the first computation, for example, you'd just write stupidMath(75, 93, 4, 18) Then the function gets executed, treating 75 as 'a', 83 as 'b', and so forth.
In your example, the function is named "above" and it takes two arguments, denoted 'x' and 'n'.
2) The "assignment operator": R is unique among major programming languages in using <- -- that's equivalent to = in most other languages, i.e., it says "the name on the left has the value on the right." Conceptually, it's just like how a variable in algebra works.
3) so the "body" of the function (the stuff in the curly brackets) first assigns the name "use" to the expression x > n. What's going on there. Well, an expression is something that the computer evaluates to get data. So remember that when you call the function, you give it values for x and n. The first thing this function does is figures out whether x is greater than n or less than n. If it's greater than n, it evaluates the expression x > n as TRUE. Otherwise, FALSE.
So if you were to define the function in your example and then call it with above(10, 5), then the first line of the body would set the local variable (don't worry right now about what a 'local' variable is) 'use' to be 'TRUE'. This is a boolean value.
Then the next line of the function is a "filter." Filtering is a long topic in R, but basically, R things of everything as a "vector," that is, a bunch of pieces of data in a row. A vector in R can be like a vector in linear algebra, i.e., (1, 2, 3, 4, 5, 99) is a vector, but it can also be of stuff other than numbers. For now let's just focus on numbers.
The wacky thing about R (one of the many wacky things about R) is that it treats a single number (a "scalar" in linear algebra terms) just as a vector with only one item in it.
Ok, so why did I just go into that? Because in lots of places in R, a vector and a scalar are interchangable.
So in your example code, instead of giving a scalar for the first argument, when we call the function we've given 'above' a vector for its first argument. R likes vectors. R really likes vectors. (Just talk to R people for a while. They're all obsessed with doing every goddmamn thing in terms of a vector.) So it's no problem to pass a vector for the first argument. But what that means is that the variable 'use' is going to be a vector too. Specifically, 'use' is going to be a vector of booleans, i.e., of TRUE or FALSE for each individual value of X.
To take a simpler version: suppose you said:
mynums <- c(5, 10)
myresult <- above(mynums, 7)
when the code runs, the first thing it's going to do is define that 'use' variable. But x is a vector now, not a scalar (the c(5,10) code said "make a vector with two elements, and fill them with the numbers '5' and '10'), so R's going to go ahead and carry out the comparison for each element of x. Since 5 is less than 7 and 10 is greater than 7, use becomes the two item-vector of boolean values (FALSE, TRUE)
Ok, now we can talk about filtering. So a vector of boolean values is called a 'logical vector.' And the code x[use] says "filter x by the stuff in the variable use." When you tell R to filter something by a logical vector, it spits back out the elements of the thing being filtered which correspond to the values of 'TRUE'
So in the example just given:
mynums <- c(5, 10)
myresult <- above(mynums, 7)
the value of myresult will just be 10. Why? Because the function filtered 'x' by the logical vector 'use,' 'x' was (5, 10), and 'use' was (FALSE, TRUE); since the second element of the logical was the only true, you only got the second element of x.
And that gets assigned to the variable myresult because myresult <- above(mynums, 7) means "assign the name myresult to the value of above(mynums, 7)"
voila.
Using R, calculate for x and y be integers ∈ [1, 1000], How many unique powers, x^y exist.
This is what I have right now, just don't know how to eliminate the duplicate numbers,
x<-1:1000
y<-1:1000
for (i in x)
{
for (j in y){
print(i^j)
}
}
A combinatorial approach to this could split the numbers from 1-1000 into equivalence classes where each number in the class is the power of some other number. For instance, we would split the numbers 1-10 into (1), (2, 4, 8), (3, 9), (5), (6), (7), (10). None of the powers of values between equivalence classes will coincide, so we can just handle each equivalence class separately.
num.unique.comb <- function(limit) {
# Count number of powers in each equivalence class (labeled by lowest val)
num.powers <- rep(0, limit)
# Handle 1 as special case
num.powers[1] <- 1
# Beyond sqrt(limit), all unhandled numbers are in own equivalence class
handled <- c(T, rep(F, limit-1))
for (base in 2:ceiling(sqrt(limit))) {
if (!handled[base]) {
# Handle all the values in 1:limit that are powers of base
num.handle <- floor(log(limit, base))
handled[base^(1:num.handle)] <- T
# Compute the powers of base that we cover
num.powers[base] <- length(unique(as.vector(outer(1:num.handle, 1:limit))))
}
}
num.powers[!handled] <- limit
# Handle sums too big for standard numeric types
library(gmp)
print(sum(as.bigz(num.powers)))
}
num.unique.comb(10)
# [1] 76
num.unique.comb(1000)
# [1] 978318
One nice property of this combinatorial approach is that it's very fast compared to a brute-force approach. For instance, it takes less than 0.1 seconds to compute with limit set to 1000. This allows us to compute the result for much larger values:
# ~0.15 seconds
num.unique.comb(10000)
# [1] 99357483
# ~4 seconds
num.unique.comb(100000)
# [1] 9981335940
# ~220 seconds
num.unique.comb(1000000)
# [1] 999439867182
This is a pretty neat result -- in under 4 minutes we can compute the number of unique values within 1 trillion numbers, where each number can have up to 6 million digits!
Update: Based on this combinatorial code I've updated the OEIS entry for this sequence to include terms up to 10,000.
A brute-force approach would be to just compute all the powers and count the number of unique values:
num.unique.bf <- function(limit) {
length(unique(as.vector(sapply(1:limit, function(x) x^(1:limit)))))
}
num.unique.bf(10)
# [1] 76
A problem with this brute-force analysis is that you are dealing with large numbers that will create numerical issues. For instance:
1000^1000
# [1] Inf
As a result we get an inaccurate value:
# Wrong due to numerical issues!
num.unique.bf(1000)
# [1] 119117
However, a package like the gmp can enable us to compute even numbers as large as 1000^1000. My computer has trouble storing all 1 million numbers in memory at once, so I'll write them to a file (size for n=1000 is 1.2 GB on my computer) and then compute the number of unique values in that file:
library(gmp)
num.unique.bf2 <- function(limit) {
sink("foo.txt")
for (x in 1:limit) {
vals <- as.bigz(x)^(1:limit)
for (idx in 1:limit) {
cat(paste0(as.character(vals[idx]), "\n"))
}
}
sink()
as.numeric(system("sort foo.txt | uniq | wc -l", intern=T))
}
num.unique.bf2(10)
# [1] 76
num.unique.bf2(1000)
# [1] 978318
A quick visit to the OEIS (click the link for the first 1000 values) shows that this is correct. This approach is rather slow (roughly 40 minutes on my computer), and combinatorial approaches should be significantly faster.
Just starting to program in R... Got stumped on this one, perhaps because I don't know where to begin.
Define a random variable to be equal to the number of trials before there is a match. So if you have a list of numbers, (4,5,7,11,3,11,12,8,8,1....), the first value of the random variable is 6 because by then there are two 11's.(4,5,7,11,3,11) The second value is 3 because then you have 2 8's..12,8,8.
The code below creates the list of numbers, u, by simulating from a uniform distribution.
Thank-you for any help or pointers. I've included a full description of the problem I am solving below if anyone is interested (trying to learn by coding a statistics text).
set.seed(1); u = matrix(runif(1000), nrow=1000)
u[u > 0 & u <= 1/12] <- 1
u[u > 1/12 & u <= 2/12] <- 2
u[u > 2/12 & u <= 3/12] <- 3
u[u > 3/12 & u <= 4/12] <- 4
u[u > 4/12 & u <= 5/12] <- 5
u[u > 5/12 & u <= 6/12] <- 6
u[u > 6/12 & u <= 7/12] <- 7
u[u > 7/12 & u <= 8/12] <- 8
u[u > 8/12 & u <= 9/12] <- 9
u[u > 9/12 & u <= 10/12] <- 10
u[u > 10/12 & u <= 11/12] <- 11
u[u > 11/12 & u < 12/12] <- 12
table(u); u[1:10,]
Example 2.6-3 Concepts in Probability and Stochastic Modeling, Higgins
Suppose we were to ask people at random in which month they were born. Let the random variable X denote the number of people we would need to ask before we found two people born in the same month. The possible values for X are 2,3,...13. That is, at least two people must be asked in order to have a match and no more than 13 need to be asked. With the simplifying assumption that every month is an equally likely candidate for a response, a computer simulation was used to estimate the probabilitiy mass function of X. The simulation generated birth months until a match was found. Based on 1000 repetitions of this experiment, the following empirical distribution and sample statistics were obtained...
R has a steep initial learning curve. I don't think it's fair to assume this is your homework, and yes, it's possible to find solutions if you know what you're looking for. However, I remember it being difficult at times to research problems online simply because I didn't know what to search for (I wasn't familiar enough with the terminology).
Below is an explanation of one approach to solving the problem in R. Read the commented code and try and figure out exactly what it's doing. Still, I would recommend working through a good beginner resource. From memory, a good one to get up and running is icebreakeR, but there are many out there...
# set the number of simulations
nsim <- 10000
# Create a matrix, with nsim columns, and fill it with something.
# The something with which you'll populate it is a random sample,
# with replacement, of month names (held in a built-in vector called
# 'month.abb'). We're telling the sample function that it should take
# 13*nsim samples, and these will be used to fill the matrix, which
# has nsim columns (and hence 13 rows). We've chosen to take samples
# of length 13, because as your textbook states, 13 is the maximum
# number of month names necessary for a month name to be duplicated.
mat <- matrix(sample(month.abb, 13*nsim, replace=TRUE), ncol=nsim)
# If you like, take a look at the first 10 columns
mat[, 1:10]
# We want to find the position of the first duplicated value for each column.
# Here's one way to do this, but it might be a bit confusing if you're just
# starting out. The 'apply' family of functions is very useful for
# repeatedly applying a function to columns/rows/elements of an object.
# Here, 'apply(mat, 2, foo)' means that for each column (2 represents columns,
# 1 would apply to rows, and 1:2 would apply to every cell), do 'foo' to that
# column. Our function below extends this a little with a custom function. It
# says: for each column of mat in turn, call that column 'x' and perform
# 'match(1, duplicated(x))'. This match function will return the position
# of the first '1' in the vector 'duplicated(x)'. The vector 'duplicated(x)'
# is a logical (boolean) vector that indicates, for each element of x,
# whether that element has already occurred earlier in the vector (i.e. if
# the month name has already occurred earlier in x, the corresponding element
# of duplicated(x) will be TRUE (which equals 1), else it will be false (0).
# So the match function returns the position of the first duplicated month
# name (well, actually the second instance of that month name). e.g. if
# x consists of 'Jan', 'Feb', 'Jan', 'Mar', then duplicated(x) will be
# FALSE, FALSE, TRUE, FALSE, and match(1, duplicated(x)) will return 3.
# Referring back to your textbook problem, this is x, a realisation of the
# random variable X.
# Because we've used the apply function, the object 'res' will end up with
# nsim realisations of X, and these can be plotted as a histogram.
res <- apply(mat, 2, function(x) match(1, duplicated(x)))
hist(res, breaks=seq(0.5, 13.5, 1))