Introduction
This second programming assignment will require you to write an R function that is able to cache potentially timeconsuming computations. For example, taking the mean of a numeric vector is typically a fast operation. However, for a very long vector, it may take too long to compute the mean, especially if it has to be computed repeatedly (e.g. in a loop). If the contents of a vector are not changing, it may make sense to cache the value of the mean so that when we need it again, it can be looked up in the cache rather than recomputed. In this Programming Assignment you will take advantage of the scoping rules of the R language and how they can be manipulated to preserve state inside of an R object.
Example: Caching the Mean of a Vector
In this example we introduce the <<
operator which can be used to
assign a value to an object in an environment that is different from the
current environment. Below are two functions that are used to create a
special object that stores a numeric vector and caches its mean.
The first function, makeVector
creates a special "vector", which is
really a list containing a function to
 set the value of the vector
 get the value of the vector
 set the value of the mean
 get the value of the mean
makeVector < function(x = numeric()) {
m < NULL
set < function(y) {
x << y
m << NULL
}
get < function() x
setmean < function(mean) m << mean
getmean < function() m
list(set = set, get = get,
setmean = setmean,
getmean = getmean)
}
The following function calculates the mean of the special "vector"
created with the above function. However, it first checks to see if the
mean has already been calculated. If so, it get
s the mean from the
cache and skips the computation. Otherwise, it calculates the mean of
the data and sets the value of the mean in the cache via the setmean
function.
cachemean < function(x, ...) {
m < x$getmean()
if(!is.null(m)) {
message("getting cached data")
return(m)
}
data < x$get()
m < mean(data, ...)
x$setmean(m)
m
}
Assignment: Caching the Inverse of a Matrix
Matrix inversion is usually a costly computation and there may be some benefit to caching the inverse of a matrix rather than computing it repeatedly (there are also alternatives to matrix inversion that we will not discuss here). Your assignment is to write a pair of functions that cache the inverse of a matrix.
Write the following functions:

makeCacheMatrix
: This function creates a special "matrix" object that can cache its inverse. 
cacheSolve
: This function computes the inverse of the special "matrix" returned bymakeCacheMatrix
above. If the inverse has already been calculated (and the matrix has not changed), thencacheSolve
should retrieve the inverse from the cache.
Computing the inverse of a square matrix can be done with the solve
function in R. For example, if X
is a square invertible matrix, then
solve(X)
returns its inverse.
For this assignment, assume that the matrix supplied is always invertible.
In order to complete this assignment, you must do the following:
 Fork the GitHub repository containing the stub R files at https://github.com/rdpeng/ProgrammingAssignment2 to create a copy under your own account.
 Clone your forked GitHub repository to your computer so that you can edit the files locally on your own machine.
 Edit the R file contained in the git repository and place your solution in that file (please do not rename the file).
 Commit your completed R file into YOUR git repository and push your git branch to the GitHub repository under your account.
 Submit to Coursera the URL to your GitHub repository that contains the completed R code for the assignment.
Grading
This assignment will be graded via peer assessment.