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This second programming assignment will require you to write an Rfunction that is able to cache potentially time-consuming computations.For example, taking the mean of a numeric vector is typically a fastoperation. However, for a very long vector, it may take too long tocompute 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 makesense to cache the value of the mean so that when we need it again, itcan be looked up in the cache rather than recomputed. In thisProgramming Assignment you will take advantage of the scoping rules ofthe R language and how they can be manipulated to preserve state insideof an R object.

In this example we introduce the<<- operator which can be used toassign a value to an object in an environment that is different from thecurrent environment. Below are two functions that are used to create aspecial object that stores a numeric vector and caches its mean.

The first function,makeVector creates a special "vector", which isreally a list containing a function to

1. set the value of the vector
2. get the value of the vector
3. set the value of the mean
4. 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 themean has already been calculated. If so, itgets the mean from thecache and skips the computation. Otherwise, it calculates the mean ofthe data and sets the value of the mean in the cache via thesetmeanfunction.

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}

Matrix inversion is usually a costly computation and there may be somebenefit to caching the inverse of a matrix rather than computing itrepeatedly (there are also alternatives to matrix inversion that we willnot discuss here). Your assignment is to write a pair of functions thatcache the inverse of a matrix.

Write the following functions:

1. makeCacheMatrix: This function creates a special "matrix" objectthat can cache its inverse.
2. cacheSolve: This function computes the inverse of the special"matrix" returned bymakeCacheMatrix above. If the inverse hasalready 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 thesolvefunction in R. For example, ifX is a square invertible matrix, thensolve(X) returns its inverse.

For this assignment, assume that the matrix supplied is alwaysinvertible.

In order to complete this assignment, you must do the following:

1. Fork the GitHub repository containing the stub R files athttps://github.com/rdpeng/ProgrammingAssignment2to create a copy under your own account.
2. Clone your forked GitHub repository to your computer so that you canedit the files locally on your own machine.
3. Edit the R file contained in the git repository and place yoursolution in that file (please do not rename the file).
4. Commit your completed R file into YOUR git repository and push yourgit branch to the GitHub repository under your account.
5. Submit to Coursera the URL to your GitHub repository that containsthe completed R code for the assignment.

This assignment will be graded via peer assessment.