## The two functions here are for calculating the inverse of a

## matrix. What is special here is that, if the inverse of the

## matrix is calculated once before, the functions directly

## read the result from cache instead of recalculating.

## The makeCacheMatrix() funtion here creates an artificial list

## including a function to: I.set the value of the matrix; II.

## get the value of the matrix; III. set the value of the inverse;

## IV. get the value of the inverse.

makeCacheMatrix <- function(x = matrix()) {

m <- NULL ##Set an empty value to 'm'

set <- function(y) {

x <<- y ## Assign the value 'y' to the R object

## 'x' which is in its own environment,

## one that is different from the current

## one ("<<-"s below are likewise).

m <<- NULL

}

get <- function() x ## Assign function(x) to 'get'

setinverse <- function(inverse) m <<- inverse

getinverse <- function() m ## Assign function(m) to 'getinverse'

list(set = set, get = get,

setinverse = setinverse,

getinverse = getinverse) ## Lastly, return four functions

}

## The cacheSolve() function here calculates the inverse of

## the list of 4 functions generated from the makeCacheMatrix()

## function. Specifically, it first does some checking. If the

## inverse has been calculated, it copies the result from the

## cache memory and drops calculation.If not yet, it calculates

## the inverse of the matrix typed in and sets the solve() value

## in the cache with the setinverse function.

cacheSolve <- function(x, ...) {

m <- x$getinverse()

if(!is.null(m)) { ## if the cache memory is not NULL,

message("getting cached data") ## to expedite the caculation,

return(m) ## we return the value stored in cache

} ## without any computation.

data <- x$get() ## if there is not any values stored

m <- solve(data, ...) ## in cache, we calculate the inverse

x$setinverse(m) ## using the solve() function in R.

m ## Return a matrix that is the inverse of 'x'

}