cachematrix.R

## Put comments here that give an overall description of what your
## functions do

## makeCacheMatrix(): Creates a special matrix object that is able
## to cache the inverse of itself

makeCacheMatrix <- function(x = matrix()) {
  ## @x: a square invertible matrix
  ## return: a list containing functions to
  ##              1. set the matrix
  ##              2. get the matrix
  ##              3. set the inverse
  ##              4. get the inverse
  ##         this list is used as the input to cacheSolve()
  
  inv <- NULL
  set <- function(y) {
    # use `<<-` to assign a value to an object in an environment 
    # different from the current environment. 
    x <<- y
    inv <<- NULL
  }
  get <- function() x
  setinv <- function(inverse) inv <<- inverse 
  getinv <- function() inv
  list(set=set, get=get, 
       setinv=setinv,
       getinv=getinv)
}


## cacheSolve(): computes the inverse of the "matrix" returned by 
## makeCacheMatrix(). It checks if the inverse has already been calculated 
## and the matrix has not changed, if it has the function retrieves the 
## inverse from the cache itself and dose not do the computation.

cacheSolve <- function(x, ...) {
        ## Return a matrix that is the inverse of 'x'
  ## @x: output of makeCacheMatrix()
  ## return: inverse of the original matrix input to makeCacheMatrix()
  
  inv <- x$getinv()
  
  # if the inverse has already been calculated
  if (!is.null(inv)){
    # get it from the cache and skips the computation. 
    message("getting cached data")
    return(inv)
  }
  
  # otherwise, calculates the inverse 
  mat.data <- x$get()
  inv <- solve(mat.data, ...)
  
  # sets the value of the inverse in the cache via the setinv function.
  x$setinv(inv)
  
  return(inv)
}