cachematrix.R

## makeCacheMatrix(): creates a special "matrix" object that can cache its inverse.
## cacheSolve(): computes the inverse of the "matrix" returned by makeCacheMatrix(). 
## If the inverse has already been calculated and the matrix has not changed, it'll retrieves the inverse 
## from the cache directly.

## makeCacheMatrix 
## @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()

makeCacheMatrix <- function(x = matrix()) {
  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
## @x: output of makeCacheMatrix()
## return: inverse of the original matrix input to makeCacheMatrix()

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