## The functions in this file allow for faster inversing of matrices by preventing

## duplication of work. Unfortunately, they are very memory intense for large

## matrices, due to the requirement of 3 matrices per matrix. A matrix is passed

## into makeCacheMatrix as x<-makeCacheMatrix(x). Details of the output can be

## found below. Similarly, x<-cacheSolve(x) will check for a chached inverse of

## the matrix created by the 1st function. If it exists, it is returned. Otherwise

## the matrix is inverted and the inverse cached. Details can be found below.

library(compare)

## This function creates a list out of the matrix. The first element is the matrix

## itself, the second is a check matrix to make sure the function hasn't been

## modified, and the third is the inverse to be cached. Elements 2:3 are 0s.

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

cache<-as.matrix(0,0)

changecheck<-matrix(0,nrow=nrow(x),ncol=ncol(x))

y<-list(x,changecheck,cache)

}

## The correct way to call this function is x<-cacheSolve(x). Using the compare

## package, the 1st and 2nd elements of the passed list are checked to see if

## they are identical or mathematically equivalent. If they are, then the function

## has already been run, and is still accurate. The cached inverse of the matrix

## is retrieved. Otherwise, the 2nd element is set to be equal to the value of

## the 1st, and the 3rd element is set equal to the inverse.

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

q<-compare(x[[2]],x[[1]])

if (as.logical(q[1])){

message("Retrieving cached inverse")

print(x[[3]])

invisible(return(x))

}

else{

message("Calculating and caching inverse")

x[[2]]<-x[[1]]

x[[3]]<-solve(x[[2]])

print(x[[3]])

invisible(return(x))

}

}