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Programming_Assignment_2_-_Lexical_Scoping.md

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1. Introduction

This report was prepared to test functions created as part of Programming Assignment 2 for R Programming course (Coursera, Data Science Specialization, Johns Hopkins Bloomberg School of Public Health).

Credit for test scenarios goes to Alan E. Berger, who generously explained basic algebra which stands behind operations with matrices.

2. Definition of inverse matrix

#Lets define m1 as simple 2*2 matrix
m1 <- matrix(c(1/2, -1/4, -1, 3/4), nrow = 2, ncol = 2)
m1

##       [,1]  [,2]
## [1,]  0.50 -1.00
## [2,] -0.25  0.75

# Lets define n1 also as simple 2*2 matrix
n1 <- matrix(c(6,2,8,4), nrow = 2, ncol = 2)
n1

##      [,1] [,2]
## [1,]    6    8
## [2,]    2    4

# Matrix m1 is said to be inverse of matrix n1 (and vice versa) 
# if their multiplication produces identity matrix i1
i1 <- matrix(c(1,0,0,1), nrow = 2, ncol = 2)
i1

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

# Lets check whether n1 is in fact inverse of m1 (and vice versa)
i2<-m1 %*% n1
i2

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

i2==i1

##      [,1] [,2]
## [1,] TRUE TRUE
## [2,] TRUE TRUE

i3<-n1 %*% m1
i3

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

i3==i1

##      [,1] [,2]
## [1,] TRUE TRUE
## [2,] TRUE TRUE

3. Calculation of inverse matrix

# Lets now calculate inverse matrix for m1 via solve() 
# to make sure that result is the same as n1
# We will do the same thing for n1

n2 <- solve(m1)
n2

##      [,1] [,2]
## [1,]    6    8
## [2,]    2    4

n2==n1

##      [,1] [,2]
## [1,] TRUE TRUE
## [2,] TRUE TRUE

m2<-solve(n1)
m2

##       [,1]  [,2]
## [1,]  0.50 -1.00
## [2,] -0.25  0.75

m2==m1

##       [,1]  [,2]
## [1,]  TRUE FALSE
## [2,] FALSE FALSE

Hmmm, in the last example it seems that R does not agree that matrices m1 and m2 are equal. Lets find out why. Full answer to this question is presented here stackoverflow, but in short here is how m2 matrix really looks like. Not quite what we expected, yeah?

sprintf("%.54f",m2)

## [1] "0.500000000000000000000000000000000000000000000000000000" 
## [2] "-0.249999999999999972244424384371086489409208297729492187"
## [3] "-0.999999999999999888977697537484345957636833190917968750"
## [4] "0.749999999999999888977697537484345957636833190917968750"

We can use all.equal() function to make our test again and receive proper answer.

isTRUE(all.equal(m1, m2))

## [1] TRUE

4. Testing makeCacheMatrix() and cacheSolve()

# To run the tests you must have both functions loaded
# in your environment

setwd("D:/Coursera/Data Science Specialization/2. R Programming/Programming Assignment 2/ProgrammingAssignment2")

source("cachematrix.r")

# myInvMatrix must be equal to n1
myMatrix <- makeCacheMatrix(m1)
myInvMatrix_m <- cacheSolve(myMatrix)
myInvMatrix_m

##      [,1] [,2]
## [1,]    6    8
## [2,]    2    4

myInvMatrix_m == n1

##      [,1] [,2]
## [1,] TRUE TRUE
## [2,] TRUE TRUE

# Calling cacheSolve() second time should retrieve cached matrix
# and not recalculate it

cacheSolve(myMatrix)

## getting cached data

##      [,1] [,2]
## [1,]    6    8
## [2,]    2    4

# We can put inside our fuction new matrix
# by using set function and get inverse of it

myMatrix$set(n1)
myInvMatrix_n <- cacheSolve(myMatrix)
myInvMatrix_n

##       [,1]  [,2]
## [1,]  0.50 -1.00
## [2,] -0.25  0.75

myInvMatrix_n == m1

##       [,1]  [,2]
## [1,]  TRUE FALSE
## [2,] FALSE FALSE

# Okay, and what about this? Seems better.
isTRUE(all.equal(m1, myInvMatrix_n))

## [1] TRUE

# Later we can retrieve our cached matrix
# in the same way we did previously

cacheSolve(myMatrix)

## getting cached data

##       [,1]  [,2]
## [1,]  0.50 -1.00
## [2,] -0.25  0.75