Data605 Assignment - 2

2. Problem set 2 Matrix factorization is a very important problem. There are supercomputers built just to do matrix factorizations. Every second you are on an airplane, matrices are being factorized. Radars that track ights use a technique called Kalman filtering. At the heart of Kalman Filtering is a Matrix Factorization operation. Kalman Filters are solving linear systems of equations when they track your ight using radars. Write an R function to factorize a square matrix A into LU or LDU, whichever you prefer. Please submit your response in an R Markdown document using our class naming convention, E.g. LFulton_Assignment2_PS2.png You don’t have to worry about permuting rows of A and you can assume that A is less than 5x5, if you need to hard-code any variables in your code. If you doing the entire assignment in R, then please submit only one markdown document for both the problems.

factorize.function <- function(A)
{
  n <- nrow(A)
  L <- diag(n)
  for(k in 1:(n-1))
  {
    for (i in (k+1):n) {
      L[i,k] <- A[i,k]/A[k,k]
      A[i,] <- A[i,] - L[i,k] * A[k,]
    }
  }
  print("U is: ")
  print(A)
  print("L is: ")
  print(L)
}

test1_matrix <- matrix(c(2,6,1,8), 2,2)
test1_matrix

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

factorize.function(test1_matrix)

## [1] "U is: "
##      [,1] [,2]
## [1,]    2    1
## [2,]    0    5
## [1] "L is: "
##      [,1] [,2]
## [1,]    1    0
## [2,]    3    1

test2_matrix <- matrix(c(1,2,3,1,1,1,2,0,1), 3,3)
test2_matrix

##      [,1] [,2] [,3]
## [1,]    1    1    2
## [2,]    2    1    0
## [3,]    3    1    1

factorize.function(test2_matrix)

## [1] "U is: "
##      [,1] [,2] [,3]
## [1,]    1    1    2
## [2,]    0   -1   -4
## [3,]    0    0    3
## [1] "L is: "
##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    2    1    0
## [3,]    3    2    1