DATA 605 - Assignment 2, Problem Set 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 flights 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 flight 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.

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.

fctR <- function(A){
  n <- nrow(A)   #number of rows in A
  U <- A
  L <- diag(n)   #assign diagonal to L

  for (j in c(1:n)){
    for(i in c(2:n)){
      if(i > j){
        the_row <- U[j,]
        multiplier <- U[i, j] / the_row[j]
        U[i,] <- U[i,] - (multiplier * the_row)
        L[i,j] <- multiplier
      }
    }
  }
  
  return(list(L=L, U=U))
}

library(Matrix)

## Warning: package 'Matrix' was built under R version 3.5.3

library(matrixcalc)


A <- matrix(c(1,2,3,4,5,6,7,8,9), nrow = 3, byrow = TRUE)

A

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    4    5    6
## [3,]    7    8    9

factors <- fctR(A)
factors$L

##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    4    1    0
## [3,]    7    2    1

factors$U

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0   -3   -6
## [3,]    0    0    0

luA <- lu.decomposition(A)
luA$L

##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    4    1    0
## [3,]    7    2    1

luA <- lu.decomposition(A)
luA$U

##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    0   -3   -6
## [3,]    0    0    0