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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.

library(matrixcalc)
library(Matrix)

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

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))
}

A <- matrix(c(2,1,-6,4,-4,-9,-4,3,5),3,3)
A;

##      [,1] [,2] [,3]
## [1,]    2    4   -4
## [2,]    1   -4    3
## [3,]   -6   -9    5

factors <- fctR(A)
factors$L

##      [,1] [,2] [,3]
## [1,]  1.0  0.0    0
## [2,]  0.5  1.0    0
## [3,] -3.0 -0.5    1

factors$U

##      [,1] [,2] [,3]
## [1,]    2    4 -4.0
## [2,]    0   -6  5.0
## [3,]    0    0 -4.5

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

##      [,1] [,2] [,3]
## [1,]  1.0  0.0    0
## [2,]  0.5  1.0    0
## [3,] -3.0 -0.5    1

luA$U

##      [,1] [,2] [,3]
## [1,]    2    4 -4.0
## [2,]    0   -6  5.0
## [3,]    0    0 -4.5