IS 605 FUNDAMENTALS OF COMPUTATIONAL MATHEMATICS - 2015

Problem Set 1

1. (AA’)‘= (A’)‘x A’ = AA’ , which is true only for symmetric matrices.
2. A’A = AA’ => A = A’ is only true for summetric matrices, like I.

Problem Set 2

3. LU factorization

get_LU <- function(A) # for function Ax=b
{
  n <-nrow(A)
  #browser()
  
  A_Orig <- A
  L <- diag(n)
  U <- diag(n)
  
  
  for (i in 2:n)
  {
    for (j in i:n)
    {
      E <- diag(n)
      E[j,i-1] <- -(A[j,i-1] / A[i-1, i-1])
      L <- L %*% solve(E)
      
      U <- E %*% U
      A <- E %*% A
    }
  }
  
  #Upper
  U <- U %*% A_Orig
  
  #Test
  L%*%U == A_Orig
  
}

get_LU(matrix(c(1,2,3,1,1,1,2,0,1),nrow=3))

##      [,1] [,2] [,3]
## [1,] TRUE TRUE TRUE
## [2,] TRUE TRUE TRUE
## [3,] TRUE TRUE TRUE