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Computing Matrix Algebra with R and Rcpp

LU Matrix Factorization

Explore how to implement LU matrix factorization using R and C++ libraries RcppArmadillo and RcppEigen. Understand the decomposition into lower and upper triangular matrices, validate results by reconstructing original matrices, and gain practical skills to solve linear systems efficiently.

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What is LU factorization?

An LU factorization refers to the factorization of a square matrix 𝐴𝐴, with proper row and/or column orderings or permutations, into two factors:

  • A lower triangular matrix 𝐿𝐿
  • An upper triangular matrix π‘ˆπ‘ˆ

𝐴=πΏπ‘ˆπ΄ = πΏπ‘ˆ

R
A <- matrix(c(1, -1, 2, 2), 
            nrow = 2, 
            byrow = TRUE)
fA <- LU(A = A) # compute the LU factorization of A
A_ <- fA$P%*%fA$L%*%fA$U # retrieve the original matrix A by multiplying all factors
fA
A_

Explanation

The resulting LU factors are:

fA <- LU(A = A)
P, L, U factors

P=[1001]    L=[1021]    U=[1βˆ’104] P= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \space\space\space\space L= \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \space\space\space\space U= \begin{bmatrix} 1 & -1 \\ 0 & 4 \end{bmatrix} ...