Matrix Addition

Learn matrix addition using R, Rcpp, Armadillo, and Eigen.

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Addition is defined over matrices having the same dimensions.

𝑅=𝐴+𝐵𝑅 = 𝐴 + 𝐵

𝑟𝑖𝑗=𝑎𝑖𝑗+𝑏𝑖𝑗𝑟_{𝑖𝑗} = 𝑎_{𝑖𝑗} + 𝑏_{𝑖𝑗}

Here, both 𝐴𝐴 and 𝐵𝐵 are (𝑚×𝑛)(𝑚 \times 𝑛) matrices.

Matrix addition in R

In base R, the binary operator +, executes the addition.

Let’s implement the following example:

Addition: A + B = R

[1−32522]  +  [17−5384]  =  [24−38106]\begin{bmatrix} 1 & -3 & 2 \\ 5 & 2 & 2 \\ \end{bmatrix} \space \space + \space\space \begin{bmatrix} 1 & 7 & -5 \\ 3 & 8 & 4 \\ \end{bmatrix} \space\space = \space\space \begin{bmatrix} 2 & 4 & -3 \\ 8 & 10 & 6 \end{bmatrix}

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A <- matrix(c(1, -3, 2, 5, 2, 2), 
            ncol = 3, 
            nrow = 2, 
            byrow = TRUE
            ) 
            
B <- matrix(c(1, 7, -5, 3, 8, 4), 
            ncol = 3, 
            nrow = 2, 
            byrow = TRUE
            ) 
            
R = A + B # Adding Matrices A and B
cat("Matrix A:\n") 
A
cat("\nMatrix B:\n") 
B
cat("\nMatrix R:\n") 
R

In the code above, in line 13, we have computed the sum of matrices AA and BB in variable R. We then print all three matrices in lines 16, 19, and 22, respectively.

Matrix addition algorithm

Implementing the addition operation is really simple. The algorithm performs the addition element by element, using two nested loops along the rows and the columns.

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main.r
main.cpp
#include <Rcpp.h>
using namespace Rcpp;
// [[Rcpp::export]]
NumericMatrix mat_add(NumericMatrix A, NumericMatrix B) {
int m = A.nrow();
int n = A.ncol();
  if (m != B.nrow() | n != B.ncol()) {
      stop("error: matrices dimensions do not match");
    }
    
    NumericMatrix R(m,n);
      int i = 0, j = 0;
      for (i = 0; i < m; i++) {     // loop over rows
        for (j = 0; j < n; j++) {   // loop over columns
          R(i,j) = A(i,j) + B(i,j); // elementwise addition 
        }
      } 
  return(R);
}

When we compare the matrix addition implementation with the results of the R addition, we can confirm that it is correct.

all.equal(mat_add(A, B), R)

# [1] TRUE

Matrix addition in Armadillo

Adding matrices using Armadillo is just as simple as it is in R.

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main.r
main.cpp
#include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]
using namespace Rcpp;
// [[Rcpp::export]]
// performing matrix addition using thr Armadillo library
arma::mat mat_add_A(arma::mat A, arma::mat B) {
  arma::mat R = A + B;
return(R);
}

By comparing the results of the matrix addition using Armadillo C++ library with the results of the R addition, we see that the results are correct.

all.equal(mat_add_A(A, B), R)

## [1] TRUE

Matrix addition in Eigen

Now, let’s look at matrix addition using the Eigen library, which is similar to R and Armadillo.

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main.r
main.cpp
#include <RcppEigen.h>
// [[Rcpp::depends(RcppEigen)]]
using namespace Rcpp;
using Eigen::MatrixXd;
// [[Rcpp::export]]
// performing matrix additon using the Eigen library
MatrixXd mat_add_E(MatrixXd A, MatrixXd B) {
  MatrixXd R = A + B;
return(R); 
}

We can see that our results are correct by comparing the matrix addition results using the Eigen C++ library with the results from using R.

all.equal(mat_add_E(A, B), R)

## [1] TRUE