Inverse of sum of two matrices - c++

Question

I am trying to implement a code to compute the inverse of a sum of two matrices. My algorithm is recursive, and I need to use a loop for() I tried to do in R, but my code is very slow. Then, I am trying to do using RcppArmadillo, but my code is very very slow. I think I am doing some thing wrong. Let me show my R code.
mySolveR <- function(A,B){
ncol = dim(B)[1]
ZERO.B <- Matrix(0,ncol = ncol, nrow = ncol)
invCi <- A
for(i in 1:ncol){
ZERO.B[,i] <- B[,i]
gi <- 1/(1 + sum(diag(ZERO.B%*%invCi)))
invCi <- invCi - gi*(invCi%*%ZERO.B%*%invCi)
ZERO.B[,i] <- 0
}
return(invCi)}
And now my C++ code using RcppArmadillo.
src <- '
Rcpp::NumericMatrix Ac(A); // creates Rcpp matrix from SEXP
Rcpp::NumericMatrix Bc(B);
int n = Ac.nrow(), k = Ac.ncol();
arma::mat A(Ac.begin(), n, k, false); // reuses memory and avoids extra copy
arma::mat B(Bc.begin(), n, k, false);
arma::mat Z(n,k);
Z.zeros();
arma::mat invCi = A;
for( int i = 0 ; i < n ; i++){
Z.col(i) = B.col(i);
double gi = 1/(1 + trace(Z*invCi));
invCi = invCi - gi*(invCi*Z*invCi);
Z.zeros() ;
}
return wrap(invCi);'
I am using the inline package to compile my function.
mySolveCpp <- cxxfunction(signature(A = "numeric", B = "numeric"),
src, plugin="RcppArmadillo")
Now consider the following easy example,
A <- diag(5)
B <- matrix(c(1,-1,0,0,0, -1, 2, -1,0,0, 0,-1,2,-1,0,
0,0,-1,2,-1, 0,0,0,-1,1),5,5)
Using my function to compute the inverse of A + B
mySolveCpp(A,B)
mySolveR(A,B)
You can see my functions work well, in this small example. But I would like to apply this algorithm for a matrix around 15000 x 15000. In this case my R code does not work and my C++ code is very slow, spends hours to compute the inverse. I would like to know if is possible to improve my C++ code to deal with big matrix, as 15000 x 15000.
Best

Answer 1

Have you tried solve()?
A <- diag(5)
B <- matrix(c(1,-1,0,0,0, -1, 2, -1,0,0, 0,-1,2,-1,0,0,0,-1,2,-1, 0,0,0,-1,1),5,5)
solve(A+B)
For sparse Matrix objects:
As=Matrix(A)
Bs=Matrix(B)
solve(As+Bs)
5 x 5 Matrix of class "dsyMatrix"
[,1] [,2] [,3] [,4] [,5]
[1,] 0.61818182 0.23636364 0.09090909 0.03636364 0.01818182
[2,] 0.23636364 0.47272727 0.18181818 0.07272727 0.03636364
[3,] 0.09090909 0.18181818 0.45454545 0.18181818 0.09090909
[4,] 0.03636364 0.07272727 0.18181818 0.47272727 0.23636364
[5,] 0.01818182 0.03636364 0.09090909 0.23636364 0.61818182

Answer 2

I'm more comfortable with Eigen and can get some speed-up without changing the algorithm:
src2 <- '
using Eigen::Map;
using Eigen::MatrixXd;
using Rcpp::as;
const Map<MatrixXd> A(as<Map<MatrixXd> >(AA));
const Map<MatrixXd> B(as<Map<MatrixXd> >(BB));
const int n = A.rows(), k = A.cols();
MatrixXd Z(n,k), C(n,k);
const MatrixXd Z0 = Z.setZero();
MatrixXd invCi = A;
double gi;
for( int i = 0 ; i < n ; i++){
Z.col(i) = B.col(i);
C = Z*invCi;
gi = 1/(1 + C.trace());
invCi -= gi*(invCi*C);
Z=Z0;
}
return wrap(invCi);'
mySolveCpp2 <- cxxfunction(signature(AA = "matrix", BB = "matrix"),
src2, plugin="RcppEigen")
set.seed(42)
A <- matrix(rnorm(1e4), 1e2)
B <- matrix(rnorm(1e4), 1e2)
all.equal(
mySolveCpp(A,B),
mySolveCpp2(A,B))
#[1] TRUE
library(microbenchmark)
microbenchmark(mySolveCpp(A,B),
mySolveCpp2(A,B), times=10)
#Unit: milliseconds
# expr min lq median uq max neval
# mySolveCpp(A, B) 129.51222 129.62216 132.68336 136.67307 137.43591 10
# mySolveCpp2(A, B) 46.76913 47.26311 47.96435 50.12505 61.82288 10