Why is this naive matrix multiplication faster than base R's?

Question

In R, matrix multiplication is very optimized, i.e. is really just a call to BLAS/LAPACK. However, I\'m surprised this very naive C++ code for matrix-vector multiplication s

Answer 1

Josh's answer explains why R's matrix multiplication is not as fast as this naive approach. I was curious to see how much one could gain using RcppArmadillo. The code is simple enough:

arma_code <- 
  "arma::vec arma_mm(const arma::mat& m, const arma::vec& v) {
       return m * v;
   };"
arma_mm = cppFunction(code = arma_code, depends = "RcppArmadillo")

Benchmark:

> microbenchmark::microbenchmark(my_mm(m,v), m %*% v, arma_mm(m,v), times = 10)
Unit: milliseconds
          expr      min       lq      mean    median        uq       max neval
   my_mm(m, v) 71.23347 75.22364  90.13766  96.88279  98.07348  98.50182    10
       m %*% v 92.86398 95.58153 106.00601 111.61335 113.66167 116.09751    10
 arma_mm(m, v) 41.13348 41.42314  41.89311  41.81979  42.39311  42.78396    10

So RcppArmadillo gives us nicer syntax and better performance.

Curiosity got the better of me. Here a solution for using BLAS directly:

blas_code = "
NumericVector blas_mm(NumericMatrix m, NumericVector v){
  int nRow = m.rows();
  int nCol = m.cols();
  NumericVector ans(nRow);
  char trans = 'N';
  double one = 1.0, zero = 0.0;
  int ione = 1;
  F77_CALL(dgemv)(&trans, &nRow, &nCol, &one, m.begin(), &nRow, v.begin(),
           &ione, &zero, ans.begin(), &ione);
  return ans;
}"
blas_mm <- cppFunction(code = blas_code, includes = "#include <R_ext/BLAS.h>")

Benchmark:

Unit: milliseconds
          expr      min       lq      mean    median        uq       max neval
   my_mm(m, v) 72.61298 75.40050  89.75529  96.04413  96.59283  98.29938    10
       m %*% v 95.08793 98.53650 109.52715 111.93729 112.89662 128.69572    10
 arma_mm(m, v) 41.06718 41.70331  42.62366  42.47320  43.22625  45.19704    10
 blas_mm(m, v) 41.58618 42.14718  42.89853  42.68584  43.39182  44.46577    10

Armadillo and BLAS (OpenBLAS in my case) are almost the same. And the BLAS code is what R does in the end as well. So 2/3 of what R does is error checking etc.

Answer 2

To add another point to Ralf Stubner's solution, then you can use the following C++ version to

1. do multiple columns at the same time to avoid re-reading the output vector many times.
2. add __restrict__ to potentially allow for vector operations (likely does not matter here as it is only reads I guess).

#include <Rcpp.h>
using namespace Rcpp;

inline void mat_vec_mult_vanilla
(double const * __restrict__ m, 
 double const * __restrict__ v, 
 double * __restrict__ const res, 
 size_t const dn, size_t const dm) noexcept {
  for(size_t j = 0; j < dm; ++j, ++v){
    double * r = res;
    for(size_t i = 0; i < dn; ++i, ++r, ++m)
      *r += *m * *v;
  }
}

inline void mat_vec_mult
(double const * __restrict__ const m, 
 double const * __restrict__ const v, 
 double * __restrict__ const res, 
 size_t const dn, size_t const dm) noexcept {
  size_t j(0L);
  double const * vj = v,
               * mi = m;
  constexpr size_t const ncl(8L);
  {
    double const * mvals[ncl];
    size_t const end_j = dm - (dm % ncl),
                   inc = ncl * dn;
    for(; j < end_j; j += ncl, vj += ncl, mi += inc){
      double *r = res;
      mvals[0] = mi;
      for(size_t i = 1; i < ncl; ++i)
        mvals[i] = mvals[i - 1L] + dn;
      for(size_t i = 0; i < dn; ++i, ++r)
        for(size_t ii = 0; ii < ncl; ++ii)
          *r += *(vj + ii) * *mvals[ii]++;
    }
  }
  
  mat_vec_mult_vanilla(mi, vj, res, dn, dm - j);
}

// [[Rcpp::export("mat_vec_mult", rng = false)]]
NumericVector mat_vec_mult_cpp(NumericMatrix m, NumericVector v){
  size_t const dn = m.nrow(), 
               dm = m.ncol();
  NumericVector res(dn);
  mat_vec_mult(&m[0], &v[0], &res[0], dn, dm);
  return res;
}

// [[Rcpp::export("mat_vec_mult_vanilla", rng = false)]]
NumericVector mat_vec_mult_vanilla_cpp(NumericMatrix m, NumericVector v){
  size_t const dn = m.nrow(), 
               dm = m.ncol();
  NumericVector res(dn);
  mat_vec_mult_vanilla(&m[0], &v[0], &res[0], dn, dm);
  return res;
}

The result with -O3 in my Makevars file and gcc-8.3 is

set.seed(1)
dn <- 10001L
dm <- 10001L
m <- matrix(rnorm(dn * dm), dn, dm)
lv <- rnorm(dm)

all.equal(drop(m %*% lv), mat_vec_mult(m = m, v = lv))
#R> [1] TRUE
all.equal(drop(m %*% lv), mat_vec_mult_vanilla(m = m, v = lv))
#R> [1] TRUE

bench::mark(
  R              = m %*% lv, 
  `OP's version` = my_mm(m = m, v = lv), 
  `BLAS`         = blas_mm(m = m, v = lv),
  `C++ vanilla`  = mat_vec_mult_vanilla(m = m, v = lv), 
  `C++`          = mat_vec_mult(m = m, v = lv), check = FALSE)
#R> # A tibble: 5 x 13
#R>   expression        min   median `itr/sec` mem_alloc `gc/sec` n_itr  n_gc total_time result memory                 time          gc               
#R>   <bch:expr>   <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl> <int> <dbl>   <bch:tm> <list> <list>                 <list>        <list>           
#R> 1 R             147.9ms    151ms      6.57    78.2KB        0     4     0      609ms <NULL> <Rprofmem[,3] [2 × 3]> <bch:tm [4]>  <tibble [4 × 3]> 
#R> 2 OP's version   56.9ms   57.1ms     17.4     78.2KB        0     9     0      516ms <NULL> <Rprofmem[,3] [2 × 3]> <bch:tm [9]>  <tibble [9 × 3]> 
#R> 3 BLAS           90.1ms   90.7ms     11.0     78.2KB        0     6     0      545ms <NULL> <Rprofmem[,3] [2 × 3]> <bch:tm [6]>  <tibble [6 × 3]> 
#R> 4 C++ vanilla    57.2ms   57.4ms     17.4     78.2KB        0     9     0      518ms <NULL> <Rprofmem[,3] [2 × 3]> <bch:tm [9]>  <tibble [9 × 3]> 
#R> 5 C++              51ms   51.4ms     19.3     78.2KB        0    10     0      519ms <NULL> <Rprofmem[,3] [2 × 3]> <bch:tm [10]> <tibble [10 × 3]>

So a slight improvement. The result may though be very dependent on the BLAS version. The version I used is

sessionInfo()
#R> #...
#R> Matrix products: default
#R> BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
#R> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
#R> ...

The whole file I Rcpp::sourceCpp()ed is

#include <Rcpp.h>
#include <R_ext/BLAS.h>
using namespace Rcpp;

inline void mat_vec_mult_vanilla
(double const * __restrict__ m, 
 double const * __restrict__ v, 
 double * __restrict__ const res, 
 size_t const dn, size_t const dm) noexcept {
  for(size_t j = 0; j < dm; ++j, ++v){
    double * r = res;
    for(size_t i = 0; i < dn; ++i, ++r, ++m)
      *r += *m * *v;
  }
}

inline void mat_vec_mult
(double const * __restrict__ const m, 
 double const * __restrict__ const v, 
 double * __restrict__ const res, 
 size_t const dn, size_t const dm) noexcept {
  size_t j(0L);
  double const * vj = v,
               * mi = m;
  constexpr size_t const ncl(8L);
  {
    double const * mvals[ncl];
    size_t const end_j = dm - (dm % ncl),
                   inc = ncl * dn;
    for(; j < end_j; j += ncl, vj += ncl, mi += inc){
      double *r = res;
      mvals[0] = mi;
      for(size_t i = 1; i < ncl; ++i)
        mvals[i] = mvals[i - 1L] + dn;
      for(size_t i = 0; i < dn; ++i, ++r)
        for(size_t ii = 0; ii < ncl; ++ii)
          *r += *(vj + ii) * *mvals[ii]++;
    }
  }
  
  mat_vec_mult_vanilla(mi, vj, res, dn, dm - j);
}

// [[Rcpp::export("mat_vec_mult", rng = false)]]
NumericVector mat_vec_mult_cpp(NumericMatrix m, NumericVector v){
  size_t const dn = m.nrow(), 
               dm = m.ncol();
  NumericVector res(dn);
  mat_vec_mult(&m[0], &v[0], &res[0], dn, dm);
  return res;
}

// [[Rcpp::export("mat_vec_mult_vanilla", rng = false)]]
NumericVector mat_vec_mult_vanilla_cpp(NumericMatrix m, NumericVector v){
  size_t const dn = m.nrow(), 
               dm = m.ncol();
  NumericVector res(dn);
  mat_vec_mult_vanilla(&m[0], &v[0], &res[0], dn, dm);
  return res;
}

// [[Rcpp::export(rng = false)]]
NumericVector my_mm(NumericMatrix m, NumericVector v){
  int nRow = m.rows();
  int nCol = m.cols();
  NumericVector ans(nRow);
  double v_j;
  for(int j = 0; j < nCol; j++){
    v_j = v[j];
    for(int i = 0; i < nRow; i++){
      ans[i] += m(i,j) * v_j;
    }
  }
  return(ans);
}

// [[Rcpp::export(rng = false)]]
NumericVector blas_mm(NumericMatrix m, NumericVector v){
  int nRow = m.rows();
  int nCol = m.cols();
  NumericVector ans(nRow);
  char trans = 'N';
  double one = 1.0, zero = 0.0;
  int ione = 1;
  F77_CALL(dgemv)(&trans, &nRow, &nCol, &one, m.begin(), &nRow, v.begin(),
           &ione, &zero, ans.begin(), &ione);
  return ans;
}

/*** R
set.seed(1)
dn <- 10001L
dm <- 10001L
m <- matrix(rnorm(dn * dm), dn, dm)
lv <- rnorm(dm)

all.equal(drop(m %*% lv), mat_vec_mult(m = m, v = lv))
all.equal(drop(m %*% lv), mat_vec_mult_vanilla(m = m, v = lv))

bench::mark(
  R              = m %*% lv, 
  `OP's version` = my_mm(m = m, v = lv), 
  `BLAS`         = blas_mm(m = m, v = lv),
  `C++ vanilla`  = mat_vec_mult_vanilla(m = m, v = lv), 
  `C++`          = mat_vec_mult(m = m, v = lv), check = FALSE)
*/

Answer 3

A quick glance in names.c (here in particular) points you to do_matprod, the C function that is called by %*% and which is found in the file array.c. (Interestingly, it turns out, that both crossprod and tcrossprod dispatch to that same function as well). Here is a link to the code of do_matprod.

Scrolling through the function, you can see that it takes care of a number of things your naive implementation does not, including:

1. Keeps row and column names, where that makes sense.
2. Allows for dispatch to alternative S4 methods when the two objects being operated on by a call to %*% are of classes for which such methods have been provided. (That's what's happening in this portion of the function.)
3. Handles both real and complex matrices.
4. Implements a series of rules for how to handle multiplication of a matrix and a matrix, a vector and a matrix, a matrix and a vector, and a vector and a vector. (Recall that under cross-multiplication in R, a vector on the LHS is treated as a row vector, whereas on the RHS, it is treated as a column vector; this is the code that makes that so.)

Near the end of the function, it dispatches to either of matprod or or cmatprod. Interestingly (to me at least), in the case of real matrices, if either matrix might contain NaN or Inf values, then matprod dispatches (here) to a function called simple_matprod which is about as simple and straightforward as your own. Otherwise, it dispatches to one of a couple of BLAS Fortran routines which, presumably are faster, if uniformly 'well-behaved' matrix elements can be guaranteed.