Sparse matrix-vector multiplication (SpMV) is a bottleneck computation in many scientific and engineering computations. Mathematically, SpMV can be expressed as

where is a sparse matrix and , , and are dense vectors. The computation can also be expressed in index notation as

You can use the TACO C++ library to easily and efficiently compute SpMV, as shown here:

// On Linux and MacOS, you can compile and run this program like so:
//   g++ -std=c++11 -O3 -DNDEBUG -DTACO -I ../../include -L../../build/lib spmv.cpp -o spmv -ltaco
//   LD_LIBRARY_PATH=../../build/lib ./spmv
#include <random>
#include "taco.h"
using namespace taco;
int main(int argc, char* argv[]) {
  std::default_random_engine gen(0);
  std::uniform_real_distribution<double> unif(0.0, 1.0);
  // Predeclare the storage formats that the inputs and output will be stored as.
  // To define a format, you must specify whether each dimension is dense or sparse 
  // and (optionally) the order in which dimensions should be stored. The formats 
  // declared below correspond to compressed sparse row (csr) and dense vector (dv). 
  Format csr({Dense,Sparse});
  Format  dv({Dense});

  // Load a sparse matrix from file (stored in the Matrix Market format) and 
  // store it as a compressed sparse row matrix. Matrices correspond to order-2 
  // tensors in taco. The matrix in this example can be downloaded from:
  Tensor<double> A = read("pwtk.mtx", csr);

  // Generate a random dense vector and store it in the dense vector format. 
  // Vectors correspond to order-1 tensors in taco.
  Tensor<double> x({A.getDimension(1)}, dv);
  for (int i = 0; i < x.getDimension(0); ++i) {
    x.insert({i}, unif(gen));

  // Generate another random dense vetor and store it in the dense vector format..
  Tensor<double> z({A.getDimension(0)}, dv);
  for (int i = 0; i < z.getDimension(0); ++i) {
    z.insert({i}, unif(gen));

  // Declare and initializing the scaling factors in the SpMV computation. 
  // Scalars correspond to order-0 tensors in taco.
  Tensor<double> alpha(42.0);
  Tensor<double> beta(33.0);

  // Declare the output matrix to be a sparse matrix with the same dimensions as 
  // input matrix B, to be also stored as a doubly compressed sparse row matrix.
  Tensor<double> y({A.getDimension(0)}, dv);
  // Define the SpMV computation using index notation.
  IndexVar i, j;
  y(i) = alpha() * (A(i,j) * x(j)) + beta() * z(i);
  // At this point, we have defined how entries in the output vector should be 
  // computed from entries in the input matrice and vectorsbut have not actually 
  // performed the computation yet. To do so, we must first tell taco to generate 
  // code that can be executed to compute the SpMV operation.
  // We can now call the functions taco generated to assemble the indices of the 
  // output vector and then actually compute the SpMV.
  // Write the output of the computation to file (stored in the FROSTT format).
  write("y.tns", y);

You can also use the TACO Python library to perform the same computation, as demonstrated here:

import pytaco as pt
from pytaco import compressed, dense
import numpy as np

# Define formats for storing the sparse matrix and dense vectors
csr = pt.format([dense, compressed])
dv  = pt.format([dense])

# Load a sparse matrix stored in the matrix market format) and store it 
# as a CSR matrix.  The matrix in this example can be downloaded from:
A ="pwtk.mtx", csr)

# Generate two random vectors using NumPy and pass them into TACO
x = pt.from_array(np.random.uniform(size=A.shape[1]))
z = pt.from_array(np.random.uniform(size=A.shape[0]))

# Declare the result to be a dense vector
y = pt.tensor([A.shape[0]], dv)

# Declare index vars
i, j = pt.get_index_vars(2)

# Define the SpMV computation
y[i] = A[i, j] * x[j] + z[i]

# Perform the SpMV computation and write the result to file
pt.write("y.tns", y)

When you run the above Python program, TACO will generate code under the hood that efficiently performs the computation in one shot. This lets TACO avoid materializing the intermediate matrix-vector product, thus reducing the amount of memory accesses and speeding up the computation.