Sparse matrix-vector multiplication (SpMV) is a bottleneck operation in many scientific and engineering computations. Mathematically, the operation demonstrated in this example can be expressed as $y = \alpha Ax + \beta z$, where $x$, $y$, and $z$ are dense vectors, $A$ is a sparse matrix, and $\alpha$ and $\beta$ are scalar values. This operation can also be expressed in index notation as

y(i) = alpha * A(i,j) * x(j) + beta * z(i)


You can use the taco C++ library to easily and efficiently compute the SpMV as demonstrated 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 -ltaco spmv.cpp -o spmv
//   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.

// 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.getDimensions()[1]}, dv);
for (int i = 0; i < x.getDimensions()[0]; ++i) {
x.insert({i}, unif(gen));
}
x.pack();

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

// 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.getDimensions()[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.
y.compile();

// We can now call the functions taco generated to assemble the indices of the
// output vector and then actually compute the SpMV.
y.assemble();
y.compute();

// Write the output of the computation to file (stored in the FROSTT format).
write("y.tns", y);
}


Under the hood, when you run the above C++ program, taco generates the imperative code shown below to compute the SpMV. taco is able to evaluate this compound operation efficiently with a single kernel that avoids materializing the intermediate matrix-vector product.

for (int iA = 0; iA < 217918; iA++) {
double tj = 0;
for (int A2_pos = A.d2.pos[iA]; A2_pos < A.d2.pos[(iA + 1)]; A2_pos++) {
int jA = A.d2.idx[A2_pos];
tj += A.vals[A2_pos] * x.vals[jA];
}
y.vals[iA] = (alpha.vals[0] * tj) + (beta.vals[0] * z.vals[iA]);
}