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VexCL is a C++ vector expression template library for OpenCL/CUDA
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VexCL is a vector expression template library for OpenCL/CUDA. It has been created for ease of GPGPU development with C++. VexCL strives to reduce amount of boilerplate code needed to develop GPGPU applications. The library provides convenient and intuitive notation for vector arithmetic, reduction, sparse matrix-vectork products, etc. Multi-device and even multi-platform computations are supported. The source code of the library is distributed under very permissive MIT license.

The code is available at

Doxygen-generated documentation:

Slides from VexCL talks:

Other talks may be found at

Table of contents

Selecting backend

VexCL provides the following backends:

  • OpenCL, built on top of Khronos C++ API. The backend is selected when VEXCL_BACKEND_OPENCL macro is defined, or by default. Link with on unix-like systems or with OpenCL.dll on Windows.
  • Boost.compute. The backend is also based on OpenCL, but uses core functionality of the Boost.compute library instead of somewhat outdated Khronos C++ API. The additional advantage is the increased interoperability between VexCL and Boost.Compute. The backend is selected when VEXCL_BACKEND_COMPUTE macro is defined. Link with and make sure that Boost.Compute headers are in the include path.
  • CUDA, uses NVIDIA CUDA technology. The backend is selected when VEXCL_BACKEND_CUDA macro is defined. Link with For the CUDA backend to work, CUDA Toolkit has to be installed, and NVIDIA CUDA compiler driver nvcc has to be in executable PATH and usable at runtime.

Context initialization

VexCL transparently works with multiple compute devices that are present in the system. A VexCL context is initialized with a device filter, which is just a functor that takes a reference to vex::device and returns a bool. Several standard filters are provided, but one can easily add a custom functor. Filters may be combined with logical operators. All compute devices that satisfy the provided filter are added to the created context. In the example below all GPU devices that support double precision arithmetic are selected:

#include <iostream>
#include <stdexcept>
#include <vexcl/vexcl.hpp>

int main() {
    vex::Context ctx( vex::Filter::GPU && vex::Filter::DoublePrecision );

    if (!ctx) throw std::runtime_error("No devices available.");

    // Print out list of selected devices:
    std::cout << ctx << std::endl;

One of the most convenient filters is vex::Filter::Env which selects compute devices based on environment variables. It allows to switch compute device without need to recompile the program.

Memory allocation

The vex::vector<T> class constructor accepts a const reference to std::vector<vex::command_queue>. A vex::Context instance may be conveniently converted to this type, but it is also possible to initialize the command queues elsewhere (e.g. with the OpenCL backend vex::command_queue is typedefed to cl::CommandQueue), thus completely eliminating the need to create a vex::Context. Each command queue in the list should uniquely identify a single compute device.

The contents of the created vector will be partitioned across all devices that were present in the queue list. The size of each partition will be proportional to the device bandwidth, which is measured the first time the device is used. All vectors of the same size are guaranteed to be partitioned consistently, which minimizes inter-device communication.

In the example below, three device vectors of the same size are allocated. Vector A is copied from host vector a, and the other vectors are created uninitialized:

const size_t n = 1024 * 1024;
vex::Context ctx( vex::Filter::Any );

std::vector<double> a(n, 1.0);

vex::vector<double> A(ctx, a);
vex::vector<double> B(ctx, n);
vex::vector<double> C(ctx, n);

Assuming that the current system has an NVIDIA and an AMD GPUs along with an Intel CPU installed, possible partitioning may look as in the following figure:


Copies between host and devices

The function vex::copy() allows one to copy data between host and device memory spaces. There are two forms of the function -- a simple one and an STL-like one:

std::vector<double> h(n);       // Host vector.
vex::vector<double> d(ctx, n);  // Device vector.

// Simple form:
vex::copy(h, d);    // Copy data from host to device.
vex::copy(d, h);    // Copy data from device to host.

// STL-like form:
vex::copy(h.begin(), h.end(), d.begin()); // Copy data from host to device.
vex::copy(d.begin(), d.end(), h.begin()); // Copy data from device to host.

The STL-like variant can copy sub-ranges of the vectors, or copy data from/to raw host pointers.

Vectors also overload the array subscript operator, operator[], so that users may directly read or write individual vector elements. This operation is highly ineffective and should be used with caution. Iterators allow for element access as well, so that STL algorithms may in principle be used with device vectors. This would be very slow but may be used as a temporary building block.

Another option for host-device data transfer is mapping device memory buffer to a host array. The mapped array then may be transparently read or written. The method vector::map(unsigned d) maps the d-th partition of the vector and returns the mapped array:

vex::vector<double> X(ctx, N);
auto mapped_ptr =; // Unmapped automatically when goes out of scope
for(size_t i = 0; i < X.part_size(0); ++i)
    mapped_ptr[i] = host_function(i);

Vector expressions

VexCL allows the use of convenient and intuitive notation for vector operations. In order to be used in the same expression, all vectors have to be compatible:

  • Have same size;
  • Span same set of compute devices.

If these conditions are satisfied, then vectors may be combined with rich set of available expressions. Vector expressions are processed in parallel across all devices they were allocated on. One should keep in mind that in case several command queues are used, then the queues of the vector that is being assigned to will be employed. Each vector expression results in the launch of a single compute kernel. The kernel is automatically generated and launched the first time the expression is encountered in the program. If the VEXCL_SHOW_KERNELS macro is defined, then the sources of all generated kernels will be dumped to the standard output. For example, the expression:

X = 2 * Y - sin(Z);

will lead to the launch of the following compute kernel:

kernel void vexcl_vector_kernel(
    ulong n,
    global double * prm_1,
    int prm_2,
    global double * prm_3,
    global double * prm_4
    for(size_t idx = get_global_id(0); idx < n; idx += get_global_size(0)) {
        prm_1[idx] = ( ( prm_2 * prm_3[idx] ) - sin( prm_4[idx] ) );

Here and in the rest of examples X, Y, and Z are compatible instances of vex::vector<double>; it is also assumed that OpenCL backend is selected.

VexCL is able to cache the compiled kernels offline. The compiled binaries are stored in $HOME/.vexcl on Linux and MacOSX, and in %APPDATA%\vexcl on Windows systems. In order to enable this functionality for OpenCL backend, the user has to define the VEXCL_CACHE_KERNELS macro. NVIDIA OpenCL implementation does the caching already, but on AMD or Intel platforms this may lead to dramatic decrease of program initialization time (e.g. VexCL tests take around 20 seconds to complete without kernel caches, and 2 seconds when caches are available). In case of the CUDA backend the offline caching is always enabled.

Builtin operations

VexCL expressions may combine device vectors and scalars with arithmetic, logic, or bitwise operators as well as with builtin OpenCL functions. If some builtin operator or function is unavailable, it should be considered a bug. Please do not hesitate to open an issue in this case.

Z = sqrt(2 * X) + pow(cos(Y), 2.0);


As you have seen above, 2 in the expression 2 * Y - sin(Z) is passed to the generated compute kernel as an int parameter (prm_2). Sometimes this is desired behaviour, because the same kernel will be reused for the expressions 42 * Z - sin(Y) or a * Y - sin(Y) (where a is an integer variable). But this may lead to a slight overhead if an expression involves true constant that will always have same value. The macro VEX_CONSTANT allows one to define such constants for use in vector expressions. Compare the generated kernel for the following example with the kernel above:


X = two() * Y - sin(Z);
kernel void vexcl_vector_kernel(
    ulong n,
    global double * prm_1,
    global double * prm_3,
    global double * prm_4
    for(size_t idx = get_global_id(0); idx < n; idx += get_global_size(0)) {
        prm_1[idx] = ( ( ( 2 ) * prm_3[idx] ) - sin( prm_4[idx] ) );

VexCL provides some predefined constants in the vex::constants namespace that correspond to boost::math::constants (e.g. vex::constants::pi()).

Element indices

The function vex::element_index(size_t offset = 0) allows one to use the index of each vector element inside vector expressions. The numbering is continuous across the compute devices and starts with an optional offset.

// Linear function:
double x0 = 0.0, dx = 1.0 / (X.size() - 1);
X = x0 + dx * vex::element_index();

// Single period of sine function:
Y = sin(vex::constants::two_pi() * vex::element_index() / Y.size());

User-defined functions

Users may define custom functions for use in vector expressions. One has to define the function signature and the function body. The body may contain any number of lines of valid OpenCL or CUDA code, depending on the selected backend. The most convenient way to define a function is via the VEX_FUNCTION macro:

VEX_FUNCTION(double, squared_radius, (double, x)(double, y),
    return x * x + y * y;
Z = sqrt(squared_radius(X, Y));

The first macro parameter here defines the function return type, the second parameter is the function name, the third parameter defines function arguments in form of a preprocessor sequence. Each element of the sequence is a tuple of argument type and name. The rest of the macro is the function body (compare this with how functions are defined in C/C++). The resulting squared_radius function object is stateless; only its type is used for kernel generation. Hence, it is safe to define commonly used functions at the global scope.

Note that any valid vector expression may be passed as a function parameter, including nested function calls:

Z = squared_radius(sin(X + Y), cos(X - Y));

Another version of the macro takes the function body directly as a string:

VEX_FUNCTION_S(double, squared_radius, (double, x)(double, y),
    "return x * x + y * y;"
Z = sqrt(squared_radius(X, Y));

In case the function that is being defined calls other custom function inside its body, one can use the version of the VEX_FUNCTION macro that takes sequence of parent function names as the fourth parameter:

VEX_FUNCTION(double, bar, (double, x),
        double s = sin(x);
        return s * s;
VEX_FUNCTION(double, baz, (double, x),
        double c = cos(x);
        return c * c;
VEX_FUNCTION_D(double, foo, (double, x)(double, y), (bar)(baz),
        return bar(x - y) * baz(x + y);

Similarly to VEX_FUNCTION_S, there is a version called VEX_FUNCTION_DS (or VEX_FUNCTION_SD) that takes the function body as a string parameter.

Custom functions may be used not only for convenience, but also for performance reasons. The example with squared_radius could in principle be rewritten as:

Z = sqrt(X * X + Y * Y);

The drawback of the latter variant is that X and Y will be passed to the kernel and read twice (see next section for an explanation).

Note that prior to release 1.2 of VexCL the VEX_FUNCTION macro had different interface. That version is considered deprecated but is still available as VEX_FUNCTION_V1.

Another example of using a custom function is type-casting a vector. This has the advantage of beeing backend independent.

VEX_FUNCTION(float, make_float, (int, i),
    return (float)i;
b = make_float(a) / 2;

Another option would be to use builtin OpenCL functions convert_*' (vex::convert_*),but those are obviously only available for OpenCL backend.

Tagged terminals

The last example of the previous section is ineffective because the compiler cannot tell if any two terminals in an expression tree are actually referring to the same data. But programmers often have this information. VexCL allows one to pass this knowledge to compiler by tagging terminals with unique tags. By doing this, the programmer guarantees that any two terminals with matching tags are referencing same data.

Below is a more effective variant of the above example:

using vex::tag;
Z = sqrt(tag<1>(X) * tag<1>(X) + tag<2>(Y) * tag<2>(Y));

Here, the generated kernel will have one parameter for each of the vectors X and Y.

Temporary values

Some expressions may have several occurences of the same subexpression. Unfortunately, VexCL is not able to determine these cases without the programmer's help. For example, let's look at the following expression:

Y = log(X) * (log(X) + Z);

Here, log(X) would be computed twice. One could tag vector X as in:

auto x = vex::tag<1>(X);
Y = log(x) * (log(x) + Z);

and hope that the backend compiler is smart enough to reuse result of log(x) (e.g. NVIDIA's compiler is smart enough to do this). But it is also possible to explicitly ask VexCL to store result of a subexpression in a local variable and reuse it. The vex::make_temp() function template serves this purpose:

auto tmp1 = vex::make_temp<1>( sin(X) );
auto tmp2 = vex::make_temp<2>( cos(X) );
Y = (tmp1 - tmp2) * (tmp1 + tmp2);

Any valid vector or multivector expression (but not additive expressions, such as sparse matrix-vector products) may be wrapped into a make_temp() call.

Random number generation

VexCL provides a counter-based random number generators from Random123 suite, in which Nth random number is obtained by applying a stateless mixing function to N instead of the conventional approach of using N iterations of a stateful transformation. This technique is easily parallelizable and is well suited for use in GPGPU applications.

For integral types, the generated values span the complete range; for floating point types, the generated values lie in the interval [0,1].

In order to use a random number sequence in a vector expression, the user has to declare an instance of either vex::Random or vex::RandomNormal class template as in the following example:

vex::Random<double, vex::random::threefry> rnd;

// X will contain random numbers from [-1, 1]:
X = 2 * rnd(vex::element_index(), std::rand()) - 1;

Note that vex::element_index() here provides the random number generator with a sequence position N.


vex::permutation() allows the use of a permuted vector in a vector expression. The function accepts a vector expression that returns integral values (indices). The following example reverses X and assigns it to Y:

vex::vector<size_t> I(ctx, N);
I = N - 1 - vex::element_index();
auto reverse = vex::permutation(I)

Y = reverse(X);

The drawback of the above approach is that you have to store and access an index vector. Sometimes this is a necessary evil, but in this simple example we can do better. In the following snippet a lightweight expression is used to construct the same permutation:

auto reverse = vex::permutation( N - 1 - vex::element_index() );
Y = reverse(X);

Note that any valid vector expression may be used as an index, including user-defined functions.

Permutation operations are only supported in single-device contexts.


An instance of the vex::slicer<NDIM> class allows one to conveniently access sub-blocks of multi-dimensional arrays that are stored in vex::vector in row-major order. The constructor of the class accepts the dimensions of the array to be sliced. The following example extracts every other element from interval [100, 200) of a one-dimensional vector X:

vex::vector<double> X(ctx, n);
vex::vector<double> Y(ctx, 50);

vex::slicer<1> slice(vex::extents[n]);

Y = slice[vex::range(100, 2, 200)](X);

And the example below shows how to work with a two-dimensional matrix:

using vex::range;

vex::vector<double> X(ctx, n * n); // n-by-n matrix stored in row-major order.
vex::vector<double> Y(ctx, n);

// vex::extents is a helper object similar to boost::multi_array::extents
vex::slicer<2> slice(vex::extents[n][n]);

Y = slice[42](X);          // Put 42-nd row of X into Y.
Y = slice[range()][42](X); // Put 42-nd column of X into Y.

slice[range()][10](X) = Y; // Put Y into 10-th column of X.

// Assign sub-block [10,20)x[30,40) of X to Z:
vex::vector<double> Z = slice[range(10, 20)][range(30, 40)](X);
assert(Z.size() == 100);

Slicing is only supported in single-device contexts.

Reducing multidimensional expressions

vex::reduce() function allows one to reduce a multidimensional expression along one or more dimensions. The result is again a vector expression. The supported reduction operations are SUM, MIN, and MAX. The function takes three arguments: the shape of the expression to reduce (with the slowest changing dimension in the front), the expression to reduce, and the dimension(s) to reduce along. The latter are specified as indices into the shape array.

In the following example we find maximum absolute value of each row in a two-dimensional matrix and assign the result to a vector:

vex::vector<double> A(ctx, N * M);
vex::vector<double> x(ctx, N);

x = vex::reduce<vex::MAX>(vex::extents[N][M], fabs(A), vex::extents[1]);

Expression reduction is only supported in single-device contexts.


vex::reshape(expr, dst_dims, src_dims) function is a powerful primitive that allows one to conveniently manipulate multidimensional data. It takes three arguments -- an arbitrary vector expression expr to reshape, the dimensions dst_dims of the final result (with the slowest changing dimension in the front), and the dimensions src_dims of the expression, which are specified as indices into dst_dims. The function returns a vector expression that could be assigned to a vector or participate in a larger expression. The dimensions may be conveniently specified with help of vex::extents object.

Here is an example of transposing a two-dimensional matrix of size NxM:

vex::vector<double> A(ctx, N * M);
vex::vector<double> B = vex::reshape(A,
                            vex::extents[M][N], // new shape
                            vex::extents[1][0]  // A is shaped as [N][M]

If the source expression lacks some of the destination dimensions, then those will be introduced by replicating the available data. For example, to make a two-dimensional matrix from a one-dimensional vector by copying the vector to each row of the matrix, one could do the following:

vex::vector<double> x(ctx, N);
vex::vector<double> y(ctx, M);
vex::vector<double> A(ctx, M * N);

// Copy x into rows of A:
A = vex::reshape(x, vex::extents[M][N], vex::extents[1]);
// Now, copy y into columns of A:
A = vex::reshape(x, vex::extents[M][N], vex::extents[0]);

Here is a more realistic example of a dense matrix-matrix multiplication. Elements of a matrix product C = A * B are defined as C[i][j] = sum_k(A[i][k] * B[k][j]). Let's assume that matrix A has shape [N][L], and matrix B is shaped as [L][M]. Then matrix C has dimensions [N][M]. In order to implement the multiplication we extend matrices A and B to the shape of [N][L][M], multiply the resulting expressions, and reduce the product along the middle dimension L:

vex::vector<double> A(ctx, N * L);
vex::vector<double> B(ctx, L * M);
vex::vector<double> C(ctx, N * M);

C = vex::reduce<vex::SUM>(
        vex::reshape(A, vex::extents[N][L][M], vex::extents[0][1]) *
        vex::reshape(B, vex::extents[N][L][M], vex::extents[1][2]),

This of course would not be as efficient as a carefully crafted custom implementation or a call to a vendor BLAS function. Still, the fact that the result is a vector expression (and hence may be a part of a still larger expression) could be more important sometimes.

Reshaping is only supported in single-device contexts.

Tensor product

Given two tensors (arrays of dimension greater than or equal to one), A and B, and a list of axes pairs (where each pair represents corresponding axes from two tensors), the tensor product operation sums the products of A's and B's elements over the given axes. In VexCL this is implemented as vex::tensordot() operation (compare with python's numpy.tensordot).

For example, the above matrix-matrix product may be implemented much more efficiently with tensordot():

using vex::_;

vex::slicer<2> Adim(vex::extents[N][M]);
vex::slicer<2> Bdim(vex::extents[M][L]);

C = vex::tensordot(Adim[_](A), Bdim[_](B), vex::axes_pairs(1, 0));

tensordot() is only available for single-device contexts.

Scattered data interpolation with multilevel B-Splines

VexCL provides an implementation of the MBA algorithm based on paper by Lee, Wolberg, and Shin ([S. Lee, G. Wolberg, and S. Y. Shin. Scattered data interpolation with multilevel B-Splines. IEEE Transactions on Visualization and Computer Graphics, 3:228–244, 1997][bsplines]). This is a fast algorithm for scattered N-dimensional data interpolation and approximation. Multilevel B-splines are used to compute a C2-continuously differentiable surface through a set of irregularly spaced points. The algorithm makes use of a coarse-to-fine hierarchy of control lattices to generate a sequence of bicubic B-spline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using B-spline refinement to reduce the sum of these functions into one equivalent B-spline function. High-fidelity reconstruction is possible from a selected set of sparse and irregular samples.

The algorithm is first prepared on a CPU. After that, it may be used in vector expressions. Here is an example in 2D:

// Coordinates of data points:
std::vector< std::array<double,2> > coords = {
    {0.0, 0.0},
    {0.0, 1.0},
    {1.0, 0.0},
    {1.0, 1.0},
    {0.4, 0.4},
    {0.6, 0.6}

// Data values:
std::vector<double> values = {
    0.2, 0.0, 0.0, -0.2, -1.0, 1.0

// Bounding box:
std::array<double, 2> xmin = {-0.01, -0.01};
std::array<double, 2> xmax = { 1.01,  1.01};

// Initial grid size:
std::array<size_t, 2> grid = {5, 5};

// Algorithm setup.
vex::mba<2> surf(ctx, xmin, xmax, coords, values, grid);

// x and y are coordinates of arbitrary 2D points:
// vex::vector<double> x, y, z;

// Get interpolated values:
z = surf(x, y);

Fast Fourier Transform

VexCL provides an implementation of the Fast Fourier Transform (FFT) that accepts arbitrary vector expressions as input, allows one to perform multidimensional transforms (of any number of dimensions), and supports arbitrary sized vectors:

vex::FFT<double, cl_double2> fft(ctx, n);
vex::FFT<cl_double2, double> ifft(ctx, n, vex::fft::inverse);

vex::vector<double> rhs(ctx, n), u(ctx, n), K(ctx, n);

// Solve Poisson equation with FFT:
u = ifft( K * fft(rhs) );

The restriction of the FFT is that it currently only supports contexts with a single compute device.


An instance of vex::Reductor<T, OP> allows one to reduce an arbitrary vector expression to a single value of type T. Supported reduction operations are SUM, MIN, and MAX. Reductor objects receive a list of command queues at construction and should only be applied to vectors residing on the same compute devices.

In the following example an inner product of two vectors is computed:

vex::Reductor<double, vex::SUM> sum(ctx);

double s = sum(x * y);

And here is an easy way to compute an approximate value of π with Monte-Carlo method:

VEX_FUNCTION(double, squared_radius, (double, x)(double, y),
    return x * x + y * y;

vex::Reductor<size_t, vex::SUM> sum(ctx);
vex::Random<double, vex::random::threefry> rnd;

X = 2 * rnd(vex::element_index(), std::rand()) - 1;
Y = 2 * rnd(vex::element_index(), std::rand()) - 1;

double pi = 4.0 * sum(squared_radius(X, Y) < 1) / X.size();

Sparse matrix-vector products

One of the most common operations in linear algebra is matrix-vector multiplication. An instance of vex::SpMat class holds a representation of a sparse matrix. Its constructor accepts a sparse matrix in common CRS format. In the example below a vex::SpMat is constructed from an Eigen sparse matrix:

Eigen::SparseMatrix<double, Eigen::RowMajor, int> E;

vex::SpMat<double, int> A(ctx, E.rows(), E.cols(),
    E.outerIndexPtr(), E.innerIndexPtr(), E.valuesPtr());

Matrix-vector products may be used in vector expressions. The only restriction is that the expressions have to be additive. This is due to the fact that the matrix representation may span several compute devices. Hence, a matrix-vector product operation may require several kernel launches and inter-device communication.

// Compute residual value for a system of linear equations:
Z = Y - A * X;

This restriction may be lifted for single-device contexts. In this case VexCL does not need to worry about inter-device communication. Hence, it is possible to inline matrix-vector product into a normal vector expression with the help of vex::make_inline():

residual = sum(Y - vex::make_inline(A * X));
Z = sin(vex::make_inline(A * X));

Stencil convolutions

Stencil convolution is another common operation that may be used, for example, to represent a signal filter, or a (one-dimensional) differential operator. VexCL implements two stencil kinds. The first one is a simple linear stencil that holds linear combination coefficients. The example below computes the moving average of a vector with a 5-point window:

vex::stencil<double> S(ctx, /*coefficients:*/{0.2, 0.2, 0.2, 0.2, 0.2}, /*center:*/2);

Y = X * S;

Users may also define custom stencil operators. This may be of use if, for example, the operator is nonlinear. The definition of a stencil operator looks very similar to a definition of a custom function. The only difference is that the stencil operator constructor accepts a vector of command queues. The following example implements the nonlinear operator y(i) = sin(x(i) - x(i - 1)) + sin(x(i+1) - sin(x(i)):

VEX_STENCIL_OPERATOR(S, /*return type:*/double, /*window width:*/3, /*center:*/1,
    "return sin(X[0] - X[-1]) + sin(X[1] - X[0]);", ctx);

Z = S(Y);

The current window is available inside the body of the operator through the X array, which is indexed relative to the stencil center.

Stencil convolution operations, similar to the matrix-vector products, are only allowed in additive expressions.

Raw pointers

Unfortunately, describing two dimensional stencils (e.g. discretization of the Laplace operator) would not be effective, because the stencil width would be too large. One can solve this problem by using a raw_pointer(const vector<T>&) with a subscript operator. For the sake of simplicity, the example below implements a 3-point laplace operator for a one-dimensional vector; but this could be easily extended onto a two-dimensional case:

VEX_CONSTANT(zero, 0);
VEX_CONSTANT(one,  1);
VEX_CONSTANT(two,  2);

size_t N   = x.size();
auto   ptr = vex::raw_pointer(x);

auto i     = vex::make_temp<1>( vex::element_index() );
auto left  = vex::make_temp<2>( if_else(i > zero(),    i - one(), i) );
auto right = vex::make_temp<3>( if_else(i + one() < N, i + one(), i) );

y = ptr[i] * two() - ptr[left] - ptr[right];

Similar approach could be used in order to implement an N-body problem with a user-defined function:

// Takes vector size, current element position, and pointer to a vector to sum:
VEX_FUNCTION(double, global_interaction, (size_t, n)(size_t, i)(double*, val),
    double sum = 0;
    double myval = val[i];
    for(size_t j = 0; j < n; ++j)
        if (j != i) sum += fabs(val[j] - myval);
    return sum;

y = global_interaction(x.size(), vex::element_index(), vex::raw_pointer(x));

Note that the use of raw_pointer() is limited to single-device contexts for obvious reasons.

Using constant cache in OpenCL

The OpenCL backend of VexCL allows one to use constant cache on GPUs in order to speed up read-only access to small vectors. Usually around 64Kb of constant cache multiprocessor is available. Vectors wrapped in vex::constant() will be decorated with constant keyword insted of the usual global one. For example, the following expression:

x = 2 * vex::constant(y);

will result in the OpenCL kernel below:

kernel void vexcl_vector_kernel
  ulong n,
  global int * prm_1,
  int prm_2,
  constant int * prm_3
  for(ulong idx = get_global_id(0); idx < n; idx += get_global_size(0)) {
    prm_1[idx] = ( prm_2 * prm_3[idx] );

In cases where access to arbitrary vector elements is required, vex::constant_pointer() may be used similarly to vex::raw_pointer(). The extracted pointer will be decorated with constant keyword.

Sort, scan, reduce-by-key algorithms

VexCL provides several standalone parallel primitives that may not be used as part of a vector expression. These are inclusive_scan, exclusive_scan, sort, sort_by_key, reduce_by_key. All of these functions take VexCL vectors as both input and output parameters.

Sorting and scan functions take an optional function object used for comparison and summing of elements. The functor should provide the same interface as, e.g. std::less for sorting or std::plus for summing; additionally, it should provide a VexCL function for device-side operations.

Here is an example of such an object comparing integer elements in such a way that even elements precede odd ones:

template <typename T>
struct even_first {
    #define BODY                        \
        char bit1 = 1 & a;              \
        char bit2 = 1 & b;              \
        if (bit1 == bit2) return a < b; \
        return bit1 < bit2;

    // Device version.
    VEX_FUNCTION(bool, device, (int, a)(int, b), BODY);

    // Host version.
    bool operator()(int a, int b) const { BODY }

    #undef BODY

Same functor could be created with help of VEX_DUAL_FUNCTOR macro, which takes return type, sequence of arguments (similar to VEX_FUNCTION), and the body of the functor:

template <typename T>
struct even_first {
    VEX_DUAL_FUNCTOR(bool, (int, a)(int, b),
        char bit1 = 1 & a;
        char bit2 = 1 & b;
        if (bit1 == bit2) return a < b;
        return bit1 < bit2;

Note that VexCL already provides vex::less<T>, vex::less_equal<T>, vex::greater<T>, vex::greater_equal<T>, and vex::plus<T>.

The need to provide both host-side and device-side parts of the functor comes from the fact that multidevice vectors are first sorted partially on each of the compute devices they are allocated on and then merged on the host.

Sorting algorithms may also take tuples of keys/values (in fact, any Boost.Fusion sequence will do). One will have to explicitly specify the comparison functor in this case. Both host and device variants of the comparison functor should take 2n arguments, where n is the number of keys. The first n arguments correspond to the left set of keys, and the second n arguments correspond to the right set of keys. Here is an example that sorts values by a tuple of two keys:

vex::vector<int>    keys1(ctx, n);
vex::vector<float>  keys2(ctx, n);
vex::vector<double> vals (ctx, n);

struct {
    VEX_FUNCTION(bool, device, (int, a1)(float, a2)(int, b1)(float, b2),
            return (a1 == b1) ? (a2 < b2) : (a1 < b1);
    bool operator()(int a1, float a2, int b1, float b2) const {
        return std::make_tuple(a1, a2) < std::tuple(b1, b2);
} comp;

vex::sort_by_key(std::tie(keys1, keys2), vals, comp);


The class template vex::multivector<T,N> allows one to store several equally sized device vectors and perform computations on all components synchronously. Each operation is delegated to the underlying vectors, but usually results in the launch of a single fused kernel. Expressions may include values of std::array<T,N> type, where N is equal to the number of multivector components. Each component gets the corresponding element of std::array<> when the expression is applied. Similarly, the array subscript operator or reduction of a multivector returns an std::array<T,N>. In order to access k-th component of a multivector, one can use the overloaded operator():

VEX_FUNCTION(bool, between, (double, a)(double, b)(double, c),
    return a <= b && b <= c;

vex::Reductor<double, vex::SUM> sum(ctx);
vex::SpMat<double> A(ctx, ... );
std::array<double, 2> v = {6.0, 7.0};

vex::multivector<double, 2> X(ctx, N), Y(ctx, N);

// ...

X = sin(v * Y + 1);             // X(k) = sin(v[k] * Y(k) + 1);
v = sum( between(0, X, Y) );    // v[k] = sum( between( 0, X(k), Y(k) ) );
X = A * Y;                      // X(k) = A * Y(k);

Some operations can not be expressed with simple multivector arithmetic. For example, an operation of two dimensional rotation mixes components in the right hand side expressions:

y0 = x0 * cos(alpha) - x1 * sin(alpha);
y1 = x0 * sin(alpha) + x1 * cos(alpha);

This may in principle be implemented as:

double alpha;
vex::multivector<double, 2> X(ctx, N), Y(ctx, N);

Y(0) = X(0) * cos(alpha) - X(1) * sin(alpha);
Y(1) = X(0) * sin(alpha) + X(1) * cos(alpha);

But this would result in two kernel launches. VexCL allows one to assign a tuple of expressions to a multivector, which will lead to the launch of a single fused kernel:

Y = std::tie( X(0) * cos(alpha) - X(1) * sin(alpha),
              X(0) * sin(alpha) + X(1) * cos(alpha) );

Converting generic C++ algorithms to OpenCL/CUDA

CUDA and OpenCL differ in their handling of compute kernels compilation. In NVIDIA's framework the compute kernels are compiled to PTX code together with the host program. In OpenCL the compute kernels are compiled at runtime from high-level C-like sources, adding an overhead which is particularly noticeable for smaller sized problems. This distinction leads to higher initialization cost of OpenCL programs, but at the same time it allows one to generate better optimized kernels for the problem at hand. VexCL exploits this possibility with help of its kernel generator mechanism. Moreover, VexCL's CUDA backend uses the same technique to generate and compile CUDA kernels at runtime.

An instance of vex::symbolic<T> dumps to an output stream any arithmetic operations it is being subjected to. For example, this code snippet:

vex::symbolic<double> x = 6, y = 7;
x = sin(x * y);

results in the following output:

double var1 = 6;
double var2 = 7;
var1 = sin( ( var1 * var2 ) );

Kernel generator

The symbolic type allows one to record a sequence of arithmetic operations made by a generic C++ algorithm. To illustrate the idea, consider the generic implementation of a 4th order Runge-Kutta ODE stepper:

template <class state_type, class SysFunction>
void runge_kutta_4(SysFunction sys, state_type &x, double dt) {
    state_type k1 = dt * sys(x);
    state_type k2 = dt * sys(x + 0.5 * k1);
    state_type k3 = dt * sys(x + 0.5 * k2);
    state_type k4 = dt * sys(x + k3);

    x += (k1 + 2 * k2 + 2 * k3 + k4) / 6;

This function takes a system function sys, state variable x, and advances x by time step dt. For example, to model the equation dx/dt = sin(x), one has to provide the following system function:

template <class state_type>
state_type sys_func(const state_type &x) {
    return sin(x);

The following code snippet makes one hundred RK4 iterations for a single double value on a CPU:

double x = 1, dt = 0.01;

for(int step = 0; step < 100; ++step)
    runge_kutta_4(sys_func<double>, x, dt);

Let's now generate the kernel for a single RK4 step and apply the kernel to a vex::vector<double> (by doing this we essentially simultaneously solve a large number of identical ODEs with different initial conditions).

// Set recorder for expression sequence.
std::ostringstream body;

// Create symbolic variable.
typedef vex::symbolic<double> sym_state;
sym_state sym_x(sym_state::VectorParameter);

// Record expression sequience for a single RK4 step.
double dt = 0.01;
runge_kutta_4(sys_func<sym_state>, sym_x, dt);

// Build kernel from the recorded sequence.
auto kernel = vex::generator::build_kernel(ctx, "rk4_stepper", body.str(), sym_x);

// Create initial state.
const size_t n = 1024 * 1024;
vex::vector<double> x(ctx, n);
x = 10.0 * vex::element_index() / n;

// Make 100 RK4 steps.
for(int i = 0; i < 100; i++) kernel(x);

This approach has some obvious restrictions. Namely, the C++ code has to be embarrassingly parallel and is not allowed to contain any branching or data-dependent loops. Nevertheless, the kernel generation facility may save a substantial amount of both human and machine time when applicable.

Function generator

VexCL also provides a user-defined function generator which takes a function signature and generic function object, and returns custom VexCL function ready to be used in vector expressions. Let's rewrite the above example using an autogenerated function for a Runge-Kutta stepper. First, we need to implement generic functor:

struct rk4_stepper {
    double dt;

    rk4_stepper(double dt) : dt(dt) {}

    template <class state_type>
    state_type operator()(const state_type &x) const {
        state_type new_x = x;
        runge_kutta_4(sys_func<state_type>, new_x, dt);
        return new_x;

Now we can generate and apply the custom function:

double dt = 0.01;
rk4_stepper stepper(dt);

// Generate custom VexCL function:
auto rk4 = vex::generator::make_function<double(double)>(stepper);

// Create initial state.
const size_t n = 1024 * 1024;
vex::vector<double> x(ctx, n);
x = 10.0 * vex::element_index() / n;

// Use the function to advance initial state:
for(int i = 0; i < 100; i++) x = rk4(x);

Note that both runge_kutta_4() and rk4_stepper may be reused for host-side computations.

It is very easy to generate a VexCL function from a Boost.Phoenix lambda expression (since Boost.Phoenix lambdas are themselves generic functors):

using namespace boost::phoenix::arg_names;
using vex::generator::make_function;

auto squared_radius = make_function<double(double, double)>(arg1 * arg1 + arg2 * arg2);

Z = squared_radius(X, Y);

Custom kernels

As Kozma Prutkov repeatedly said, "One cannot embrace the unembraceable". So in order to be usable, VexCL has to support custom kernels. vex::vector::operator()(uint k) returns a cl::Buffer that holds vector data on the k-th compute device. If the result depends on the neighboring points, one has to keep in mind that these points are possibly located on a different compute device. In this case the exchange of these halo points has to be addressed manually.

The following example builds and launches a custom kernel for each device in the context:

std::vector<vex::backend::kernel> kernel;

// Compile and store the kernels for later use.
for(uint d = 0; d < ctx.size(); d++) {
            kernel void dummy(ulong n, global float *x) {
                for(size_t i = get_global_id(0); i < n; i += get_global_size(0))
                    x[i] = 4.2;

// Apply the kernels to the vector partitions on each device.
for(uint d = 0; d < ctx.size(); d++) {


Interoperability with other libraries

Since VexCL is built upon standard Khronos OpenCL C++ bindings, it is easily interoperable with other OpenCL libraries. In particular, VexCL provides some glue code for the ViennaCL, Boost.compute and CLOGS libraries.

  • ViennaCL (The Vienna Computing Library) is a scientific computing library written in C++. It provides OpenCL, CUDA, and OpenMP compute backends. It is possible to use ViennaCL's generic solvers with VexCL types. See examples/viennacl/solvers.cpp for an example.
  • Boost.compute is a GPU/parallel-computing library for C++ based on OpenCL. The core library is a thin C++ wrapper over the OpenCL C API and provides access to compute devices, contexts, command queues and memory buffers. On top of the core library is a generic, STL-like interface providing common algorithms along with common containers. vexcl/external/boost_compute.hpp provides an example of using Boost.compute algorithms with VexCL vectors. Namely, it implements parallel sort and inclusive scan primitives on top of the corresponding Boost.compute algorithms.
  • CLOGS is a parallel primitives library implementing exclusive scan and radix sort in OpenCL. It uses auto-tuning to provide high performance for large problem sizes. In particular, the exclusive scan has much higher performance than the generic implementation in VexCL. vexcl/external/clogs.hpp provides wrappers to use CLOGS functionality with VexCL vectors. This interface currently does not benefit from the VexCL kernel cache, and so performance may be poor for small problem sizes.

Supported compilers

VexCL makes heavy use of C++11 features, so your compiler has to be modern enough. The compilers that have been tested and supported:

  • GCC v4.6 and higher.
  • Clang v3.1 and higher.
  • Microsoft Visual C++ 2010 and higher.

VexCL uses standard OpenCL bindings for C++ from Khronos group. The cl.hpp file should be included with the OpenCL implementation on your system, but it is also provided with the library.


  • D. Demidov, K. Ahnert, K. Rupp, and P. Gottchling. "Programming CUDA and OpenCL: A Case Study Using Modern C++ Libraries." SIAM Journal on Scientific Computing 35.5 (2013): C453-C472. DOI: 10.1137/120903683.
  • K. Ahnert, D. Demidov, and M. Mulansky. "Solving Ordinary Differential Equations on GPUs." Numerical Computations with GPUs. Springer International Publishing, 2014. 125-157. DOI: 10.1007/978-3-319-06548-9_7.

This work is a joint effort of Supercomputer Center of Russian Academy of Sciences (Kazan branch) and Kazan Federal University. It is partially supported by RFBR grants No 12-07-0007 and 12-01-00333a.

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