Ginkgo  Generated from pipelines/130473384 branch based on develop. Ginkgo version 1.1.1
A numerical linear algebra library targeting many-core architectures
The nine-pt-stencil-solver program

The 9-point stencil example..

This example depends on simple-solver, three-pt-stencil-solver, poisson-solver.

Table of contents
  1. Introduction
  2. The commented program
  1. Results
  2. The plain program

Introduction

This example solves a 2D Poisson equation:

[ \Omega = (0,1)^2 \ \Omega_b = [0,1]^2 \text{ (with boundary)} \ \partial\Omega = \Omega_b \backslash \Omega \ u : \Omega_b -> R \ u'' = f \in \Omega \ u = u_D \in \partial\Omega \ ]

using a finite difference method on an equidistant grid with K discretization points (K can be controlled with a command line parameter). The discretization may be done by any order Taylor polynomial. For an equidistant grid with K "inner" discretization points ((x1,y1), \ldots, (xk,y1),(x1,y2), \ldots, (xk,yk,z1)) step size (h = 1 / (K + 1)) and a stencil (\in \mathbb{R}^{3 \times 3}), the formula produces a system of linear equations

(\sum_{a,b=-1}^1 stencil(a,b) * u_{(i+a,j+b} = -f_k h^2), on any inner node with a neighborhood of inner nodes

On any node, where neighbor is on the border, the neighbor is replaced with a (-stencil(a,b) * u_{i+a,j+b}) and added to the right hand side vector. For example a node with a neighborhood of only edge nodes may look like this

[ \sum_{a,b=-1}^(1,0) stencil(a,b) * u_{(i+a,j+b} = -f_k h^2 - \sum_{a=-1}^1 stencil(a,1) * u_{(i+a,j+1} ]

which is then solved using Ginkgo's implementation of the CG method preconditioned with block-Jacobi. It is also possible to specify on which executor Ginkgo will solve the system via the command line. The function f is set to (f(x,y) = 6x + 6y) (making the solution (u(x,y) = x^3

  • y^3)), but that can be changed in the main function. Also the stencil values for the core, the faces, the edge and the corners can be changed when passing additional parameters.

The intention of this is to show how generation of stencil values and the right hand side vector changes when increasing the dimension.

About the example

The commented program

/ *****************************<DESCRIPTION>***********************************
This example solves a 2D Poisson equation:
\Omega = (0,1)^2
\Omega_b = [0,1]^2 (with boundary)
\partial\Omega = \Omega_b \backslash \Omega
u : \Omega_b -> R
u'' = f in \Omega
u = u_D on \partial\Omega
using a finite difference method on an equidistant grid with `K` discretization
points (`K` can be controlled with a command line parameter). The discretization
may be done by any order Taylor polynomial.
For an equidistant grid with K "inner" discretization points (x1,y1), ...,
(xk,y1),(x1,y2), ..., (xk,yk) step size h = 1 / (K + 1) and a stencil \in
\R^{3 x 3}, the formula produces a system of linear equations
\sum_{a,b=-1}^1 stencil(a,b) * u_{(i+a,j+b} = -f_k h^2, on any inner node with
a neighborhood of inner nodes
On any node, where neighbor is on the border, the neighbor is replaced with a
'-stencil(a,b) * u_{i+a,j+b}' and added to the right hand side vector. For
example a node with a neighborhood of only edge nodes may look like this
\sum_{a,b=-1}^(1,0) stencil(a,b) * u_{(i+a,j+b} = -f_k h^2 - \sum_{a=-1}^1
stencil(a,1) * u_{(i+a,j+1}
which is then solved using Ginkgo's implementation of the CG method
preconditioned with block-Jacobi. It is also possible to specify on which
executor Ginkgo will solve the system via the command line.
The function `f` is set to `f(x,y) = 6x + 6y` (making the solution `u(x,y) = x^3
+ y^3`), but that can be changed in the `main` function. Also the stencil values
for the core, the faces, the edge and the corners can be changed when passing
additional parameters.
The intention of this is to show how generation of stencil values and the right
hand side vector changes when increasing the dimension.
*****************************<DESCRIPTION>********************************** /
#include <array>
#include <chrono>
#include <ginkgo/ginkgo.hpp>
#include <iostream>
#include <map>
#include <string>
#include <vector>

Stencil values. Ordering can be seen in the main function Can also be changed by passing additional parameter when executing

constexpr double default_alpha = 10.0 / 3.0;
constexpr double default_beta = -2.0 / 3.0;
constexpr double default_gamma = -1.0 / 6.0;
/ * Possible alternative default values are for example
* default_alpha = 8.0;
* default_beta = -1.0;
* default_gamma = -1.0;
* /

Creates a stencil matrix in CSR format for the given number of discretization points.

void generate_stencil_matrix(int dp, int *row_ptrs, int *col_idxs,
double *values, double *coefs)
{
int pos = 0;
const size_t dp_2 = dp * dp;
row_ptrs[0] = pos;
for (int k = 0; k < dp; ++k) {
for (int i = 0; i < dp; ++i) {
const size_t index = i + k * dp;
for (int j = -1; j <= 1; ++j) {
for (int l = -1; l <= 1; ++l) {
const int64_t offset = l + 1 + 3 * (j + 1);
if ((k + j) >= 0 && (k + j) < dp && (i + l) >= 0 &&
(i + l) < dp) {
values[pos] = coefs[offset];
col_idxs[pos] = index + l + dp * j;
++pos;
}
}
}
row_ptrs[index + 1] = pos;
}
}
}

Generates the RHS vector given f and the boundary conditions.

template <typename Closure, typename ClosureT>
void generate_rhs(int dp, Closure f, ClosureT u, double *rhs, double *coefs)
{
const size_t dp_2 = dp * dp;
const auto h = 1.0 / (dp + 1.0);
for (int i = 0; i < dp; ++i) {
const auto yi = (i + 1) * h;
for (int j = 0; j < dp; ++j) {
const auto xi = (j + 1) * h;
const auto index = i * dp + j;
rhs[index] = -f(xi, yi) * h * h;
}
}

Iterating over the edges to add boundary values and adding the overlapping 3x1 to the rhs

for (size_t i = 0; i < dp; ++i) {
const auto xi = (i + 1) * h;
const auto index_top = i;
const auto index_bot = i + dp * (dp - 1);
rhs[index_top] -= u(xi - h, 0.0) * coefs[0];
rhs[index_top] -= u(xi, 0.0) * coefs[1];
rhs[index_top] -= u(xi + h, 0.0) * coefs[2];
rhs[index_bot] -= u(xi - h, 1.0) * coefs[6];
rhs[index_bot] -= u(xi, 1.0) * coefs[7];
rhs[index_bot] -= u(xi + h, 1.0) * coefs[8];
}
for (size_t i = 0; i < dp; ++i) {
const auto yi = (i + 1) * h;
const auto index_left = i * dp;
const auto index_right = i * dp + (dp - 1);
rhs[index_left] -= u(0.0, yi - h) * coefs[0];
rhs[index_left] -= u(0.0, yi) * coefs[3];
rhs[index_left] -= u(0.0, yi + h) * coefs[6];
rhs[index_right] -= u(1.0, yi - h) * coefs[2];
rhs[index_right] -= u(1.0, yi) * coefs[5];
rhs[index_right] -= u(1.0, yi + h) * coefs[8];
}

remove the double corner values

rhs[0] += u(0.0, 0.0) * coefs[0];
rhs[(dp - 1)] += u(1.0, 0.0) * coefs[2];
rhs[(dp - 1) * dp] += u(0.0, 1.0) * coefs[6];
rhs[dp * dp - 1] += u(1.0, 1.0) * coefs[8];
}

Prints the solution u.

void print_solution(int dp, const double *u)
{
for (int i = 0; i < dp; ++i) {
for (int j = 0; j < dp; ++j) {
std::cout << u[i * dp + j] << ' ';
}
std::cout << '\n';
}
std::cout << std::endl;
}

Computes the 1-norm of the error given the computed u and the correct solution function correct_u.

template <typename Closure>
double calculate_error(int dp, const double *u, Closure correct_u)
{
const auto h = 1.0 / (dp + 1);
auto error = 0.0;
for (int j = 0; j < dp; ++j) {
const auto xi = (j + 1) * h;
for (int i = 0; i < dp; ++i) {
using std::abs;
const auto yi = (i + 1) * h;
error +=
abs(u[i * dp + j] - correct_u(xi, yi)) / abs(correct_u(xi, yi));
}
}
return error;
}
void solve_system(const std::string &executor_string,
unsigned int discretization_points, int *row_ptrs,
int *col_idxs, double *values, double *rhs, double *u,
double accuracy)
{

Some shortcuts

using val_array = gko::Array<double>;
using idx_array = gko::Array<int>;
const auto &dp = discretization_points;
const size_t dp_2 = dp * dp;

Figure out where to run the code

const auto omp = gko::OmpExecutor::create();
std::map<std::string, std::shared_ptr<gko::Executor>> exec_map{
{"omp", omp},
{"cuda", gko::CudaExecutor::create(0, omp)},
{"hip", gko::HipExecutor::create(0, omp)},
{"reference", gko::ReferenceExecutor::create()}};

executor where Ginkgo will perform the computation

const auto exec = exec_map.at(executor_string); // throws if not valid

executor where the application initialized the data

const auto app_exec = exec_map["omp"];

Tell Ginkgo to use the data in our application

Matrix: we have to set the executor of the matrix to the one where we want SpMVs to run (in this case exec). When creating array views, we have to specify the executor where the data is (in this case app_exec).

If the two do not match, Ginkgo will automatically create a copy of the data on exec (however, it will not copy the data back once it is done

  • here this is not important since we are not modifying the matrix).
auto matrix = mtx::create(
exec, gko::dim<2>(dp_2),
val_array::view(app_exec, (3 * dp - 2) * (3 * dp - 2), values),
idx_array::view(app_exec, (3 * dp - 2) * (3 * dp - 2), col_idxs),
idx_array::view(app_exec, dp_2 + 1, row_ptrs));

RHS: similar to matrix

auto b = vec::create(exec, gko::dim<2>(dp_2, 1),
val_array::view(app_exec, dp_2, rhs), 1);

Solution: we have to be careful here - if the executors are different, once we compute the solution the array will not be automatically copied back to the original memory locations. Fortunately, whenever apply is called on a linear operator (e.g. matrix, solver) the arguments automatically get copied to the executor where the operator is, and copied back once the operation is completed. Thus, in this case, we can just define the solution on app_exec, and it will be automatically transferred to/from exec if needed.

auto x = vec::create(app_exec, gko::dim<2>(dp_2, 1),
val_array::view(app_exec, dp_2, u), 1);

Generate solver

auto solver_gen =
cg::build()
.with_criteria(
gko::stop::Iteration::build().with_max_iters(dp_2).on(exec),
.with_reduction_factor(accuracy)
.on(exec))
.with_preconditioner(bj::build().on(exec))
.on(exec);
auto solver = solver_gen->generate(gko::give(matrix));

Solve system

solver->apply(gko::lend(b), gko::lend(x));
}
int main(int argc, char *argv[])
{
if (argc < 2) {
std::cerr << "Usage: " << argv[0] << " DISCRETIZATION_POINTS [executor]"
<< " [stencil_alpha] [stencil_beta] [stencil_gamma]"
<< std::endl;
std::exit(-1);
}
const int discretization_points = argc >= 2 ? std::atoi(argv[1]) : 100;
const auto executor_string = argc >= 3 ? argv[2] : "reference";
const double alpha_c = argc >= 4 ? std::atof(argv[3]) : default_alpha;
const double beta_c = argc >= 5 ? std::atof(argv[4]) : default_beta;
const double gamma_c = argc >= 6 ? std::atof(argv[5]) : default_gamma;

clang-format off

std::array<double, 9> coefs{
gamma_c, beta_c, gamma_c,
beta_c, alpha_c, beta_c,
gamma_c, beta_c, gamma_c};

clang-format on

const auto dp = discretization_points;
const size_t dp_2 = dp * dp;

problem:

auto correct_u = [](double x, double y) { return x * x * x + y * y * y; };
auto f = [](double x, double y) { return 6 * x + 6 * y; };

matrix

std::vector<int> row_ptrs(dp_2 + 1);
std::vector<int> col_idxs((3 * dp - 2) * (3 * dp - 2));
std::vector<double> values((3 * dp - 2) * (3 * dp - 2));

right hand side

std::vector<double> rhs(dp_2);

solution

std::vector<double> u(dp_2, 0.0);
generate_stencil_matrix(dp, row_ptrs.data(), col_idxs.data(), values.data(),
coefs.data());

looking for solution u = x^3: f = 6x, u(0) = 0, u(1) = 1

generate_rhs(dp, f, correct_u, rhs.data(), coefs.data());
auto start_time = std::chrono::steady_clock::now();
solve_system(executor_string, dp, row_ptrs.data(), col_idxs.data(),
values.data(), rhs.data(), u.data(), 1e-12);
auto stop_time = std::chrono::steady_clock::now();
double runtime_duration =
std::chrono::duration_cast<std::chrono::nanoseconds>(stop_time -
start_time)
.count() *
1e-6;
print_solution(dp, u.data());
std::cout << "The average relative error is "
<< calculate_error(dp, u.data(), correct_u) / dp_2 << std::endl;
std::cout << "The runtime is " << std::to_string(runtime_duration) << " ms"
<< std::endl;
}

Results

The expected output of the relative error at K=10 should be

The average relative error is 1.45687e-13

Comments about programming and debugging

The plain program

/*******************************<GINKGO LICENSE>******************************
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LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
******************************<GINKGO LICENSE>*******************************/
/*****************************<DESCRIPTION>***********************************
This example solves a 2D Poisson equation:
\Omega = (0,1)^2
\Omega_b = [0,1]^2 (with boundary)
\partial\Omega = \Omega_b \backslash \Omega
u : \Omega_b -> R
u'' = f in \Omega
u = u_D on \partial\Omega
using a finite difference method on an equidistant grid with `K` discretization
points (`K` can be controlled with a command line parameter). The discretization
may be done by any order Taylor polynomial.
For an equidistant grid with K "inner" discretization points (x1,y1), ...,
(xk,y1),(x1,y2), ..., (xk,yk) step size h = 1 / (K + 1) and a stencil \in
\R^{3 x 3}, the formula produces a system of linear equations
\sum_{a,b=-1}^1 stencil(a,b) * u_{(i+a,j+b} = -f_k h^2, on any inner node with
a neighborhood of inner nodes
On any node, where neighbor is on the border, the neighbor is replaced with a
'-stencil(a,b) * u_{i+a,j+b}' and added to the right hand side vector. For
example a node with a neighborhood of only edge nodes may look like this
\sum_{a,b=-1}^(1,0) stencil(a,b) * u_{(i+a,j+b} = -f_k h^2 - \sum_{a=-1}^1
stencil(a,1) * u_{(i+a,j+1}
which is then solved using Ginkgo's implementation of the CG method
preconditioned with block-Jacobi. It is also possible to specify on which
executor Ginkgo will solve the system via the command line.
The function `f` is set to `f(x,y) = 6x + 6y` (making the solution `u(x,y) = x^3
+ y^3`), but that can be changed in the `main` function. Also the stencil values
for the core, the faces, the edge and the corners can be changed when passing
additional parameters.
The intention of this is to show how generation of stencil values and the right
hand side vector changes when increasing the dimension.
*****************************<DESCRIPTION>**********************************/
#include <array>
#include <chrono>
#include <ginkgo/ginkgo.hpp>
#include <iostream>
#include <map>
#include <string>
#include <vector>
constexpr double default_alpha = 10.0 / 3.0;
constexpr double default_beta = -2.0 / 3.0;
constexpr double default_gamma = -1.0 / 6.0;
/* Possible alternative default values are for example
* default_alpha = 8.0;
* default_beta = -1.0;
* default_gamma = -1.0;
*/
void generate_stencil_matrix(int dp, int *row_ptrs, int *col_idxs,
double *values, double *coefs)
{
int pos = 0;
const size_t dp_2 = dp * dp;
row_ptrs[0] = pos;
for (int k = 0; k < dp; ++k) {
for (int i = 0; i < dp; ++i) {
const size_t index = i + k * dp;
for (int j = -1; j <= 1; ++j) {
for (int l = -1; l <= 1; ++l) {
const int64_t offset = l + 1 + 3 * (j + 1);
if ((k + j) >= 0 && (k + j) < dp && (i + l) >= 0 &&
(i + l) < dp) {
values[pos] = coefs[offset];
col_idxs[pos] = index + l + dp * j;
++pos;
}
}
}
row_ptrs[index + 1] = pos;
}
}
}
template <typename Closure, typename ClosureT>
void generate_rhs(int dp, Closure f, ClosureT u, double *rhs, double *coefs)
{
const size_t dp_2 = dp * dp;
const auto h = 1.0 / (dp + 1.0);
for (int i = 0; i < dp; ++i) {
const auto yi = (i + 1) * h;
for (int j = 0; j < dp; ++j) {
const auto xi = (j + 1) * h;
const auto index = i * dp + j;
rhs[index] = -f(xi, yi) * h * h;
}
}
for (size_t i = 0; i < dp; ++i) {
const auto xi = (i + 1) * h;
const auto index_top = i;
const auto index_bot = i + dp * (dp - 1);
rhs[index_top] -= u(xi - h, 0.0) * coefs[0];
rhs[index_top] -= u(xi, 0.0) * coefs[1];
rhs[index_top] -= u(xi + h, 0.0) * coefs[2];
rhs[index_bot] -= u(xi - h, 1.0) * coefs[6];
rhs[index_bot] -= u(xi, 1.0) * coefs[7];
rhs[index_bot] -= u(xi + h, 1.0) * coefs[8];
}
for (size_t i = 0; i < dp; ++i) {
const auto yi = (i + 1) * h;
const auto index_left = i * dp;
const auto index_right = i * dp + (dp - 1);
rhs[index_left] -= u(0.0, yi - h) * coefs[0];
rhs[index_left] -= u(0.0, yi) * coefs[3];
rhs[index_left] -= u(0.0, yi + h) * coefs[6];
rhs[index_right] -= u(1.0, yi - h) * coefs[2];
rhs[index_right] -= u(1.0, yi) * coefs[5];
rhs[index_right] -= u(1.0, yi + h) * coefs[8];
}
rhs[0] += u(0.0, 0.0) * coefs[0];
rhs[(dp - 1)] += u(1.0, 0.0) * coefs[2];
rhs[(dp - 1) * dp] += u(0.0, 1.0) * coefs[6];
rhs[dp * dp - 1] += u(1.0, 1.0) * coefs[8];
}
void print_solution(int dp, const double *u)
{
for (int i = 0; i < dp; ++i) {
for (int j = 0; j < dp; ++j) {
std::cout << u[i * dp + j] << ' ';
}
std::cout << '\n';
}
std::cout << std::endl;
}
template <typename Closure>
double calculate_error(int dp, const double *u, Closure correct_u)
{
const auto h = 1.0 / (dp + 1);
auto error = 0.0;
for (int j = 0; j < dp; ++j) {
const auto xi = (j + 1) * h;
for (int i = 0; i < dp; ++i) {
using std::abs;
const auto yi = (i + 1) * h;
error +=
abs(u[i * dp + j] - correct_u(xi, yi)) / abs(correct_u(xi, yi));
}
}
return error;
}
void solve_system(const std::string &executor_string,
unsigned int discretization_points, int *row_ptrs,
int *col_idxs, double *values, double *rhs, double *u,
double accuracy)
{
using val_array = gko::Array<double>;
using idx_array = gko::Array<int>;
const auto &dp = discretization_points;
const size_t dp_2 = dp * dp;
const auto omp = gko::OmpExecutor::create();
std::map<std::string, std::shared_ptr<gko::Executor>> exec_map{
{"omp", omp},
{"cuda", gko::CudaExecutor::create(0, omp)},
{"hip", gko::HipExecutor::create(0, omp)},
{"reference", gko::ReferenceExecutor::create()}};
const auto exec = exec_map.at(executor_string); // throws if not valid
const auto app_exec = exec_map["omp"];
auto matrix = mtx::create(
exec, gko::dim<2>(dp_2),
val_array::view(app_exec, (3 * dp - 2) * (3 * dp - 2), values),
idx_array::view(app_exec, (3 * dp - 2) * (3 * dp - 2), col_idxs),
idx_array::view(app_exec, dp_2 + 1, row_ptrs));
auto b = vec::create(exec, gko::dim<2>(dp_2, 1),
val_array::view(app_exec, dp_2, rhs), 1);
auto x = vec::create(app_exec, gko::dim<2>(dp_2, 1),
val_array::view(app_exec, dp_2, u), 1);
auto solver_gen =
cg::build()
.with_criteria(
gko::stop::Iteration::build().with_max_iters(dp_2).on(exec),
.with_reduction_factor(accuracy)
.on(exec))
.with_preconditioner(bj::build().on(exec))
.on(exec);
auto solver = solver_gen->generate(gko::give(matrix));
solver->apply(gko::lend(b), gko::lend(x));
}
int main(int argc, char *argv[])
{
if (argc < 2) {
std::cerr << "Usage: " << argv[0] << " DISCRETIZATION_POINTS [executor]"
<< " [stencil_alpha] [stencil_beta] [stencil_gamma]"
<< std::endl;
std::exit(-1);
}
const int discretization_points = argc >= 2 ? std::atoi(argv[1]) : 100;
const auto executor_string = argc >= 3 ? argv[2] : "reference";
const double alpha_c = argc >= 4 ? std::atof(argv[3]) : default_alpha;
const double beta_c = argc >= 5 ? std::atof(argv[4]) : default_beta;
const double gamma_c = argc >= 6 ? std::atof(argv[5]) : default_gamma;
std::array<double, 9> coefs{
gamma_c, beta_c, gamma_c,
beta_c, alpha_c, beta_c,
gamma_c, beta_c, gamma_c};
const auto dp = discretization_points;
const size_t dp_2 = dp * dp;
auto correct_u = [](double x, double y) { return x * x * x + y * y * y; };
auto f = [](double x, double y) { return 6 * x + 6 * y; };
std::vector<int> row_ptrs(dp_2 + 1);
std::vector<int> col_idxs((3 * dp - 2) * (3 * dp - 2));
std::vector<double> values((3 * dp - 2) * (3 * dp - 2));
std::vector<double> rhs(dp_2);
std::vector<double> u(dp_2, 0.0);
generate_stencil_matrix(dp, row_ptrs.data(), col_idxs.data(), values.data(),
coefs.data());
generate_rhs(dp, f, correct_u, rhs.data(), coefs.data());
auto start_time = std::chrono::steady_clock::now();
solve_system(executor_string, dp, row_ptrs.data(), col_idxs.data(),
values.data(), rhs.data(), u.data(), 1e-12);
auto stop_time = std::chrono::steady_clock::now();
double runtime_duration =
std::chrono::duration_cast<std::chrono::nanoseconds>(stop_time -
start_time)
.count() *
1e-6;
print_solution(dp, u.data());
std::cout << "The average relative error is "
<< calculate_error(dp, u.data(), correct_u) / dp_2 << std::endl;
std::cout << "The runtime is " << std::to_string(runtime_duration) << " ms"
<< std::endl;
}
gko::abs
constexpr T abs(const T &x)
Returns the absolute value of the object.
Definition: math.hpp:572
gko::matrix::Csr
CSR is a matrix format which stores only the nonzero coefficients by compressing each row of the matr...
Definition: coo.hpp:51
gko::HipExecutor::create
static std::shared_ptr< HipExecutor > create(int device_id, std::shared_ptr< Executor > master)
Creates a new HipExecutor.
gko::give
std::remove_reference< OwningPointer >::type && give(OwningPointer &&p)
Marks that the object pointed to by p can be given to the callee.
Definition: utils.hpp:231
gko::matrix::Dense
Dense is a matrix format which explicitly stores all values of the matrix.
Definition: coo.hpp:55
gko::CudaExecutor::create
static std::shared_ptr< CudaExecutor > create(int device_id, std::shared_ptr< Executor > master)
Creates a new CudaExecutor.
gko::OmpExecutor::create
static std::shared_ptr< OmpExecutor > create()
Creates a new OmpExecutor.
Definition: executor.hpp:775
gko::stop::ResidualNormReduction
The ResidualNormReduction class is a stopping criterion which stops the iteration process when the re...
Definition: residual_norm_reduction.hpp:64
gko::preconditioner::Jacobi
A block-Jacobi preconditioner is a block-diagonal linear operator, obtained by inverting the diagonal...
Definition: jacobi.hpp:207
gko::lend
std::enable_if< detail::have_ownership_s< Pointer >::value, detail::pointee< Pointer > * >::type lend(const Pointer &p)
Returns a non-owning (plain) pointer to the object pointed to by p.
Definition: utils.hpp:253
gko::dim< 2 >
gko::solver::Cg
CG or the conjugate gradient method is an iterative type Krylov subspace method which is suitable for...
Definition: cg.hpp:72
gko::Array
An Array is a container which encapsulates fixed-sized arrays, stored on the Executor tied to the Arr...
Definition: array.hpp:65