Ginkgo  Generated from pipelines/1068515030 branch based on master. Ginkgo version 1.7.0
A numerical linear algebra library targeting many-core architectures
The inverse-iteration program

The inverse iteration example..

This example depends on simple-solver, .

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

Introduction

This example shows how components available in Ginkgo can be used to implement higher-level numerical methods. The method used here will be the shifted inverse iteration method for eigenvalue computation which find the eigenvalue and eigenvector of A closest to z, for some scalar z. The method requires repeatedly solving the shifted linear system (A - zI)x = b, as well as performing matrix-vector products with the matrix A. Here is the complete pseudocode of the method:

x_0 = initial guess
for i = 0 .. max_iterations:
solve (A - zI) y_i = x_i for y_i+1
x_(i+1) = y_i / || y_i || # compute next eigenvector approximation
g_(i+1) = x_(i+1)^* A x_(i+1) # approximate eigenvalue (Rayleigh quotient)
if ||A x_(i+1) - g_(i+1)x_(i+1)|| < tol * g_(i+1): # check convergence
break

About the example

The commented program

#include <ginkgo/ginkgo.hpp>
#include <cmath>
#include <complex>
#include <fstream>
#include <iomanip>
#include <iostream>
#include <map>
#include <string>
int main(int argc, char* argv[])
{

Some shortcuts

using precision = std::complex<double>;
using real_precision = gko::remove_complex<precision>;
using vec = gko::matrix::Dense<precision>;
using real_vec = gko::matrix::Dense<real_precision>;
using mtx = gko::matrix::Csr<precision>;
using solver_type = gko::solver::Bicgstab<precision>;
using std::abs;
using std::sqrt;

Print version information

std::cout << gko::version_info::get() << std::endl;
std::cout << std::scientific << std::setprecision(8) << std::showpos;

Figure out where to run the code

if (argc == 2 && (std::string(argv[1]) == "--help")) {
std::cerr << "Usage: " << argv[0] << " [executor]" << std::endl;
std::exit(-1);
}
const auto executor_string = argc >= 2 ? argv[1] : "reference";
std::map<std::string, std::function<std::shared_ptr<gko::Executor>()>>
exec_map{
{"omp", [] { return gko::OmpExecutor::create(); }},
{"cuda",
[] {
return gko::CudaExecutor::create(0,
gko::OmpExecutor::create());
}},
{"hip",
[] {
return gko::HipExecutor::create(0, gko::OmpExecutor::create());
}},
{"dpcpp",
[] {
return gko::DpcppExecutor::create(0,
gko::OmpExecutor::create());
}},
{"reference", [] { return gko::ReferenceExecutor::create(); }}};

executor where Ginkgo will perform the computation

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

linear system solver parameters

auto system_max_iterations = 100u;
auto system_residual_goal = real_precision{1e-16};

eigensolver parameters

auto max_iterations = 20u;
auto residual_goal = real_precision{1e-8};
auto z = precision{20.0, 2.0};

Read data

auto A = share(gko::read<mtx>(std::ifstream("data/A.mtx"), exec));

Generate shifted matrix A - zI

  • we avoid duplicating memory by not storing both A and A - zI, but compute A - zI on the fly by using Ginkgo's utilities for creating linear combinations of operators
auto one = share(gko::initialize<vec>({precision{1.0}}, exec));
auto neg_one = share(gko::initialize<vec>({-precision{1.0}}, exec));
auto neg_z = gko::initialize<vec>({-z}, exec);
auto system_matrix = share(gko::Combination<precision>::create(
one, A, gko::initialize<vec>({-z}, exec),
gko::matrix::Identity<precision>::create(exec, A->get_size()[0])));

Generate solver operator (A - zI)^-1

auto solver =
solver_type::build()
.with_criteria(gko::stop::Iteration::build().with_max_iters(
system_max_iterations),
gko::stop::ResidualNorm<precision>::build()
.with_reduction_factor(system_residual_goal))
.on(exec)
->generate(system_matrix);

inverse iterations

start with guess [1, 1, ..., 1]

auto x = [&] {
auto work = vec::create(this_exec, gko::dim<2>{A->get_size()[0], 1});
const auto n = work->get_size()[0];
for (int i = 0; i < n; ++i) {
work->get_values()[i] = precision{1.0} / sqrt(n);
}
return clone(exec, work);
}();
auto y = clone(x);
auto tmp = clone(x);
auto norm = gko::initialize<real_vec>({1.0}, exec);
auto inv_norm = clone(this_exec, one);
auto g = clone(one);
for (auto i = 0u; i < max_iterations; ++i) {
std::cout << "{ ";

(A - zI)y = x

solver->apply(x, y);
system_matrix->apply(one, y, neg_one, x);
x->compute_norm2(norm);
std::cout << "\"system_residual\": "
<< clone(this_exec, norm)->get_values()[0] << ", ";
x->copy_from(y);

x = y / || y ||

x->compute_norm2(norm);
inv_norm->get_values()[0] =
real_precision{1.0} / clone(this_exec, norm)->get_values()[0];
x->scale(clone(exec, inv_norm));

g = x^* A x

A->apply(x, tmp);
x->compute_dot(tmp, g);
auto g_val = clone(this_exec, g)->get_values()[0];
std::cout << "\"eigenvalue\": " << g_val << ", ";

||Ax - gx|| < tol * g

auto v = gko::initialize<vec>({-g_val}, exec);
tmp->add_scaled(v, x);
tmp->compute_norm2(norm);
auto res_val = clone(exec->get_master(), norm)->get_values()[0];
std::cout << "\"residual\": " << res_val / g_val << " }," << std::endl;
if (abs(res_val) < residual_goal * abs(g_val)) {
break;
}
}
}

Results

This is the expected output:

{ "system_residual": +1.61736920e-14, "eigenvalue": (+2.03741410e+01,-1.17744356e-16), "residual": (+2.92231055e-01,+1.68883476e-18) },
{ "system_residual": +4.98014795e-15, "eigenvalue": (+1.94878474e+01,+1.25948378e-15), "residual": (+7.94370276e-02,-5.13395071e-18) },
{ "system_residual": +3.39296916e-15, "eigenvalue": (+1.93282121e+01,-1.19329332e-15), "residual": (+4.11149623e-02,+2.53837290e-18) },
{ "system_residual": +3.35953656e-15, "eigenvalue": (+1.92638912e+01,+3.28657016e-16), "residual": (+2.34717040e-02,-4.00445585e-19) },
{ "system_residual": +2.91474009e-15, "eigenvalue": (+1.92409166e+01,+3.65597737e-16), "residual": (+1.34709547e-02,-2.55962367e-19) },
{ "system_residual": +3.09863953e-15, "eigenvalue": (+1.92331106e+01,-1.07919176e-15), "residual": (+7.72060707e-03,+4.33212063e-19) },
{ "system_residual": +2.31198069e-15, "eigenvalue": (+1.92305014e+01,-2.89755360e-16), "residual": (+4.42106625e-03,+6.66143651e-20) },
{ "system_residual": +3.02771202e-15, "eigenvalue": (+1.92296339e+01,+8.04259901e-16), "residual": (+2.53081312e-03,-1.05848687e-19) },
{ "system_residual": +2.02954523e-15, "eigenvalue": (+1.92293461e+01,+7.81834016e-16), "residual": (+1.44862114e-03,-5.88985854e-20) },
{ "system_residual": +2.31762332e-15, "eigenvalue": (+1.92292506e+01,-1.11718775e-16), "residual": (+8.29183451e-04,+4.81741912e-21) },
{ "system_residual": +8.12541038e-15, "eigenvalue": (+1.92292190e+01,-6.55606254e-16), "residual": (+4.74636702e-04,+1.61823936e-20) },
{ "system_residual": +2.77259926e-15, "eigenvalue": (+1.92292085e+01,+4.30588140e-16), "residual": (+2.71701077e-04,-6.08403935e-21) },
{ "system_residual": +8.87888675e-14, "eigenvalue": (+1.92292051e+01,+9.67936313e-18), "residual": (+1.55539937e-04,-7.82937998e-23) },
{ "system_residual": +2.85077117e-15, "eigenvalue": (+1.92292039e+01,-4.52923128e-16), "residual": (+8.90457139e-05,+2.09737561e-21) },
{ "system_residual": +6.46865302e-14, "eigenvalue": (+1.92292035e+01,+1.58710681e-17), "residual": (+5.09805252e-05,-4.20774259e-23) },
{ "system_residual": +4.18913713e-15, "eigenvalue": (+1.92292034e+01,+1.06839590e-15), "residual": (+2.91887365e-05,-1.62175862e-21) },
{ "system_residual": +1.06421578e-11, "eigenvalue": (+1.92292034e+01,+3.26089685e-17), "residual": (+1.67126561e-05,-2.83413965e-23) },
{ "system_residual": +2.97434420e-13, "eigenvalue": (+1.92292034e+01,-7.85427712e-16), "residual": (+9.56961199e-06,+3.90876227e-22) },
{ "system_residual": +1.63230281e-11, "eigenvalue": (+1.92292033e+01,+3.69307000e-16), "residual": (+5.47975753e-06,-1.05241636e-22) },
{ "system_residual": +6.14939758e-14, "eigenvalue": (+1.92292033e+01,+1.36057865e-15), "residual": (+3.13794996e-06,-2.22028320e-22) },

Comments about programming and debugging

The plain program

/*******************************<GINKGO LICENSE>******************************
Copyright (c) 2017-2023, the Ginkgo authors
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:
1. Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
3. Neither the name of the copyright holder nor the names of its
contributors may be used to endorse or promote products derived from
this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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>*******************************/
#include <ginkgo/ginkgo.hpp>
#include <cmath>
#include <complex>
#include <fstream>
#include <iomanip>
#include <iostream>
#include <map>
#include <string>
int main(int argc, char* argv[])
{
using precision = std::complex<double>;
using real_precision = gko::remove_complex<precision>;
using vec = gko::matrix::Dense<precision>;
using real_vec = gko::matrix::Dense<real_precision>;
using mtx = gko::matrix::Csr<precision>;
using solver_type = gko::solver::Bicgstab<precision>;
using std::abs;
using std::sqrt;
std::cout << gko::version_info::get() << std::endl;
std::cout << std::scientific << std::setprecision(8) << std::showpos;
if (argc == 2 && (std::string(argv[1]) == "--help")) {
std::cerr << "Usage: " << argv[0] << " [executor]" << std::endl;
std::exit(-1);
}
const auto executor_string = argc >= 2 ? argv[1] : "reference";
std::map<std::string, std::function<std::shared_ptr<gko::Executor>()>>
exec_map{
{"omp", [] { return gko::OmpExecutor::create(); }},
{"cuda",
[] {
return gko::CudaExecutor::create(0,
gko::OmpExecutor::create());
}},
{"hip",
[] {
return gko::HipExecutor::create(0, gko::OmpExecutor::create());
}},
{"dpcpp",
[] {
return gko::DpcppExecutor::create(0,
gko::OmpExecutor::create());
}},
{"reference", [] { return gko::ReferenceExecutor::create(); }}};
const auto exec = exec_map.at(executor_string)(); // throws if not valid
auto this_exec = exec->get_master();
auto system_max_iterations = 100u;
auto system_residual_goal = real_precision{1e-16};
auto max_iterations = 20u;
auto residual_goal = real_precision{1e-8};
auto z = precision{20.0, 2.0};
auto A = share(gko::read<mtx>(std::ifstream("data/A.mtx"), exec));
auto one = share(gko::initialize<vec>({precision{1.0}}, exec));
auto neg_one = share(gko::initialize<vec>({-precision{1.0}}, exec));
auto neg_z = gko::initialize<vec>({-z}, exec);
auto system_matrix = share(gko::Combination<precision>::create(
one, A, gko::initialize<vec>({-z}, exec),
gko::matrix::Identity<precision>::create(exec, A->get_size()[0])));
auto solver =
solver_type::build()
.with_criteria(gko::stop::Iteration::build().with_max_iters(
system_max_iterations),
gko::stop::ResidualNorm<precision>::build()
.with_reduction_factor(system_residual_goal))
.on(exec)
->generate(system_matrix);
auto x = [&] {
auto work = vec::create(this_exec, gko::dim<2>{A->get_size()[0], 1});
const auto n = work->get_size()[0];
for (int i = 0; i < n; ++i) {
work->get_values()[i] = precision{1.0} / sqrt(n);
}
return clone(exec, work);
}();
auto y = clone(x);
auto tmp = clone(x);
auto norm = gko::initialize<real_vec>({1.0}, exec);
auto inv_norm = clone(this_exec, one);
auto g = clone(one);
for (auto i = 0u; i < max_iterations; ++i) {
std::cout << "{ ";
solver->apply(x, y);
system_matrix->apply(one, y, neg_one, x);
x->compute_norm2(norm);
std::cout << "\"system_residual\": "
<< clone(this_exec, norm)->get_values()[0] << ", ";
x->copy_from(y);
x->compute_norm2(norm);
inv_norm->get_values()[0] =
real_precision{1.0} / clone(this_exec, norm)->get_values()[0];
x->scale(clone(exec, inv_norm));
A->apply(x, tmp);
x->compute_dot(tmp, g);
auto g_val = clone(this_exec, g)->get_values()[0];
std::cout << "\"eigenvalue\": " << g_val << ", ";
auto v = gko::initialize<vec>({-g_val}, exec);
tmp->add_scaled(v, x);
tmp->compute_norm2(norm);
auto res_val = clone(exec->get_master(), norm)->get_values()[0];
std::cout << "\"residual\": " << res_val / g_val << " }," << std::endl;
if (abs(res_val) < residual_goal * abs(g_val)) {
break;
}
}
}
gko::matrix::Csr
CSR is a matrix format which stores only the nonzero coefficients by compressing each row of the matr...
Definition: matrix.hpp:54
gko::Combination
The Combination class can be used to construct a linear combination of multiple linear operators c1 *...
Definition: combination.hpp:59
gko::log::profile_event_category::solver
Solver events.
gko::matrix::Dense
Dense is a matrix format which explicitly stores all values of the matrix.
Definition: dense_cache.hpp:48
gko::solver::Bicgstab
BiCGSTAB or the Bi-Conjugate Gradient-Stabilized is a Krylov subspace solver.
Definition: bicgstab.hpp:78
gko::abs
constexpr xstd::enable_if_t<!is_complex_s< T >::value, T > abs(const T &x)
Returns the absolute value of the object.
Definition: math.hpp:1104
gko::clone
detail::cloned_type< Pointer > clone(const Pointer &p)
Creates a unique clone of the object pointed to by p.
Definition: utils_helper.hpp:203
gko::HipExecutor::create
static std::shared_ptr< HipExecutor > create(int device_id, std::shared_ptr< Executor > master, bool device_reset, allocation_mode alloc_mode=default_hip_alloc_mode, CUstream_st *stream=nullptr)
Creates a new HipExecutor.
gko::version_info::get
static const version_info & get()
Returns an instance of version_info.
Definition: version.hpp:168
gko::stop::ResidualNorm
The ResidualNorm class is a stopping criterion which stops the iteration process when the actual resi...
Definition: residual_norm.hpp:138
gko::dim< 2 >
gko::share
detail::shared_type< OwningPointer > share(OwningPointer &&p)
Marks the object pointed to by p as shared.
Definition: utils_helper.hpp:254
gko::CudaExecutor::create
static std::shared_ptr< CudaExecutor > create(int device_id, std::shared_ptr< Executor > master, bool device_reset, allocation_mode alloc_mode=default_cuda_alloc_mode, CUstream_st *stream=nullptr)
Creates a new CudaExecutor.
gko::OmpExecutor::create
static std::shared_ptr< OmpExecutor > create(std::shared_ptr< CpuAllocatorBase > alloc=std::make_shared< CpuAllocator >())
Creates a new OmpExecutor.
Definition: executor.hpp:1373
gko::matrix::Identity
This class is a utility which efficiently implements the identity matrix (a linear operator which map...
Definition: identity.hpp:63
gko::remove_complex
typename detail::remove_complex_s< T >::type remove_complex
Obtain the type which removed the complex of complex/scalar type or the template parameter of class b...
Definition: math.hpp:354
gko::DpcppExecutor::create
static std::shared_ptr< DpcppExecutor > create(int device_id, std::shared_ptr< Executor > master, std::string device_type="all", dpcpp_queue_property property=dpcpp_queue_property::in_order)
Creates a new DpcppExecutor.
gko::one
constexpr T one()
Returns the multiplicative identity for T.
Definition: math.hpp:803
Generated by   doxygen 1.8.16