The inverse-iteration program

The inverse iteration example.

This example depends on simple-solver, .

Table of contents
Introduction About the example The commented program	Results Comments about programming and debugging 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 <string>


int main(int argc, char *argv[])
{

Some shortcuts

using precision = std::complex<double>;
using real_precision = double;
using vec = gko::matrix::Dense<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

std::shared_ptr<gko::Executor> exec;
if (argc == 1 || std::string(argv[1]) == "reference") {
    exec = gko::ReferenceExecutor::create();
} else if (argc == 2 && std::string(argv[1]) == "omp") {
    exec = gko::OmpExecutor::create();
} else if (argc == 2 && std::string(argv[1]) == "cuda" &&
           gko::CudaExecutor::get_num_devices() > 0) {
    exec = gko::CudaExecutor::create(0, gko::OmpExecutor::create());
} else {
    std::cerr << "Usage: " << argv[0] << " [executor]" << std::endl;
    std::exit(-1);
}

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)
                           .on(exec),
                       gko::stop::ResidualNormReduction<precision>::build()
                           .with_reduction_factor(system_residual_goal)
                           .on(exec))
        .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 = clone(one);
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(lend(x), lend(y));
system_matrix->apply(lend(one), lend(y), lend(neg_one), lend(x));
x->compute_norm2(lend(norm));
std::cout << "\"system_residual\": "
          << clone(this_exec, norm)->get_values()[0] << ", ";
x->copy_from(lend(y));

x = y / || y ||

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

g = x^* A x

A->apply(lend(x), lend(tmp));
x->compute_dot(lend(tmp), lend(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(lend(v), lend(x));
        tmp->compute_norm2(lend(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": (+1.59066966e-14,+0.00000000e+00), "eigenvalue": (+2.03741410e+01,-5.42101086e-18), "residual": (+2.92231055e-01,+7.77548230e-20) },
{ "system_residual": (+6.38877157e-15,+0.00000000e+00), "eigenvalue": (+1.94878474e+01,-4.34534678e-16), "residual": (+7.94370276e-02,+1.77126506e-18) },
{ "system_residual": (+6.79215294e-15,+0.00000000e+00), "eigenvalue": (+1.93282121e+01,-3.68988781e-16), "residual": (+4.11149623e-02,+7.84912734e-19) },
{ "system_residual": (+3.54015578e-15,+0.00000000e+00), "eigenvalue": (+1.92638912e+01,+2.03949917e-16), "residual": (+2.34717040e-02,-2.48498708e-19) },
{ "system_residual": (+2.12400044e-15,+0.00000000e+00), "eigenvalue": (+1.92409166e+01,-7.59991100e-16), "residual": (+1.34709547e-02,+5.32085134e-19) },
{ "system_residual": (+3.29202859e-15,+0.00000000e+00), "eigenvalue": (+1.92331106e+01,+2.90110055e-15), "residual": (+7.72060707e-03,-1.16456760e-18) },
{ "system_residual": (+3.99088304e-15,+0.00000000e+00), "eigenvalue": (+1.92305014e+01,-3.21058733e-16), "residual": (+4.42106625e-03,+7.38109682e-20) },
{ "system_residual": (+2.02648035e-15,+0.00000000e+00), "eigenvalue": (+1.92296339e+01,+5.11222288e-16), "residual": (+2.53081312e-03,-6.72819919e-20) },
{ "system_residual": (+1.83840397e-15,+0.00000000e+00), "eigenvalue": (+1.92293461e+01,+3.51208924e-16), "residual": (+1.44862114e-03,-2.64579289e-20) },
{ "system_residual": (+1.60253167e-15,+0.00000000e+00), "eigenvalue": (+1.92292506e+01,-2.02284978e-15), "residual": (+8.29183451e-04,+8.72271932e-20) },
{ "system_residual": (+1.96758490e-15,+0.00000000e+00), "eigenvalue": (+1.92292190e+01,+8.90545453e-16), "residual": (+4.74636702e-04,-2.19814209e-20) },
{ "system_residual": (+1.53327380e-14,+0.00000000e+00), "eigenvalue": (+1.92292085e+01,-8.25871947e-17), "residual": (+2.71701077e-04,+1.16692425e-21) },
{ "system_residual": (+3.42985865e-15,+0.00000000e+00), "eigenvalue": (+1.92292051e+01,+1.63122796e-16), "residual": (+1.55539937e-04,-1.31945701e-21) },
{ "system_residual": (+3.30861071e-11,+0.00000000e+00), "eigenvalue": (+1.92292039e+01,-5.49102025e-16), "residual": (+8.90457139e-05,+2.54275643e-21) },
{ "system_residual": (+7.11155374e-14,+0.00000000e+00), "eigenvalue": (+1.92292035e+01,+1.16689376e-15), "residual": (+5.09805252e-05,-3.09367244e-21) },
{ "system_residual": (+2.68204494e-15,+0.00000000e+00), "eigenvalue": (+1.92292034e+01,-4.07084034e-17), "residual": (+2.91887365e-05,+6.17928281e-23) },
{ "system_residual": (+5.78377594e-13,+0.00000000e+00), "eigenvalue": (+1.92292034e+01,-3.38561848e-17), "residual": (+1.67126561e-05,+2.94253882e-23) },
{ "system_residual": (+6.26422040e-12,+0.00000000e+00), "eigenvalue": (+1.92292034e+01,-3.14429218e-18), "residual": (+9.56961199e-06,+1.56478953e-24) },
{ "system_residual": (+1.41104829e-12,+0.00000000e+00), "eigenvalue": (+1.92292033e+01,-6.54656730e-16), "residual": (+5.47975753e-06,+1.86557918e-22) },
{ "system_residual": (+1.97926842e-10,+0.00000000e+00), "eigenvalue": (+1.92292033e+01,+1.58008702e-16), "residual": (+3.13794996e-06,-2.57849164e-23) },

Comments about programming and debugging

The plain program

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#include <ginkgo/ginkgo.hpp>


#include <cmath>
#include <complex>
#include <fstream>
#include <iomanip>
#include <iostream>
#include <string>


int main(int argc, char *argv[])
{
    using precision = std::complex<double>;
    using real_precision = double;
    using vec = gko::matrix::Dense<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;

    std::shared_ptr<gko::Executor> exec;
    if (argc == 1 || std::string(argv[1]) == "reference") {
        exec = gko::ReferenceExecutor::create();
    } else if (argc == 2 && std::string(argv[1]) == "omp") {
        exec = gko::OmpExecutor::create();
    } else if (argc == 2 && std::string(argv[1]) == "cuda" &&
               gko::CudaExecutor::get_num_devices() > 0) {
        exec = gko::CudaExecutor::create(0, gko::OmpExecutor::create());
    } else {
        std::cerr << "Usage: " << argv[0] << " [executor]" << std::endl;
        std::exit(-1);
    }

    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)
                               .on(exec),
                           gko::stop::ResidualNormReduction<precision>::build()
                               .with_reduction_factor(system_residual_goal)
                               .on(exec))
            .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 = clone(one);
    auto inv_norm = clone(this_exec, one);
    auto g = clone(one);

    for (auto i = 0u; i < max_iterations; ++i) {
        std::cout << "{ ";
        solver->apply(lend(x), lend(y));
        system_matrix->apply(lend(one), lend(y), lend(neg_one), lend(x));
        x->compute_norm2(lend(norm));
        std::cout << "\"system_residual\": "
                  << clone(this_exec, norm)->get_values()[0] << ", ";
        x->copy_from(lend(y));
        x->compute_norm2(lend(norm));
        inv_norm->get_values()[0] =
            precision{1.0} / clone(this_exec, norm)->get_values()[0];
        x->scale(lend(clone(exec, inv_norm)));
        A->apply(lend(x), lend(tmp));
        x->compute_dot(lend(tmp), lend(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(lend(v), lend(x));
        tmp->compute_norm2(lend(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;
        }
    }
}