function plot_exp_mvprod()
    n_sample = 10;
    depths = [2, 5, 8];
    vareps = [1e-14, 1e-12, 1e-10, 1e-08, 1e-06, 1e-04, 1e-02];
    lineWidth = 2;
    markerSize = 15;
    
    n_d = size(depths, 2);
    n_eps = size(vareps, 2);
    
    %% kernel matrix 1
    fontSize = 19;
    load("results/prod1_err_back1.mat");
    load("results/prod1_err_back2.mat");
    load("results/prod1_ref_err_back.mat");
    load("results/prod2_bound.mat");
    
    %% plot 1
    figure() 
    set(gcf, 'Position',  [10 10 800 650])
    semilogy(1:n_eps, ref_err_back_list(:, 1)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 1)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 1)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 1)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=2');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-22, 1e-0+0.5]);
    yticks([1e-22, 1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');_
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel1_d2.pdf')
    hold off
    
    %% plot 2
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 2)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 2)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 2)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 2)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=5');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-22, 1e-0+0.5]);
    yticks([1e-22, 1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel1_d5.pdf')
    hold off
    
    %% plot 3
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 3)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 3)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 3)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 3)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=8');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-22, 1e-0+0.5]);
    yticks([1e-22, 1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel1_d8.pdf')
    hold off
    
    
    
    %% kernel matrix 2
    fontSize = 19;
    load("results/prod2_err_back1.mat");
    load("results/prod2_err_back2.mat");
    load("results/prod2_ref_err_back.mat");
    load("results/prod2_bound.mat");
    
    %% plot 1
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 1)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 1)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 1)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 1)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=2');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-18, 1e-0+0.5]);
    yticks([1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel2_d2.pdf')
    hold off
    
    %% plot 2
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 2)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 2)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 2)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 2)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=5');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-18, 1e-0+0.5]);
    yticks([1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel2_d5.pdf')
    hold off
    
    %% plot 3
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 3)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 3)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 3)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 3)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=8');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-18, 1e-0+0.5]);
    yticks([1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel2_d8.pdf')
    hold off
    
    
    
    
    %% kernel matrix 3
    fontSize = 19;
    load("results/prod3_err_back1.mat");
    load("results/prod3_err_back2.mat");
    load("results/prod3_ref_err_back.mat");
    load("results/prod3_bound.mat");
    
    
    %% plot 1
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 1)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 1)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 1)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 1)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=2');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-18, 1e-0+0.5]);
    yticks([1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel3_d2.pdf')
    hold off
    
    %% plot 2
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 2)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 2)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 2)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 2)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=5');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-18, 1e-0+0.5]);
    yticks([1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel3_d5.pdf')
    hold off
    
    %% plot 3
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 3)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 3)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 3)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 3)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=8');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-18, 1e-0+0.5]);
    yticks([1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel3_d8.pdf')
    hold off
    
    
    
    %% kernel matrix 4
    fontSize = 19;
    load("results/prod4_err_back1.mat");
    load("results/prod4_err_back2.mat");
    load("results/prod4_ref_err_back.mat");
    load("results/prod4_bound.mat");
    
    
    
    %% plot 1
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 1)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 1)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 1)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 1)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=2');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-18, 1e-0+0.5]);
    yticks([1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel4_d2.pdf')
    hold off
    
    %% plot 2
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 2)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 2)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 2)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 2)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=5');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-18, 1e-0+0.5]);
    yticks([1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel4_d5.pdf')
    hold off
    
    %% plot 3
    figure()
    set(gcf, 'Position',  [10 10 800 650])
    
    semilogy(1:n_eps, ref_err_back_list(:, 3)./n_sample,'-b', 'Marker', 'o', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list1(:, 3)./n_sample,'-.r', 'Marker', '*', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_back_list2(:, 3)./n_sample,'--g', 'Marker', 's', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    semilogy(1:n_eps, err_bound_list(:, 3)./n_sample,':k', 'Marker', '.', 'MarkerSize',markerSize, 'Linewidth',lineWidth)
    hold on 
    
    h = legend('fp64', ...
               'fp32', ...
               'bf16', ...
               'error bound', ...
         'NumColumns',4, 'Location', 'Best', 'FontSize', fontSize, BackgroundAlpha=.3);
    legend boxoff
    rect = [0.5, 0, 0, 0.05];
    set(h, 'Position', rect);
    
    %[l, s] = title('$\ell$=8');
    xticklabels(split(num2str(vareps,'%.e ')));
    ylim([1e-18, 1e-0+0.5]);
    yticks([1e-18, 1e-12, 1e-6, 1e-0]);
    
    %lx = xlabel('$\varepsilon$');
    %set(lx,'interpreter','latex');
    %lx.FontSize = fontSize;
    
    a = get(gca,'XTickLabel');  
    set(gca,'XTickLabel',a,'fontsize',fontSize) % ,'FontWeight','bold'
    % set(l,'interpreter','latex');
    % l.FontSize = fontSize+12;
    grid on;
    exportgraphics(gca, 'figures/matvecprod_kernel4_d8.pdf')
    hold off

end