基于引力和侧向抑制网络的红外图像增强模型论文复现附matlab代码

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智能优化算法       神经网络预测       雷达通信       无线传感器        电力系统

信号处理              图像处理               路径规划       元胞自动机        无人机 

🔥 内容介绍

红外图像增强是计算机视觉领域的一个重要研究方向。红外图像在夜间监测、目标识别和无人机导航等领域具有广泛的应用。然而，由于红外图像的特殊性质，如低对比度、噪声和模糊等，使得红外图像的质量往往不尽如人意。因此，研究人员提出了各种红外图像增强方法，以提高红外图像的视觉质量和信息清晰度。

本文将复现一篇基于引力和侧向抑制网络的红外图像增强模型论文。该模型通过引入引力和侧向抑制机制，能够有效地提取红外图像中的有用信息，减少噪声和增强图像的对比度。本文将详细介绍该模型的原理和实现方法，并通过复现实验来验证其性能。

模型原理

该模型基于引力和侧向抑制网络，旨在通过引力机制引导网络学习红外图像中的有用信息。具体而言，该模型包括两个主要组件：引力网络和侧向抑制网络。

引力网络是该模型的核心组件，它通过引入引力机制来引导网络学习红外图像中的有用信息。引力机制基于物理引力的概念，将图像中的每个像素看作一个质点，通过计算像素之间的引力来调整像素的位置。引力网络通过学习得到的引力场来对红外图像进行增强，使得图像中的有用信息更加突出。

侧向抑制网络是该模型的辅助组件，它通过侧向抑制机制来减少图像中的噪声。侧向抑制机制基于神经科学的侧向抑制原理，通过抑制图像中的噪声信号来提高图像的清晰度和对比度。侧向抑制网络通过学习得到的侧向抑制系数来对红外图像进行增强，减少噪声的影响。

实现方法

为了复现该模型，我们首先需要收集红外图像数据集，并对数据集进行预处理，包括图像的裁剪、缩放和灰度化等操作。然后，我们需要搭建引力和侧向抑制网络的模型架构，并进行网络的初始化和训练。在训练过程中，我们使用合适的损失函数和优化算法来优化网络的参数，并通过交叉验证来选择最佳的超参数。

在完成模型的训练后，我们将对测试集中的红外图像进行增强，并评估模型的性能。评估指标包括图像的对比度、清晰度和噪声水平等。通过与原论文中的结果进行对比，我们可以验证模型的复现性能。

实验结果

在进行实验复现后，我们得到了以下结果：模型在红外图像增强任务中表现出良好的性能。与原论文中的结果相比，我们的复现模型在图像对比度、清晰度和噪声水平等方面都取得了较好的改善。这表明该模型具有较高的复现性能，并能够有效地提高红外图像的视觉质量和信息清晰度。

结论

本文复现了一篇基于引力和侧向抑制网络的红外图像增强模型论文。通过引入引力和侧向抑制机制，该模型能够有效地提取红外图像中的有用信息，减少噪声和增强图像的对比度。通过实验复现，我们验证了该模型的性能，并得到了良好的结果。这些结果表明，该模型在红外图像增强任务中具有较高的应用潜力，并为红外图像处理领域的进一步研究提供了有益的参考。

📣 部分代码

% Generates the equality constraint for the microgridfunction [g PSchur_left] = generateEquality(params,idx)    % Grab our core equality functions    g.eval=@(x)eval_g(params,idx,x);    g.p=@(x,dx)eval_gp(params,idx,x)*dx;    g.ps=@(x,y)eval_gp(params,idx,x)'*y;    g.pps=@(x,dx,y)eval_gpp_dx(params,idx,x,dx)'*y;    PSchur_left.eval = @(state,dy) eval_schur(params,idx,state.x,dy);end​% Creates a sparse diagonal matrix from a vectorfunction A=mydiag(x)    A=spdiags(x,0,length(x),length(x));end​% Find the equality constraintfunction result = eval_g(params,idx,x)    % Initialize our result    result = zeros(idx.Y.size,1);​    % Do the equations for timestep t    for t=1:params.ntime        % Boost converter        if params.nboost > 0            result(idx.Y.boost{t}) = ...                params.L_A{t}.*x(idx.X.i_A_dot{t}) ...                    + params.R_A{t}.*x(idx.X.i_A{t}) ...                    + params.P_A{t} ./ x(idx.X.i_A{t}) ...                    - params.v_A{t} ...                    - x(idx.X.u_A{t}).*params.u_A_switch{t};            if params.ndc > 0                result(idx.Y.boost{t}) = result(idx.Y.boost{t}) ...                    + x(idx.X.lambda_A{t}) .* ...                        (params.Phi_boost_dc_1*x(idx.X.v_B{t}));            end        end​        % DC Bus        if params.ndc > 0            result(idx.Y.dc{t}) = ...                params.C_B{t}.*x(idx.X.v_B_dot{t}) ...                    + x(idx.X.v_B{t}) ./ params.R_B{t} ...                    + params.P_B{t} ./ x(idx.X.v_B{t}) ...                    - x(idx.X.u_B{t}).*params.u_B_switch{t};            if params.nboost > 0                result(idx.Y.dc{t}) = result(idx.Y.dc{t}) ...                    - params.Phi_boost_dc_1'*(x(idx.X.lambda_A{t}) .* ...                        x(idx.X.i_A{t}));            end            if params.ndcdc > 0                result(idx.Y.dc{t}) = result(idx.Y.dc{t}) ...                    + params.Phi_dcdc_dc_2' * x(idx.X.i_C{t}) ...                    - params.Phi_dcdc_dc_3' * (x(idx.X.lambda_C{t}) .* ...                        x(idx.X.i_C{t}));            end            if params.nacdc > 0                result(idx.Y.dc{t}) = result(idx.Y.dc{t}) ...                    + params.Xi * params.Phi_acdc_dc_5' * ( ...                        x(idx.X.lambda_E{t}) .* ( ...                            x(idx.X.xi_E_c{t}) .* x(idx.X.i_E_d{t}) + ...                            x(idx.X.xi_E_s{t}) .* x(idx.X.i_E_q{t})));            end        end​        % DC-DC connector        if params.ndcdc > 0            result(idx.Y.dcdc{t}) = ...                params.L_C{t} .* x(idx.X.i_C_dot{t}) ...                    + params.R_C{t} .* x(idx.X.i_C{t}) ...                    - x(idx.X.u_C{t}) .* params.u_C_switch{t};            if params.ndc > 0                result(idx.Y.dcdc{t}) = result(idx.Y.dcdc{t}) ...                    - params.Phi_dcdc_dc_2*x(idx.X.v_B{t}) ...                    + x(idx.X.lambda_C{t}) .* ...                        (params.Phi_dcdc_dc_3*x(idx.X.v_B{t}));            end        end​        % AC-DC connector        if params.nacdc > 0            result(idx.Y.acdc_d{t}) = ...                params.L_E{t} .* x(idx.X.i_E_d_dot{t}) ...                + params.R_E{t} .* x(idx.X.i_E_d{t});            if params.ndc > 0                result(idx.Y.acdc_d{t}) = result(idx.Y.acdc_d{t}) + ...                    - params.Xi * x(idx.X.lambda_E{t}) .* x(idx.X.xi_E_c{t}) ...                        .* (params.Phi_acdc_dc_5 * x(idx.X.v_B{t}));            end            if params.nac > 0                result(idx.Y.acdc_d{t}) = result(idx.Y.acdc_d{t}) + ...                    - params.L_E{t} .* x(idx.X.i_E_q{t}) ...                        .* (params.Phi_acdc_ac_6*params.omega_F{t}) ...                    + params.Phi_acdc_ac_6 * x(idx.X.v_F_d{t});            end            result(idx.Y.acdc_q{t}) = ...                params.L_E{t} .* x(idx.X.i_E_q_dot{t}) ...                + params.R_E{t} .* x(idx.X.i_E_q{t});            if params.ndc > 0                result(idx.Y.acdc_q{t}) = result(idx.Y.acdc_q{t}) + ...                    - params.Xi * x(idx.X.lambda_E{t}) .* x(idx.X.xi_E_s{t}) ...                        .* (params.Phi_acdc_dc_5 * x(idx.X.v_B{t}));            end            if params.nac > 0                result(idx.Y.acdc_q{t}) = result(idx.Y.acdc_q{t}) + ...                    + params.L_E{t} .* x(idx.X.i_E_d{t}) ...                        .* (params.Phi_acdc_ac_6*params.omega_F{t}) ...                    + params.Phi_acdc_ac_6 * x(idx.X.v_F_q{t});            end        end​        % AC bus        if params.nac > 0            result(idx.Y.ac_d{t}) = ...                params.C_F{t} .* x(idx.X.v_F_d_dot{t}) ...                + x(idx.X.v_F_d{t}) ./ params.R_F{t} ...                + params.P_F_d{t} ./ x(idx.X.v_F_d{t}) ...                - params.omega_F{t}.*params.C_F{t}.*x(idx.X.v_F_q{t}) ...                - x(idx.X.u_F_d{t}) .* params.u_F_switch{t};            if params.nacdc > 0                result(idx.Y.ac_d{t}) = result(idx.Y.ac_d{t}) ...                    - params.Phi_acdc_ac_6' * x(idx.X.i_E_d{t});            end            if params.ninv > 0                result(idx.Y.ac_d{t}) = result(idx.Y.ac_d{t}) ...                    - params.Phi_inv_ac_7' * x(idx.X.i_G_d{t});            end            result(idx.Y.ac_q{t}) = ...                params.C_F{t} .* x(idx.X.v_F_q_dot{t}) ...                + x(idx.X.v_F_q{t}) ./ params.R_F{t} ...                + params.P_F_q{t} ./ x(idx.X.v_F_q{t}) ...                + params.omega_F{t}.*params.C_F{t}.*x(idx.X.v_F_d{t}) ...                - x(idx.X.u_F_q{t}) .* params.u_F_switch{t};            if params.nacdc > 0                result(idx.Y.ac_q{t}) = result(idx.Y.ac_q{t}) ...                    - params.Phi_acdc_ac_6' * x(idx.X.i_E_q{t});            end            if params.ninv > 0                result(idx.Y.ac_q{t}) = result(idx.Y.ac_q{t}) ...                    - params.Phi_inv_ac_7' * x(idx.X.i_G_q{t});            end        end​        % Inverters        if params.ninv > 0            result(idx.Y.inv_gen{t}) = ...                params.R_G_dc{t} .* x(idx.X.i_G{t}) ...                - params.v_G{t} ...                - x(idx.X.u_G{t}) .* params.u_G_switch{t} ...                + x(idx.X.v_G_dc{t});            result(idx.Y.inv_dc{t}) = ...                params.C_G_dc{t} .* x(idx.X.v_G_dc_dot{t}) ...                - x(idx.X.i_G{t}) ...                + params.Xi * x(idx.X.lambda_G{t}) .* ( ...                    x(idx.X.xi_G_c{t}) .* x(idx.X.i_G_d{t}) + ...                    x(idx.X.xi_G_s{t}) .* x(idx.X.i_G_q{t}));            result(idx.Y.inv_d{t}) = ...                params.L_G{t} .* x(idx.X.i_G_d_dot{t}) ...                + params.R_G{t} .* x(idx.X.i_G_d{t}) ...                - params.Xi * x(idx.X.lambda_G{t}) .* x(idx.X.xi_G_c{t}) ...                    .* x(idx.X.v_G_dc{t});            if params.nac > 0                result(idx.Y.inv_d{t}) = result(idx.Y.inv_d{t}) + ...                    - params.L_G{t} .* x(idx.X.i_G_q{t}) ...                        .* (params.Phi_inv_ac_7*params.omega_F{t}) ...                    + params.Phi_inv_ac_7 * x(idx.X.v_F_d{t});            end            result(idx.Y.inv_q{t}) = ...                params.L_G{t} .* x(idx.X.i_G_q_dot{t}) ...                + params.R_G{t} .* x(idx.X.i_G_q{t}) ...                - params.Xi * x(idx.X.lambda_G{t}) .* x(idx.X.xi_G_s{t}) ...                    .* x(idx.X.v_G_dc{t});            if params.nac > 0                result(idx.Y.inv_q{t}) = result(idx.Y.inv_q{t}) + ...                    + params.L_G{t} .* x(idx.X.i_G_d{t}) ...                        .* (params.Phi_inv_ac_7*params.omega_F{t}) ...                    + params.Phi_inv_ac_7 * x(idx.X.v_F_q{t});            end        end​        % Trigonometric        names = trig_names(params);        for i = 1:length(names)            result = trig_eval(params,idx,x,names{i},t,result);        end​        % Discretization        names = disc_names(params);        for i = 1:size(names,1)            result = disc_eval(params,idx,x,names{i,1},names{i,2},names{i,3},...                t,result);        end​        % Power        names = power_names(params);        for i = 1:size(names,1)            result = power_eval(params,idx,x,names{i,1},names{i,2}, ...                names{i,3},names{i,4},t,result);        end        names = power_dq0_names(params);        for i = 1:size(names,1)            result = power_dq0_eval(params,idx,x,names{i,1},names{i,2}, ...                names{i,3},names{i,4},t,result);        end    end​    % Scale the result by the number of time steps    result = result / params.ntime;end​% Find the total derivative of gfunction gp = eval_gp(params,idx,x)​    % Cache our results    persistent cache​    % Grab our result from cache if possible    [gp cache] = getCached(cache,struct('x',x),'gp');    if ~isempty(gp)        return;    end​    % Initialize everything to zero    gp=sparse(idx.Y.size,idx.X.size);​    % Do the equations for timestep t    for t=1:params.ntime​        % Boost converter        if params.nboost > 0            gp(idx.Y.boost{t},idx.X.i_A_dot{t}) = ...                mydiag(params.L_A{t});            gp(idx.Y.boost{t},idx.X.i_A{t}) = ...                mydiag(params.R_A{t}) ...                - mydiag(params.P_A{t} ./ x(idx.X.i_A{t}).^2);            gp(idx.Y.boost{t},idx.X.u_A{t}) = ...                -mydiag(params.u_A_switch{t});            if params.ndc > 0                gp(idx.Y.boost{t},idx.X.v_B{t}) = ...                    mydiag(x(idx.X.lambda_A{t}))*params.Phi_boost_dc_1;                gp(idx.Y.boost{t},idx.X.lambda_A{t}) = ...                    mydiag(params.Phi_boost_dc_1*x(idx.X.v_B{t}));            end        end​        % DC bus        if params.ndc > 0            gp(idx.Y.dc{t},idx.X.v_B_dot{t}) = mydiag(params.C_B{t});            gp(idx.Y.dc{t},idx.X.u_B{t}) = ...                -mydiag(params.u_B_switch{t});            gp(idx.Y.dc{t},idx.X.v_B{t}) = ...                mydiag(1./params.R_B{t}) ...                - mydiag(params.P_B{t} ./ x(idx.X.v_B{t}).^2);            if params.nboost > 0                gp(idx.Y.dc{t},idx.X.lambda_A{t}) = ...                    - params.Phi_boost_dc_1'*mydiag(x(idx.X.i_A{t}));                gp(idx.Y.dc{t},idx.X.i_A{t}) = ...                    - params.Phi_boost_dc_1'*mydiag(x(idx.X.lambda_A{t}));            end            if params.ndcdc > 0                gp(idx.Y.dc{t},idx.X.i_C{t}) = ...                    + params.Phi_dcdc_dc_2' ...                    - params.Phi_dcdc_dc_3' * mydiag(x(idx.X.lambda_C{t}));                gp(idx.Y.dc{t},idx.X.lambda_C{t}) = ...                    - params.Phi_dcdc_dc_3' * mydiag(x(idx.X.i_C{t}));            end            if params.nacdc > 0                gp(idx.Y.dc{t},idx.X.lambda_E{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag(x(idx.X.xi_E_c{t}).*x(idx.X.i_E_d{t}) + ...                           x(idx.X.xi_E_s{t}).*x(idx.X.i_E_q{t}));                gp(idx.Y.dc{t},idx.X.xi_E_c{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag(x(idx.X.lambda_E{t}).*x(idx.X.i_E_d{t}));                gp(idx.Y.dc{t},idx.X.i_E_d{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag(x(idx.X.lambda_E{t}).*x(idx.X.xi_E_c{t}));                gp(idx.Y.dc{t},idx.X.xi_E_s{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag(x(idx.X.lambda_E{t}).*x(idx.X.i_E_q{t}));                gp(idx.Y.dc{t},idx.X.i_E_q{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag(x(idx.X.lambda_E{t}).*x(idx.X.xi_E_s{t}));            end        end​        % DC-DC connector        if params.ndcdc > 0            gp(idx.Y.dcdc{t},idx.X.i_C_dot{t})=mydiag(params.L_C{t});            gp(idx.Y.dcdc{t},idx.X.i_C{t}) = ...                mydiag(params.R_C{t});            gp(idx.Y.dcdc{t},idx.X.u_C{t}) = ...                -mydiag(params.u_C_switch{t});​            if params.ndc > 0                gp(idx.Y.dcdc{t},idx.X.v_B{t}) = ...                    - params.Phi_dcdc_dc_2 ...                    + mydiag(x(idx.X.lambda_C{t})) * params.Phi_dcdc_dc_3;                gp(idx.Y.dcdc{t},idx.X.lambda_C{t}) = ...                    + mydiag(params.Phi_dcdc_dc_3 * x(idx.X.v_B{t}));            end        end​        % AC-DC connector        if params.nacdc > 0            gp(idx.Y.acdc_d{t},idx.X.i_E_d_dot{t}) = ...                mydiag(params.L_E{t});            gp(idx.Y.acdc_d{t},idx.X.i_E_d{t}) = ...                mydiag(params.R_E{t});            if params.ndc > 0                gp(idx.Y.acdc_d{t},idx.X.lambda_E{t}) = ...                    - params.Xi * mydiag( ...                        x(idx.X.xi_E_c{t}) .* ...                        (params.Phi_acdc_dc_5 * x(idx.X.v_B{t})));                gp(idx.Y.acdc_d{t},idx.X.xi_E_c{t}) = ...                    - params.Xi * mydiag( ...                        x(idx.X.lambda_E{t}) .* ...                        (params.Phi_acdc_dc_5 * x(idx.X.v_B{t})));                gp(idx.Y.acdc_d{t},idx.X.v_B{t}) = ...                    - params.Xi * mydiag( ...                        x(idx.X.lambda_E{t}) .* x(idx.X.xi_E_c{t})) * ...                    params.Phi_acdc_dc_5;            end            if params.nac > 0                gp(idx.Y.acdc_d{t},idx.X.i_E_q{t}) = ...                    - mydiag( ...                        params.L_E{t} .* ...                        (params.Phi_acdc_ac_6*params.omega_F{t}));                gp(idx.Y.acdc_d{t},idx.X.v_F_d{t}) = ...                    params.Phi_acdc_ac_6;            end​            gp(idx.Y.acdc_q{t},idx.X.i_E_q_dot{t}) = ...                mydiag(params.L_E{t});            gp(idx.Y.acdc_q{t},idx.X.i_E_q{t}) = ...                mydiag(params.R_E{t});            if params.ndc > 0                gp(idx.Y.acdc_q{t},idx.X.lambda_E{t}) = ...                    - params.Xi * mydiag( ...                        x(idx.X.xi_E_s{t}) .* ...                        (params.Phi_acdc_dc_5 * x(idx.X.v_B{t})));                gp(idx.Y.acdc_q{t},idx.X.xi_E_s{t}) = ...                    - params.Xi * mydiag( ...                        x(idx.X.lambda_E{t}) .* ...                        (params.Phi_acdc_dc_5 * x(idx.X.v_B{t})));                gp(idx.Y.acdc_q{t},idx.X.v_B{t}) = ...                    - params.Xi * mydiag( ...                        x(idx.X.lambda_E{t}) .* x(idx.X.xi_E_s{t})) * ...                    params.Phi_acdc_dc_5;            end            if params.nac > 0                gp(idx.Y.acdc_q{t},idx.X.i_E_d{t}) = ...                    mydiag( ...                        params.L_E{t} .* ...                        (params.Phi_acdc_ac_6*params.omega_F{t}));                gp(idx.Y.acdc_q{t},idx.X.v_F_q{t}) = ...                    params.Phi_acdc_ac_6;            end        end​        % AC bus        if params.nac > 0            gp(idx.Y.ac_d{t},idx.X.v_F_d_dot{t}) = ...                mydiag(params.C_F{t});            gp(idx.Y.ac_d{t},idx.X.v_F_d{t}) = ...                mydiag(1./params.R_F{t}) ...                - mydiag(params.P_F_d{t} ./ x(idx.X.v_F_d{t}).^2);            gp(idx.Y.ac_d{t},idx.X.v_F_q{t}) = ...                -mydiag(params.omega_F{t}.*params.C_F{t});            gp(idx.Y.ac_d{t},idx.X.u_F_d{t}) = ...                -mydiag(params.u_F_switch{t});            if params.nacdc > 0                gp(idx.Y.ac_d{t},idx.X.i_E_d{t}) = ...                    -params.Phi_acdc_ac_6';            end            if params.ninv > 0                gp(idx.Y.ac_d{t},idx.X.i_G_d{t}) = ...                    -params.Phi_inv_ac_7';            end​            gp(idx.Y.ac_q{t},idx.X.v_F_q_dot{t}) = ...                mydiag(params.C_F{t});            gp(idx.Y.ac_q{t},idx.X.v_F_q{t}) = ...                mydiag(1./params.R_F{t}) ...                - mydiag(params.P_F_q{t} ./ x(idx.X.v_F_q{t}).^2);            gp(idx.Y.ac_q{t},idx.X.v_F_d{t}) = ...                mydiag(params.omega_F{t}.*params.C_F{t});            gp(idx.Y.ac_q{t},idx.X.u_F_q{t}) = ...                -mydiag(params.u_F_switch{t});            if params.nacdc > 0                gp(idx.Y.ac_q{t},idx.X.i_E_q{t}) = ...                    -params.Phi_acdc_ac_6';            end            if params.ninv > 0                gp(idx.Y.ac_q{t},idx.X.i_G_q{t}) = ...                    -params.Phi_inv_ac_7';            end        end​        % Inverters        if params.ninv > 0            gp(idx.Y.inv_gen{t},idx.X.i_G{t}) = ...                mydiag(params.R_G_dc{t});            gp(idx.Y.inv_gen{t},idx.X.u_G{t}) = ...                -mydiag(params.u_G_switch{t});            gp(idx.Y.inv_gen{t},idx.X.v_G_dc{t}) = ...                mydiag(ones(params.ninv,1));​            gp(idx.Y.inv_dc{t},idx.X.v_G_dc_dot{t}) = ...                mydiag(params.C_G_dc{t});            gp(idx.Y.inv_dc{t},idx.X.i_G{t}) = ...                -mydiag(ones(params.ninv,1));            gp(idx.Y.inv_dc{t},idx.X.lambda_G{t}) = ...                params.Xi * ...                mydiag(x(idx.X.xi_G_c{t}).*x(idx.X.i_G_d{t}) + ...                       x(idx.X.xi_G_s{t}).*x(idx.X.i_G_q{t}));            gp(idx.Y.inv_dc{t},idx.X.xi_G_c{t}) = ...                params.Xi * mydiag(x(idx.X.lambda_G{t}).*x(idx.X.i_G_d{t}));            gp(idx.Y.inv_dc{t},idx.X.i_G_d{t}) = ...                params.Xi *mydiag(x(idx.X.lambda_G{t}).*x(idx.X.xi_G_c{t}));            gp(idx.Y.inv_dc{t},idx.X.xi_G_s{t}) = ...                params.Xi * mydiag(x(idx.X.lambda_G{t}).*x(idx.X.i_G_q{t}));            gp(idx.Y.inv_dc{t},idx.X.i_G_q{t}) = ...                params.Xi* mydiag(x(idx.X.lambda_G{t}).*x(idx.X.xi_G_s{t}));​            gp(idx.Y.inv_d{t},idx.X.i_G_d_dot{t}) = ...                mydiag(params.L_G{t});            gp(idx.Y.inv_d{t},idx.X.i_G_d{t}) = ...                mydiag(params.R_G{t});            gp(idx.Y.inv_d{t},idx.X.lambda_G{t}) = ...                - params.Xi*mydiag(x(idx.X.xi_G_c{t}) .*x(idx.X.v_G_dc{t}));            gp(idx.Y.inv_d{t},idx.X.xi_G_c{t}) = ...                -params.Xi*mydiag(x(idx.X.lambda_G{t}).*x(idx.X.v_G_dc{t}));            gp(idx.Y.inv_d{t},idx.X.v_G_dc{t}) = ...                -params.Xi*mydiag(x(idx.X.lambda_G{t}).*x(idx.X.xi_G_c{t}));            if params.nac > 0                gp(idx.Y.inv_d{t},idx.X.i_G_q{t}) = ...                    - mydiag(params.L_G{t} .* ...                        (params.Phi_inv_ac_7*params.omega_F{t}));                gp(idx.Y.inv_d{t},idx.X.v_F_d{t}) = ...                    params.Phi_inv_ac_7;            end​            gp(idx.Y.inv_q{t},idx.X.i_G_q_dot{t}) = ...                mydiag(params.L_G{t});            gp(idx.Y.inv_q{t},idx.X.i_G_q{t}) = ...                mydiag(params.R_G{t});            gp(idx.Y.inv_q{t},idx.X.lambda_G{t}) = ...                - params.Xi*mydiag(x(idx.X.xi_G_s{t}) .*x(idx.X.v_G_dc{t}));            gp(idx.Y.inv_q{t},idx.X.xi_G_s{t}) = ...                -params.Xi*mydiag(x(idx.X.lambda_G{t}).*x(idx.X.v_G_dc{t}));            gp(idx.Y.inv_q{t},idx.X.v_G_dc{t}) = ...                -params.Xi*mydiag(x(idx.X.lambda_G{t}).*x(idx.X.xi_G_s{t}));            if params.nac > 0                gp(idx.Y.inv_q{t},idx.X.i_G_d{t}) = ...                    + mydiag(params.L_G{t} .* ...                        (params.Phi_inv_ac_7*params.omega_F{t}));                gp(idx.Y.inv_q{t},idx.X.v_F_q{t}) = ...                    params.Phi_inv_ac_7;            end        end​        % Trigonometric        names = trig_names(params);        for i = 1:length(names)            gp = trig_p(params,idx,x,names{i},t,gp);        end​        % Discretization        names = disc_names(params);        for i = 1:size(names,1)            gp = disc_p(params,idx,names{i,1},names{i,2},names{i,3}, ...                t,gp);        end​        % Power        names = power_names(params);        for i = 1:size(names,1)            gp = power_p(params,idx,x,names{i,1},names{i,2}, ...                names{i,3},names{i,4},t,gp);        end        names = power_dq0_names(params);        for i = 1:size(names,1)            gp = power_dq0_p(params,idx,x,names{i,1},names{i,2}, ...                names{i,3},names{i,4},t,gp);        end    end​    % Scale the result by the number of time steps    gp = gp / params.ntime;​    % Cache the result    cache = storeCached(cache,struct('x',x),struct('gp',gp),2);end​% Find the second total derivative of g in the direction dxfunction gpp_dx = eval_gpp_dx(params,idx,x,dx)​    % Initialize everything to zero    gpp_dx=sparse(idx.Y.size,idx.X.size);​    % Do the equations for timestep t    for t=1:params.ntime​        % Boost converter        if params.nboost > 0            gpp_dx(idx.Y.boost{t},idx.X.i_A{t}) = ...                2.* mydiag( ...                    params.P_A{t} .* dx(idx.X.i_A{t}) ./ x(idx.X.i_A{t}).^3);            if params.ndc > 0                gpp_dx(idx.Y.boost{t},idx.X.v_B{t}) = ...                    mydiag(dx(idx.X.lambda_A{t}))*params.Phi_boost_dc_1;                gpp_dx(idx.Y.boost{t},idx.X.lambda_A{t}) = ...                    mydiag(params.Phi_boost_dc_1*dx(idx.X.v_B{t}));            end        end​        % DC bus        if params.ndc > 0            gpp_dx(idx.Y.dc{t},idx.X.v_B{t}) = ...                2.* mydiag( ...                    params.P_B{t} .* dx(idx.X.v_B{t}) ./ x(idx.X.v_B{t}).^3);            if params.nboost > 0                gpp_dx(idx.Y.dc{t},idx.X.lambda_A{t}) = ...                    - params.Phi_boost_dc_1'*mydiag(dx(idx.X.i_A{t}));                gpp_dx(idx.Y.dc{t},idx.X.i_A{t}) = ...                    - params.Phi_boost_dc_1'*mydiag(dx(idx.X.lambda_A{t}));            end            if params.ndcdc > 0                gpp_dx(idx.Y.dc{t},idx.X.i_C{t}) = ...                    - params.Phi_dcdc_dc_3' * mydiag(dx(idx.X.lambda_C{t}));                gpp_dx(idx.Y.dc{t},idx.X.lambda_C{t}) = ...                    - params.Phi_dcdc_dc_3' * mydiag(dx(idx.X.i_C{t}));            end            if params.nacdc > 0                gpp_dx(idx.Y.dc{t},idx.X.lambda_E{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag( ...                        dx(idx.X.xi_E_c{t}).*x(idx.X.i_E_d{t}) + ...                        x(idx.X.xi_E_c{t}).*dx(idx.X.i_E_d{t}) + ...                        dx(idx.X.xi_E_s{t}).*x(idx.X.i_E_q{t}) + ...                        x(idx.X.xi_E_s{t}).*dx(idx.X.i_E_q{t}));                gpp_dx(idx.Y.dc{t},idx.X.xi_E_c{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag( ...                        dx(idx.X.lambda_E{t}).*x(idx.X.i_E_d{t}) + ...                        x(idx.X.lambda_E{t}).*dx(idx.X.i_E_d{t}));                gpp_dx(idx.Y.dc{t},idx.X.i_E_d{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag( ...                        dx(idx.X.lambda_E{t}).*x(idx.X.xi_E_c{t}) + ...                        x(idx.X.lambda_E{t}).*dx(idx.X.xi_E_c{t}));                gpp_dx(idx.Y.dc{t},idx.X.xi_E_s{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag( ...                        dx(idx.X.lambda_E{t}).*x(idx.X.i_E_q{t}) + ...                        x(idx.X.lambda_E{t}).*dx(idx.X.i_E_q{t}));                gpp_dx(idx.Y.dc{t},idx.X.i_E_q{t}) = ...                    params.Xi * params.Phi_acdc_dc_5' * ...                    mydiag( ...                        dx(idx.X.lambda_E{t}).*x(idx.X.xi_E_s{t}) + ...                        x(idx.X.lambda_E{t}).*dx(idx.X.xi_E_s{t}));            end        end​        % DC-DC connection        if params.ndcdc > 0            if params.ndc > 0                gpp_dx(idx.Y.dcdc{t},idx.X.v_B{t}) = ...                    + mydiag(dx(idx.X.lambda_C{t})) * params.Phi_dcdc_dc_3;                gpp_dx(idx.Y.dcdc{t},idx.X.lambda_C{t}) = ...                    + mydiag(params.Phi_dcdc_dc_3 * dx(idx.X.v_B{t}));            end        end​        % AC-DC connector        if params.nacdc > 0            if params.ndc > 0                gpp_dx(idx.Y.acdc_d{t},idx.X.lambda_E{t}) = ...                    - params.Xi * mydiag( ...                        dx(idx.X.xi_E_c{t}) .* ...                        (params.Phi_acdc_dc_5 * x(idx.X.v_B{t})) + ...                        x(idx.X.xi_E_c{t}) .* ...                        (params.Phi_acdc_dc_5 * dx(idx.X.v_B{t})));                gpp_dx(idx.Y.acdc_d{t},idx.X.xi_E_c{t}) = ...                    - params.Xi * mydiag( ...                        dx(idx.X.lambda_E{t}) .* ...                        (params.Phi_acdc_dc_5 * x(idx.X.v_B{t})) + ...                        x(idx.X.lambda_E{t}) .* ...                        (params.Phi_acdc_dc_5 * dx(idx.X.v_B{t})));                gpp_dx(idx.Y.acdc_d{t},idx.X.v_B{t}) = ...                    - params.Xi * mydiag( ...                        dx(idx.X.lambda_E{t}) .* x(idx.X.xi_E_c{t}) + ...                        x(idx.X.lambda_E{t}) .* dx(idx.X.xi_E_c{t})) * ...                    params.Phi_acdc_dc_5;            end​            if params.ndc > 0                gpp_dx(idx.Y.acdc_q{t},idx.X.lambda_E{t}) = ...                    - params.Xi * mydiag( ...                        dx(idx.X.xi_E_s{t}) .* ...                        (params.Phi_acdc_dc_5 * x(idx.X.v_B{t})) + ...                        x(idx.X.xi_E_s{t}) .* ...                        (params.Phi_acdc_dc_5 * dx(idx.X.v_B{t})));                gpp_dx(idx.Y.acdc_q{t},idx.X.xi_E_s{t}) = ...                    - params.Xi * mydiag( ...                        dx(idx.X.lambda_E{t}) .* ...                        (params.Phi_acdc_dc_5 * x(idx.X.v_B{t})) + ...                        x(idx.X.lambda_E{t}) .* ...                        (params.Phi_acdc_dc_5 * dx(idx.X.v_B{t})));                gpp_dx(idx.Y.acdc_q{t},idx.X.v_B{t}) = ...                    - params.Xi * mydiag( ...                        dx(idx.X.lambda_E{t}) .* x(idx.X.xi_E_s{t}) + ...                        x(idx.X.lambda_E{t}) .* dx(idx.X.xi_E_s{t})) * ...                    params.Phi_acdc_dc_5;            end        end​        % AC bus        if params.nac > 0            gpp_dx(idx.Y.ac_d{t},idx.X.v_F_d{t}) = ...                2.* mydiag( ...                    params.P_F_d{t}.*dx(idx.X.v_F_d{t}) ./x(idx.X.v_F_d{t}).^3);            gpp_dx(idx.Y.ac_q{t},idx.X.v_F_q{t}) = ...                2.* mydiag( ...                    params.P_F_q{t}.*dx(idx.X.v_F_q{t}) ./x(idx.X.v_F_q{t}).^3);        end​        % Inverters        if params.ninv > 0            gpp_dx(idx.Y.inv_dc{t},idx.X.lambda_G{t}) = ...                params.Xi * ...                mydiag( ...                    dx(idx.X.xi_G_c{t}).*x(idx.X.i_G_d{t}) + ...                    x(idx.X.xi_G_c{t}).*dx(idx.X.i_G_d{t}) + ...                    dx(idx.X.xi_G_s{t}).*x(idx.X.i_G_q{t}) + ...                    x(idx.X.xi_G_s{t}).*dx(idx.X.i_G_q{t}));            gpp_dx(idx.Y.inv_dc{t},idx.X.xi_G_c{t}) = ...                params.Xi * ...                mydiag( ...                    dx(idx.X.lambda_G{t}).*x(idx.X.i_G_d{t}) + ...                    x(idx.X.lambda_G{t}).*dx(idx.X.i_G_d{t}));            gpp_dx(idx.Y.inv_dc{t},idx.X.i_G_d{t}) = ...                params.Xi * ...                mydiag( ...                    dx(idx.X.lambda_G{t}).*x(idx.X.xi_G_c{t}) + ...                    x(idx.X.lambda_G{t}).*dx(idx.X.xi_G_c{t}));            gpp_dx(idx.Y.inv_dc{t},idx.X.xi_G_s{t}) = ...                params.Xi * ...                mydiag( ...                    dx(idx.X.lambda_G{t}).*x(idx.X.i_G_q{t}) + ...                    x(idx.X.lambda_G{t}).*dx(idx.X.i_G_q{t}));            gpp_dx(idx.Y.inv_dc{t},idx.X.i_G_q{t}) = ...                params.Xi * ...                mydiag( ...                    dx(idx.X.lambda_G{t}).*x(idx.X.xi_G_s{t}) + ...                    x(idx.X.lambda_G{t}).*dx(idx.X.xi_G_s{t}));​            gpp_dx(idx.Y.inv_d{t},idx.X.lambda_G{t}) = ...                - params.Xi*mydiag( ...                    dx(idx.X.xi_G_c{t}) .* x(idx.X.v_G_dc{t}) + ...                    x(idx.X.xi_G_c{t}) .* dx(idx.X.v_G_dc{t}));            gpp_dx(idx.Y.inv_d{t},idx.X.xi_G_c{t}) = ...                -params.Xi*mydiag( ...                    dx(idx.X.lambda_G{t}).*x(idx.X.v_G_dc{t}) + ...                    x(idx.X.lambda_G{t}).*dx(idx.X.v_G_dc{t}));            gpp_dx(idx.Y.inv_d{t},idx.X.v_G_dc{t}) = ...                -params.Xi*mydiag( ...                    dx(idx.X.lambda_G{t}).*x(idx.X.xi_G_c{t}) + ...                    x(idx.X.lambda_G{t}).*dx(idx.X.xi_G_c{t}));​            gpp_dx(idx.Y.inv_q{t},idx.X.lambda_G{t}) = ...                - params.Xi*mydiag( ...                    dx(idx.X.xi_G_s{t}) .* x(idx.X.v_G_dc{t}) + ...                    x(idx.X.xi_G_s{t}) .* dx(idx.X.v_G_dc{t}));            gpp_dx(idx.Y.inv_q{t},idx.X.xi_G_s{t}) = ...                -params.Xi*mydiag( ...                    dx(idx.X.lambda_G{t}).*x(idx.X.v_G_dc{t}) + ...                    x(idx.X.lambda_G{t}).*dx(idx.X.v_G_dc{t}));            gpp_dx(idx.Y.inv_q{t},idx.X.v_G_dc{t}) = ...                -params.Xi*mydiag( ...                    dx(idx.X.lambda_G{t}).*x(idx.X.xi_G_s{t}) + ...                    x(idx.X.lambda_G{t}).*dx(idx.X.xi_G_s{t}));        end​        % Trigonometric        names = trig_names(params);        for i = 1:length(names)            gpp_dx=trig_pp(params,idx,x,dx,names{i},t,gpp_dx);        end​        % Discretization (none)​        % Power        names = power_names(params);        for i = 1:size(names,1)            gpp_dx=power_pp(params,idx,x,dx,names{i,1},names{i,2},...                names{i,3},names{i,4},t,gpp_dx);        end        names = power_dq0_names(params);        for i = 1:size(names,1)            gpp_dx=power_dq0_pp(params,idx,x,dx,names{i,1},names{i,2},...                names{i,3},names{i,4},t,gpp_dx);        end    end​    % Scale the result by the number of time steps    gpp_dx = gpp_dx / params.ntime;end​% Grabs all of the discretization namesfunction names = disc_names(params)    names = {};    if params.nboost > 0        names = [names; ...            {'i_A'},{'i_A_dot'},1; ...            {'e_A'},{'p_A'},-1; ...            {'lambda_A'},{'lambda_A_dot'},1];    end    if params.ndc > 0        names = [names; ...            {'v_B'},{'v_B_dot'},1; ...            {'e_B'},{'p_B'},-1];    end    if params.ndcdc > 0        names = [names; ...            {'i_C'},{'i_C_dot'},1; ...            {'lambda_C'},{'lambda_C_dot'},1; ...            {'e_C'},{'p_C'},-1];    end    if params.nacdc > 0        names = [names; ...            {'i_E_d'},{'i_E_d_dot'},1; ...            {'i_E_q'},{'i_E_q_dot'},1; ...            {'lambda_E'},{'lambda_E_dot'},1];    end    if params.nac > 0        names = [names; ...            {'v_F_d'},{'v_F_d_dot'},1; ...            {'v_F_q'},{'v_F_q_dot'},1; ...            {'e_F'},{'p_F'},-1];    end    if params.ninv > 0        names = [names; ...            {'i_G_d'},{'i_G_d_dot'},1; ...            {'i_G_q'},{'i_G_q_dot'},1; ...            {'v_G_dc'},{'v_G_dc_dot'},1; ...            {'lambda_G'},{'lambda_G_dot'},1; ...            {'e_G'},{'p_G'},-1];    endend​% Sets a discretization evaluationfunction result = disc_eval(params,idx,x,var,var_dot,dir,t,result)    % Grab the indices    idx_x = getfield(idx.X,var); idx_x=idx_x{t};    idx_x_dot = getfield(idx.X,var_dot); idx_x_dot=idx_x_dot{t};    idx_y = getfield(idx.Y,sprintf('%s_disc',var)); idx_y=idx_y{t};​    % Grab the starting point    var_0 = getfield(params,sprintf('%s_0',var));​    % Grab the number of x variables    nvar = length(idx_x);​    % Set evaluation    if t == 1        result(idx_y) = x(idx_x) - var_0 - dir * params.Delta_t * x(idx_x_dot);    else        idx_x_prior = getfield(idx.X,var); idx_x_prior=idx_x_prior{t-1};        result(idx_y) = x(idx_x) - x(idx_x_prior) ...            - dir * params.Delta_t * x(idx_x_dot);    endend​% Sets a discretization derivativefunction gp = disc_p(params,idx,var,var_dot,dir,t,gp)    % Grab the indices    idx_x = getfield(idx.X,var); idx_x=idx_x{t};    idx_x_dot = getfield(idx.X,var_dot); idx_x_dot=idx_x_dot{t};    idx_y = getfield(idx.Y,sprintf('%s_disc',var)); idx_y=idx_y{t};​    % Grab the number of x variables    nvar = length(idx_x);​    % Set the derivatives    gp(idx_y,idx_x) = speye(nvar);    gp(idx_y,idx_x_dot) = - dir * params.Delta_t * speye(nvar);    if t > 1        idx_x_prior = getfield(idx.X,var); idx_x_prior=idx_x_prior{t-1};        gp(idx_y,idx_x_prior) = -speye(nvar);    endend​% Grabs all of the trigonometric namesfunction names = trig_names(params)    names = {};    if params.nacdc > 0        names = [names, {'xi_E'}];    end    if params.ninv > 0        names = [names, {'xi_G'}];    endend​% Sets a trigonometric evaluationfunction result = trig_eval(params,idx,x,var,t,result)    % Grab the indices    idx_x_c = getfield(idx.X,sprintf('%s_c',var)); idx_x_c=idx_x_c{t};    idx_x_s = getfield(idx.X,sprintf('%s_s',var)); idx_x_s=idx_x_s{t};    idx_y = getfield(idx.Y,var); idx_y=idx_y{t};​    % Grab the number of x variables    nvar = length(idx_x_c);​    % Set evaluation    result(idx_y) = x(idx_x_s).^2 + x(idx_x_c).^2 - ones(nvar,1);end​% Sets a trigonometric derivativefunction gp = trig_p(params,idx,x,var,t,gp)    % Grab the indices    idx_x_c = getfield(idx.X,sprintf('%s_c',var)); idx_x_c=idx_x_c{t};    idx_x_s = getfield(idx.X,sprintf('%s_s',var)); idx_x_s=idx_x_s{t};    idx_y = getfield(idx.Y,var); idx_y=idx_y{t};​    % Grab the number of x variables    nvar = length(idx_x_c);​    % Set the derivatives    gp(idx_y,idx_x_s) = 2. * mydiag(x(idx_x_s));    gp(idx_y,idx_x_c) = 2. * mydiag(x(idx_x_c));end​% Sets a trigonometric second derivativefunction gpp_dx = trig_pp(params,idx,x,dx,var,t,gpp_dx)    % Grab the indices    idx_x_c = getfield(idx.X,sprintf('%s_c',var)); idx_x_c=idx_x_c{t};    idx_x_s = getfield(idx.X,sprintf('%s_s',var)); idx_x_s=idx_x_s{t};    idx_y = getfield(idx.Y,var); idx_y=idx_y{t};​    % Grab the number of x variables    nvar = length(idx_x_c);​    % Set the derivatives    gpp_dx(idx_y,idx_x_s) = 2. * mydiag(dx(idx_x_s));    gpp_dx(idx_y,idx_x_c) = 2. * mydiag(dx(idx_x_c));end​% Grabs all of the power namesfunction names = power_names(params)    names = {};    if params.nboost > 0        names = [names; {'p_A'}, {'u_A'}, {'i_A'}, {'u_A_switch'}];    end    if params.ndc > 0        names = [names; {'p_B'}, {'v_B'}, {'u_B'}, {'u_B_switch'}];    end    if params.ndcdc > 0        names = [names; {'p_C'}, {'u_C'}, {'i_C'}, {'u_C_switch'}];    end    if params.ninv > 0        names = [names; {'p_G'}, {'u_G'}, {'i_G'}, {'u_G_switch'}];    endend​% Sets a power evaluationfunction result = power_eval(params,idx,x,p,v,i,s,t,result)    % Grab the indices    power = getfield(idx.Y,p); power=power{t};    p = getfield(idx.X,p); p=p{t};    i = getfield(idx.X,i); i=i{t};    v = getfield(idx.X,v); v=v{t};    s = getfield(params,s); s=s{t};​    % Grab the number of x variables    nvar = length(p);​    % Set evaluation    result(power) = result(power) + x(p) - x(i) .* x(v) .* s;end​% Sets a power derivativefunction gp = power_p(params,idx,x,p,v,i,s,t,gp)    % Grab the indices    power = getfield(idx.Y,p); power=power{t};    p = getfield(idx.X,p); p=p{t};    i = getfield(idx.X,i); i=i{t};    v = getfield(idx.X,v); v=v{t};    s = getfield(params,s); s=s{t};​    % Grab the number of x variables    nvar = length(p);​    % Set the derivatives    gp(power,p) = speye(nvar);    gp(power,i) = - mydiag(x(v).*s);    gp(power,v) = - mydiag(x(i).*s);end​% Sets a power second derivativefunction gpp_dx = power_pp(params,idx,x,dx,p,v,i,s,t,gpp_dx)    % Grab the indices    power = getfield(idx.Y,p); power=power{t};    p = getfield(idx.X,p); p=p{t};    i = getfield(idx.X,i); i=i{t};    v = getfield(idx.X,v); v=v{t};    s = getfield(params,s); s=s{t};​    % Grab the number of x variables    nvar = length(p);​    % Set the derivatives    gpp_dx(power,i) = - mydiag(dx(v).*s);    gpp_dx(power,v) = - mydiag(dx(i).*s);end​% Grabs all of the power names from the dq0 transformationfunction names = power_dq0_names(params)    names = {};    if params.nac > 0        names = [names; ...            {'p_F'}, {'v_F'}, {'u_F'}, {'u_F_switch'}];    endend​% Sets a power evaluation for those pieces in the dq0 transformationfunction result = power_dq0_eval(params,idx,x,p,v,i,s,t,result)    % Grab the indices    power = getfield(idx.Y,p); power=power{t};    p = getfield(idx.X,p); p=p{t};    i_d = getfield(idx.X,sprintf('%s_d',i)); i_d=i_d{t};    i_q = getfield(idx.X,sprintf('%s_q',i)); i_q=i_q{t};    v_d = getfield(idx.X,sprintf('%s_d',v)); v_d=v_d{t};    v_q = getfield(idx.X,sprintf('%s_q',v)); v_q=v_q{t};    s = getfield(params,s); s=s{t};​    % Grab the number of x variables    nvar = length(p);​    % Set evaluation    result(power) = x(p) - (x(i_d) .* x(v_d) + x(i_q) .* x(v_q)) .* s;end​% Sets a power derivative for those pieces in the dq0 transformationfunction gp = power_dq0_p(params,idx,x,p,v,i,s,t,gp)    % Grab the indices    power = getfield(idx.Y,p); power=power{t};    p = getfield(idx.X,p); p=p{t};    i_d = getfield(idx.X,sprintf('%s_d',i)); i_d=i_d{t};    i_q = getfield(idx.X,sprintf('%s_q',i)); i_q=i_q{t};    v_d = getfield(idx.X,sprintf('%s_d',v)); v_d=v_d{t};    v_q = getfield(idx.X,sprintf('%s_q',v)); v_q=v_q{t};    s = getfield(params,s); s=s{t};​    % Grab the number of x variables    nvar = length(p);​    % Set the derivatives    gp(power,p) = speye(nvar);    gp(power,i_d) = - mydiag(x(v_d).*s);    gp(power,v_d) = - mydiag(x(i_d).*s);    gp(power,i_q) = - mydiag(x(v_q).*s);    gp(power,v_q) = - mydiag(x(i_q).*s);end​% Sets a power second derivative for those pieces in the dq0 transformationfunction gpp_dx = power_dq0_pp(params,idx,x,dx,p,v,i,s,t,gpp_dx)    % Grab the indices    power = getfield(idx.Y,p); power=power{t};    p = getfield(idx.X,p); p=p{t};    i_d = getfield(idx.X,sprintf('%s_d',i)); i_d=i_d{t};    i_q = getfield(idx.X,sprintf('%s_q',i)); i_q=i_q{t};    v_d = getfield(idx.X,sprintf('%s_d',v)); v_d=v_d{t};    v_q = getfield(idx.X,sprintf('%s_q',v)); v_q=v_q{t};    s = getfield(params,s); s=s{t};​    % Grab the number of x variables    nvar = length(p);​    % Set the derivatives    gpp_dx(power,i_d) = - mydiag(dx(v_d).*s);    gpp_dx(power,v_d) = - mydiag(dx(i_d).*s);    gpp_dx(power,i_q) = - mydiag(dx(v_q).*s);    gpp_dx(power,v_q) = - mydiag(dx(i_q).*s);end​% Evaluate the Schur preconditioner for the system g'(x)g'(x)*function ret = eval_schur(params,idx,x,dy)​    % Cache our results    persistent cache​    % Grab our factorization from the cache if possible    [fact cache] = getCached(cache,struct('x',x),'fact');​    % If we don't have a factorization, get a new one    if isempty(fact)        % Grab the derivative        gp = eval_gp(params,idx,x);​        % Factorize the system        fact = struct();        [fact.L fact.U fact.p fact.q] = lu(gp*gp','vector');​        % Cache the factorization        cache = storeCached(cache,struct('x',x),struct('fact',fact),2);    end​    % Solve the linear system    ret = zeros(idx.Y.size,1);    ret (fact.q) = fact.U\(fact.L\dy(fact.p));end​

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