SEXP
GetMatCol(const SEXP data, const int idx)
{    
    Eigen::Ref<Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> > A = EigenXPtrToMapEigen<T>(data);
    Eigen::Matrix<T, Eigen::Dynamic, 1> Am = A.col(idx-1);
    return(wrap(Am));
}

void FunctionApproximatorGPR::train(const Eigen::Ref<const Eigen::MatrixXd>& inputs, const Eigen::Ref<const Eigen::MatrixXd>& targets)
{
  if (isTrained())  
  {
    cerr << "WARNING: You may not call FunctionApproximatorGPR::train more than once. Doing nothing." << endl;
    cerr << "   (if you really want to retrain, call reTrain function instead)" << endl;
    return;
  }
  
  assert(inputs.rows() == targets.rows());
  assert(inputs.cols()==getExpectedInputDim());

  const MetaParametersGPR* meta_parameters_gpr = 
    dynamic_cast<const MetaParametersGPR*>(getMetaParameters());
  
  double max_covar = meta_parameters_gpr->maximum_covariance();
  VectorXd sigmas = meta_parameters_gpr->sigmas();
  
  
  // Compute the gram matrix
  // In a gram matrix, every input point is itself a center
  MatrixXd centers = inputs;
  // Replicate sigmas, because they are the same for each data point/center
  MatrixXd widths = sigmas.transpose().colwise().replicate(centers.rows()); 

  MatrixXd gram(inputs.rows(),inputs.rows());
  bool normalize_activations = false;
  bool asymmetric_kernels = false;
  BasisFunction::Gaussian::activations(centers,widths,inputs,gram,normalize_activations,asymmetric_kernels);
  
  gram *= max_covar;

  setModelParameters(new ModelParametersGPR(inputs,targets,gram,max_covar,sigmas));
  
}

Eigen::ArrayXXi distmesh::utils::findUniqueEdges(Eigen::Ref<Eigen::ArrayXXi const> const triangulation) {
    // find all unique combinations
    auto const combinations = nOverK(triangulation.cols(), 2);

    // find unique edges for all combinations
    // guarantee direction of edges with lower node index to higher index
    std::set<std::array<int, 2>> uniqueEdges;
    std::array<int, 2> edge = {{0, 0}};
    for (int combination = 0; combination < combinations.rows(); ++combination)
    for (int triangle = 0; triangle < triangulation.rows(); ++triangle) {
        edge[0] = triangulation(triangle, combinations(combination, 0));
        edge[1] = triangulation(triangle, combinations(combination, 1));

        edge = edge[1] < edge[0] ? std::array<int, 2>{edge[1], edge[0]} : edge;

        uniqueEdges.insert(edge);
    }

    // copy set to eigen array
    Eigen::ArrayXXi edgeIndices(uniqueEdges.size(), 2);
    int index = 0;
    for (auto const& edge : uniqueEdges) {
        edgeIndices(index, 0) = edge[0];
        edgeIndices(index, 1) = edge[1];

        index++;
    }

    return edgeIndices;
}

void FunctionApproximatorGPR::predictVariance(const Eigen::Ref<const Eigen::MatrixXd>& inputs, MatrixXd& variances)
{
  if (!isTrained())  
  {
    cerr << "WARNING: You may not call FunctionApproximatorLWPR::predict if you have not trained yet. Doing nothing." << endl;
    return;
  }

  const ModelParametersGPR* model_parameters_gpr = static_cast<const ModelParametersGPR*>(getModelParameters());
  
  
  assert(inputs.cols()==getExpectedInputDim());
  
  unsigned int n_samples = inputs.rows();
  variances.resize(n_samples,1);
  
  MatrixXd ks;
  model_parameters_gpr->kernelActivations(inputs, ks);  

  double maximum_covariance = model_parameters_gpr->maximum_covariance();
  MatrixXd gram_inv = model_parameters_gpr->gram_inv();
  
  for (unsigned int ii=0; ii<n_samples; ii++)
    variances(ii) = maximum_covariance - (ks.row(ii)*gram_inv).dot(ks.row(ii).transpose());

}

double softmax<T>::compute_cost(const Eigen::Ref<const EigenMat> &train,
                                const Eigen::Ref<const EigenMat> &weight,
                                const Eigen::Ref<const EigenMat> &ground_truth)
{    
    compute_hypothesis(train, weight);
    double const NSamples = static_cast<double>(train.cols());
    return  -1.0 * (hypothesis_.array().log() *
                    ground_truth.array()).sum() / NSamples +
            weight.array().pow(2.0).sum() * params_.lambda_ / 2.0;
}

// create initial points distribution
Eigen::ArrayXXd distmesh::utils::createInitialPoints(
    Functional const& distanceFunction, double const initialPointDistance,
    Functional const& elementSizeFunction, Eigen::Ref<Eigen::ArrayXXd const> const boundingBox,
    Eigen::Ref<Eigen::ArrayXXd const> const fixedPoints) {
    // extract dimension of mesh
    unsigned const dimension = boundingBox.cols();

    // initially distribute points evenly in complete bounding box
    Eigen::ArrayXi pointsPerDimension(dimension);
    for (int dim = 0; dim < dimension; ++dim) {
        pointsPerDimension(dim) = ceil((boundingBox(1, dim) - boundingBox(0, dim)) /
            (initialPointDistance * (dim == 0 ? 1.0 : sqrt(3.0) / 2.0)));
    }

    Eigen::ArrayXXd points(pointsPerDimension.prod(), dimension);
    for (int point = 0; point < points.rows(); ++point)
    for (int dim = 0; dim < dimension; ++dim) {
        int const pointIndex = (point / std::max(pointsPerDimension.topRows(dim).prod(), 1)) %
            pointsPerDimension(dim);

        points(point, dim) = boundingBox(0, dim) + (double)pointIndex * initialPointDistance *
            (dim == 0 ? 1.0 : sqrt(3.0) / 2.0);

        if (dim > 0) {
            points(point, dim - 1) += pointIndex % 2 != 0 ? initialPointDistance / 2.0 : 0.0;
        }
    }

    // reject points outside of region defined by distance function
    points = selectMaskedArrayElements<double>(points,
        distanceFunction(points) < constants::geometryEvaluationThreshold * initialPointDistance);

    // clear duplicate points
    Eigen::Array<bool, Eigen::Dynamic, 1> isUniquePoint =
        Eigen::Array<bool, Eigen::Dynamic, 1>::Constant(points.rows(), true);
    for (int i = 0; i < fixedPoints.rows(); ++i)
    for (int j = 0; j < points.rows(); ++j) {
        isUniquePoint(j) &= !(fixedPoints.row(i) == points.row(j)).all();
    }
    points = selectMaskedArrayElements<double>(points, isUniquePoint);

    // calculate probability to keep points
    Eigen::ArrayXd probability = 1.0 / elementSizeFunction(points).pow(dimension);
    probability /= probability.maxCoeff();

    // reject points with wrong probability
    points = selectMaskedArrayElements<double>(points,
        0.5 * (1.0 + Eigen::ArrayXd::Random(points.rows())) < probability);

    // combine fixed and variable points to one array
    Eigen::ArrayXXd finalPoints(points.rows() + fixedPoints.rows(), dimension);
    finalPoints << fixedPoints, points;

    return finalPoints;
}

int softmax<T>::predict(Eigen::Ref<const EigenMat> const &input)
{    
    CV_Assert(input.cols() == 1);
    compute_hypothesis(input, weight_);
    probability_ = (hypothesis_ * input.transpose()).
            rowwise().sum();
    EigenMat::Index max_row = 0, max_col = 0;
    probability_.maxCoeff(&max_row, &max_col);

    return max_row;
}

// determine boundary edges of given triangulation
Eigen::ArrayXi distmesh::utils::boundEdges(
    Eigen::Ref<Eigen::ArrayXXi const> const triangulation,
    Eigen::Ref<Eigen::ArrayXXi const> const _edges,
    Eigen::Ref<Eigen::ArrayXXi const> const _edgeIndices) {
    // create a new edge list, if none was given
    Eigen::ArrayXXi edges;
    if (_edges.rows() == 0) {
        edges = utils::findUniqueEdges(triangulation);
    }
    else {
        edges = _edges;
    }

    // get edge indices for each triangle in triangulation
    Eigen::ArrayXXi edgeIndices;
    if (_edgeIndices.rows() == 0) {
        edgeIndices = utils::getTriangulationEdgeIndices(triangulation, edges);
    }
    else {
        edgeIndices = _edgeIndices;
    }

    // find edges, which only appear once in triangulation
    std::set<int> uniqueEdges;
    std::vector<int> boundaryEdges;
    for (int triangle = 0; triangle < triangulation.rows(); ++triangle)
    for (int edge = 0; edge < triangulation.cols(); ++edge) {
        auto const edgeIndex = edgeIndices(triangle, edge);

        // insert edge in set to get info about multiple appearance
        if (!std::get<1>(uniqueEdges.insert(edgeIndex))) {
            // find edge in vector and delete it
            auto const it = std::find(boundaryEdges.begin(), boundaryEdges.end(), edgeIndex);
            if (it != boundaryEdges.end()) {
                boundaryEdges.erase(it);
            }
        }
        else {
            boundaryEdges.push_back(edgeIndex);
        }
    }

    // convert stl vector to eigen array
    Eigen::ArrayXi boundary(boundaryEdges.size());
    for (int edge = 0; edge < boundary.rows(); ++edge) {
        boundary(edge) = boundaryEdges[edge];
    }
    
    return boundary;
}

//**************************************************************************************************
Eigen::MatrixRXd wholeBodyReach::pinvDampedEigen(const Eigen::Ref<Eigen::MatrixRXd> &A, double damp)
{
    // allocate memory
    int m = A.rows(), n = A.cols(), k = m<n?m:n;
    VectorXd SpinvD = VectorXd::Zero(k);
    // compute decomposition
    JacobiSVD<MatrixRXd> svd(A, ComputeThinU | ComputeThinV);    // default Eigen SVD
    VectorXd sv = svd.singularValues();
    // compute pseudoinverse of singular value matrix
    double damp2 = damp*damp;
    for (int c=0;c<k; c++)
        SpinvD(c) = sv(c) / (sv(c)*sv(c) + damp2);
    // compute damped pseudoinverse
    return svd.matrixV() * SpinvD.asDiagonal() * svd.matrixU().transpose();
}

void FunctionApproximatorGMR::predictVariance(const Eigen::Ref<const Eigen::MatrixXd>& inputs, Eigen::MatrixXd& variances)
{
  ENTERING_REAL_TIME_CRITICAL_CODE
  variances.resize(inputs.rows(),getExpectedOutputDim());
  predict(inputs,empty_prealloc_,variances);
  EXITING_REAL_TIME_CRITICAL_CODE
}

void DrawArrow(const Eigen::Ref<const Eigen::Vector3d>& pt, const Eigen::Ref<const Eigen::Vector3d>& dir, double length, double thickness, double arrowthickness) {
	Eigen::Vector3d normDir = dir.normalized();

	if(arrowthickness==-1) arrowthickness=2*thickness;
	double arrowlength = 2*arrowthickness;

	GLUquadricObj *c;
	c = gluNewQuadric();
	gluQuadricDrawStyle(c, GLU_FILL);
	gluQuadricNormals(c, GLU_SMOOTH);

	glPushMatrix();
	glTranslated(pt[0], pt[1], pt[2]);
	glRotated(acos(normDir[2])*180/M_PI, -normDir[1], normDir[0], 0);
	gluCylinder(c, thickness, thickness, length-arrowlength, 16, 16);

	// arrowhed
	glPushMatrix();
	glTranslated(0, 0, length-arrowlength);
	gluCylinder(c, arrowthickness, 0.0, arrowlength, 10, 10);
	glPopMatrix();

	glPopMatrix();

	gluDeleteQuadric(c);
}

// convert kartesian to polar coordinates
Eigen::ArrayXd mpFlow::math::polar(Eigen::Ref<Eigen::ArrayXd const> const point) {
    // calc radius
    double angle = 0.0;
    double radius = sqrt(point.square().sum());

    // calc angle
    if (point(0) > 0.0) {
        angle = atan(point(1) / point(0));
    }
    else if ((point(0) < 0.0) && (point(1) >= 0.0)) {
        angle = atan(point(1) / point(0)) + M_PI;
    }
    else if ((point(0) < 0.0) && (point(1) < 0.0)) {
        angle = atan(point(1) / point(0)) - M_PI;
    }
    else if ((point(0) == 0.0) && (point(1) > 0.0)) {
        angle = M_PI / 2.0;
    }
    else if ((point(0) == 0.0) && (point(1) < 0.0)) {
        angle = - M_PI / 2.0;
    }
    else {
        angle = 0.0;
    }

    Eigen::ArrayXd result(2);
    result << radius, angle;
    return result;
}

//**************************************************************************************************
Eigen::MatrixRXd wholeBodyReach::nullSpaceProjector(const Eigen::Ref<MatrixRXd> A, double tol)
{
    // allocate memory
    int m = A.rows(), n = A.cols(), k = m<n?m:n;
    MatrixRXd Spinv = MatrixRXd::Zero(k,k);
    // compute decomposition
    JacobiSVD<MatrixRXd> svd(A, ComputeThinU | ComputeThinV);    // default Eigen SVD
    VectorXd sv = svd.singularValues();
    // compute pseudoinverse of singular value matrix
    for (int c=0;c<k; c++)
        if ( sv(c)> tol)
            Spinv(c,c) = 1/sv(c);
    // compute pseudoinverse
    MatrixRXd N = MatrixRXd::Identity(n,n);
    N -= svd.matrixV() * Spinv  * svd.matrixU().transpose() * A;
    return N;
}

void assign (Eigen::Ref<Eigen::Vector3d> result, const geometry_msgs::Quaternion &q) {
	result << q.x, q.y, q.z;
	if (result.isMuchSmallerThan(1)) {
		result = Eigen::Vector3d::Zero();
	} else {
		double angle = 2. * acos(q.w);
		result *= angle / sin(0.5 * angle);
	}
}

void softmax<T>::train(const Eigen::Ref<const EigenMat> &train,
                       const std::vector<int> &labels)
{
#ifdef OCV_TEST_SOFTMAX
    gradient_check();
#endif

    auto const UniqueLabels = get_unique_labels(labels);
    auto const NumClass = UniqueLabels.size();
    weight_ = EigenMat::Random(NumClass, train.rows());
    grad_ = EigenMat::Zero(NumClass, train.rows());
    auto const TrainCols = static_cast<int>(train.cols());
    EigenMat const GroundTruth = get_ground_truth(static_cast<int>(NumClass),
                                                  TrainCols,
                                                  UniqueLabels,
                                                  labels);

    std::random_device rd;
    std::default_random_engine re(rd());
    int const Batch = (get_batch_size(TrainCols));
    int const RandomSize = TrainCols != Batch ?
                TrainCols - Batch - 1 : 0;
    std::uniform_int_distribution<int>
            uni_int(0, RandomSize);
    for(size_t i = 0; i != params_.max_iter_; ++i){
        auto const Cols = uni_int(re);
        auto const &TrainBlock =
                train.block(0, Cols, train.rows(), Batch);
        auto const &GTBlock =
                GroundTruth.block(0, Cols, NumClass, Batch);
        auto const Cost = compute_cost(TrainBlock, weight_, GTBlock);
        if(std::abs(params_.cost_ - Cost) < params_.epsillon_ ||
                Cost < 0){
            break;
        }
        params_.cost_ = Cost;
        compute_gradient(TrainBlock, weight_, GTBlock);
        weight_.array() -= grad_.array() * params_.lrate_;//*/
    }
}

// check whether points lies inside or outside of polygon
Eigen::ArrayXd distmesh::utils::pointsInsidePoly(
    Eigen::Ref<Eigen::ArrayXXd const> const points,
    Eigen::Ref<Eigen::ArrayXXd const> const polygon) {
    Eigen::ArrayXd inside = Eigen::ArrayXd::Zero(points.rows());

    for (int i = 0, j = polygon.rows() - 1; i < polygon.rows(); j = i++) {
        inside = (((points.col(1) < polygon(i, 1)) != (points.col(1) < polygon(j, 1))) &&
            (points.col(0) < (polygon(j, 0) - polygon(i, 0)) * (points.col(1) - polygon(i, 1)) /
            (polygon(j, 1) - polygon(i, 1)) + polygon(i, 0))).select(1.0 - inside, inside);
    }

    return inside;
}

void FunctionApproximatorGPR::predict(const Eigen::Ref<const Eigen::MatrixXd>& inputs, MatrixXd& outputs)
{
  if (!isTrained())  
  {
    cerr << "WARNING: You may not call FunctionApproximatorLWPR::predict if you have not trained yet. Doing nothing." << endl;
    return;
  }

  const ModelParametersGPR* model_parameters_gpr = static_cast<const ModelParametersGPR*>(getModelParameters());
  
  assert(inputs.cols()==getExpectedInputDim());
  unsigned int n_samples = inputs.rows();
  
  outputs.resize(n_samples,1);
  
  MatrixXd ks(n_samples,n_samples);
  model_parameters_gpr->kernelActivations(inputs, ks);
  
  
  VectorXd weights = model_parameters_gpr->weights();
  for (unsigned int ii=0; ii<n_samples; ii++)
    outputs(ii) = ks.row(ii).dot(weights);
  
}

void LinearElasticIsotropic<DisplacementDim>::computeConstitutiveRelation(
    double const t,
    ProcessLib::SpatialPosition const& x,
    double const aperture0,
    Eigen::Ref<Eigen::VectorXd const>
        sigma0,
    Eigen::Ref<Eigen::VectorXd const>
    /*w_prev*/,
    Eigen::Ref<Eigen::VectorXd const>
        w,
    Eigen::Ref<Eigen::VectorXd const>
    /*sigma_prev*/,
    Eigen::Ref<Eigen::VectorXd>
        sigma,
    Eigen::Ref<Eigen::MatrixXd>
        C,
    typename FractureModelBase<DisplacementDim>::MaterialStateVariables&
        material_state_variables)
{
    material_state_variables.reset();

    const int index_ns = DisplacementDim - 1;
    C.setZero();
    for (int i = 0; i < index_ns; i++)
        C(i, i) = _mp.shear_stiffness(t, x)[0];

    sigma.noalias() = C * w;

    double const aperture = w[index_ns] + aperture0;

    sigma.coeffRef(index_ns) =
        _mp.normal_stiffness(t, x)[0] * w[index_ns] *
        logPenalty(aperture0, aperture, _penalty_aperture_cutoff);

    C(index_ns, index_ns) =
        _mp.normal_stiffness(t, x)[0] *
        logPenaltyDerivative(aperture0, aperture, _penalty_aperture_cutoff);

    sigma.noalias() += sigma0;

    // correction for an opening fracture
    if (_tension_cutoff && sigma[index_ns] > 0)
    {
        C.setZero();
        sigma.setZero();
        material_state_variables.setTensileStress(true);
    }
}

    /** Draw shape triangles */
    void drawShapeTriangulation(cv::Mat& canvas, Eigen::Ref<RowVectorX const> shape, Eigen::Ref<RowVectorXi const> triangleIds, const cv::Scalar &color)
    {
        const auto &s = shape.cast<float>();
        
        for (int j = 0; j < triangleIds.cols() / 3; j++) {

            int id1 = triangleIds(0, j * 3 + 0);
            int id2 = triangleIds(0, j * 3 + 1);
            int id3 = triangleIds(0, j * 3 + 2);

            float x1 = s(0, id1 * 2 + 0);
            float y1 = s(0, id1 * 2 + 1);

            float x2 = s(0, id2 * 2 + 0);
            float y2 = s(0, id2 * 2 + 1);

            float x3 = s(0, id3 * 2 + 0);
            float y3 = s(0, id3 * 2 + 1);

            cv::line(canvas, cv::Point2f(x1, y1), cv::Point2f(x2, y2), color, 1, CV_AA);
            cv::line(canvas, cv::Point2f(x2, y2), cv::Point2f(x3, y3), color, 1, CV_AA);
            cv::line(canvas, cv::Point2f(x3, y3), cv::Point2f(x1, y1), color, 1, CV_AA);
        }
    }

    void jacobian(const Eigen::Ref<const Eigen::VectorXd> &x, Eigen::Ref<Eigen::MatrixXd> out) const override
    {
        const unsigned int offset = 3 * links_ * chainNum_;
        out.setZero();

        Eigen::VectorXd plus(3 * (links_ + 1));
        plus.head(3 * links_) = x.segment(offset, 3 * links_);
        plus.tail(3) = Eigen::VectorXd::Zero(3);

        Eigen::VectorXd minus(3 * (links_ + 1));
        minus.head(3) = offset_;
        minus.tail(3 * links_) = x.segment(offset, 3 * links_);

        const Eigen::VectorXd diagonal = plus - minus;

        for (unsigned int i = 0; i < links_; i++)
            out.row(i).segment(3 * i + offset, 3) = diagonal.segment(3 * i, 3).normalized();

        out.block(1, offset, links_ - 1, 3 * links_ - 3) -= out.block(1, offset + 3, links_ - 1, 3 * links_ - 3);
    }