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aka_math_tmpl.hh
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			/**
			* @file aka_math_tmpl.hh
			*
			* @author Ramin Aghababaei <ramin.aghababaei@epfl.ch>
			* @author Guillaume Anciaux <guillaume.anciaux@epfl.ch>
			* @author Alejandro M. Aragón <alejandro.aragon@epfl.ch>
			* @author David Simon Kammer <david.kammer@epfl.ch>
			* @author Daniel Pino Muñoz <daniel.pinomunoz@epfl.ch>
			* @author Mathilde Radiguet <mathilde.radiguet@epfl.ch>
			* @author Nicolas Richart <nicolas.richart@epfl.ch>
			* @author Leonardo Snozzi <leonardo.snozzi@epfl.ch>
			* @author Peter Spijker <peter.spijker@epfl.ch>
			* @author Marco Vocialta <marco.vocialta@epfl.ch>
			*
			* @date creation: Wed Aug 04 2010
			* @date last modification: Wed Oct 21 2015
			*
			* @brief Implementation of the inline functions of the math toolkit
			*
			* @section LICENSE
			*
			* Copyright (©) 2010-2012, 2014, 2015 EPFL (Ecole Polytechnique Fédérale de
			* Lausanne) Laboratory (LSMS - Laboratoire de Simulation en Mécanique des
			* Solides)
			*
			* Akantu is free software: you can redistribute it and/or modify it under the
			* terms of the GNU Lesser General Public License as published by the Free
			* Software Foundation, either version 3 of the License, or (at your option) any
			* later version.
			*
			* Akantu is distributed in the hope that it will be useful, but WITHOUT ANY
			* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
			* A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
			* details.
			*
			* You should have received a copy of the GNU Lesser General Public License
			* along with Akantu. If not, see <http://www.gnu.org/licenses/>.
			*
			*/

			} // akantu

			#include <cmath>
			#include <cstring>
			#include <typeinfo>

			#include "aka_blas_lapack.hh"

			namespace akantu {

			/* -------------------------------------------------------------------------- */
			inline void Math::matrix_vector(UInt im, UInt in, Real * A, Real * x, Real * y,
			Real alpha) {
			#ifdef AKANTU_USE_BLAS
			/// y = alphaop(A)x + beta*y
			char tran_A = 'N';
			int incx = 1;
			int incy = 1;
			double beta = 0.;
			int m = im;
			int n = in;

			aka_gemv(&tran_A, &m, &n, &alpha, A, &m, x, &incx, &beta, y, &incy);

			#else
			memset(y, 0, im * sizeof(Real));
			for (UInt i = 0; i < im; ++i) {
			for (UInt j = 0; j < in; ++j) {
			y[i] += A[i + j * im] * x[j];
			}
			y[i] *= alpha;
			}
			#endif
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::matrixt_vector(UInt im, UInt in, Real * A, Real * x, Real * y,
			Real alpha) {
			#ifdef AKANTU_USE_BLAS
			/// y = alphaop(A)x + beta*y
			char tran_A = 'T';
			int incx = 1;
			int incy = 1;
			double beta = 0.;
			int m = im;
			int n = in;

			aka_gemv(&tran_A, &m, &n, &alpha, A, &m, x, &incx, &beta, y, &incy);
			#else
			memset(y, 0, in * sizeof(Real));
			for (UInt i = 0; i < im; ++i) {
			for (UInt j = 0; j < in; ++j) {
			y[j] += A[j * im + i] * x[i];
			}
			y[i] *= alpha;
			}
			#endif
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::matrix_matrix(UInt im, UInt in, UInt ik, Real * A, Real * B,
			Real * C, Real alpha) {
			#ifdef AKANTU_USE_BLAS
			/// C := alphaop(A)op(B) + beta*C
			char trans_a = 'N';
			char trans_b = 'N';
			double beta = 0.;
			int m = im, n = in, k = ik;

			aka_gemm(&trans_a, &trans_b, &m, &n, &k, &alpha, A, &m, B, &k, &beta, C, &m);
			#else
			memset(C, 0, im * in * sizeof(Real));
			for (UInt j = 0; j < in; ++j) {
			UInt _jb = j * ik;
			UInt _jc = j * im;
			for (UInt i = 0; i < im; ++i) {
			for (UInt l = 0; l < ik; ++l) {
			UInt _la = l * im;
			C[i + _jc] += A[i + _la] * B[l + _jb];
			}
			C[i + _jc] *= alpha;
			}
			}
			#endif
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::matrixt_matrix(UInt im, UInt in, UInt ik, Real * A, Real * B,
			Real * C, Real alpha) {
			#ifdef AKANTU_USE_BLAS
			/// C := alphaop(A)op(B) + beta*C
			char trans_a = 'T';
			char trans_b = 'N';
			double beta = 0.;
			int m = im, n = in, k = ik;

			aka_gemm(&trans_a, &trans_b, &m, &n, &k, &alpha, A, &k, B, &k, &beta, C, &m);
			#else
			memset(C, 0, im * in * sizeof(Real));
			for (UInt j = 0; j < in; ++j) {
			UInt _jc = j * im;
			UInt _jb = j * ik;
			for (UInt i = 0; i < im; ++i) {
			UInt _ia = i * ik;
			for (UInt l = 0; l < ik; ++l) {
			C[i + _jc] += A[l + _ia] * B[l + _jb];
			}
			C[i + _jc] *= alpha;
			}
			}
			#endif
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::matrix_matrixt(UInt im, UInt in, UInt ik, Real * A, Real * B,
			Real * C, Real alpha) {
			#ifdef AKANTU_USE_BLAS
			/// C := alphaop(A)op(B) + beta*C
			char trans_a = 'N';
			char trans_b = 'T';
			double beta = 0.;
			int m = im, n = in, k = ik;

			aka_gemm(&trans_a, &trans_b, &m, &n, &k, &alpha, A, &m, B, &n, &beta, C, &m);
			#else
			memset(C, 0, im * in * sizeof(Real));
			for (UInt j = 0; j < in; ++j) {
			UInt _jc = j * im;
			for (UInt i = 0; i < im; ++i) {
			for (UInt l = 0; l < ik; ++l) {
			UInt _la = l * im;
			UInt _lb = l * in;
			C[i + _jc] += A[i + _la] * B[j + _lb];
			}
			C[i + _jc] *= alpha;
			}
			}
			#endif
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::matrixt_matrixt(UInt im, UInt in, UInt ik, Real * A, Real * B,
			Real * C, Real alpha) {
			#ifdef AKANTU_USE_BLAS
			/// C := alphaop(A)op(B) + beta*C
			char trans_a = 'T';
			char trans_b = 'T';
			double beta = 0.;
			int m = im, n = in, k = ik;

			aka_gemm(&trans_a, &trans_b, &m, &n, &k, &alpha, A, &k, B, &n, &beta, C, &m);
			#else
			memset(C, 0, im * in * sizeof(Real));
			for (UInt j = 0; j < in; ++j) {
			UInt _jc = j * im;
			for (UInt i = 0; i < im; ++i) {
			UInt _ia = i * ik;
			for (UInt l = 0; l < ik; ++l) {
			UInt _lb = l * in;
			C[i + _jc] += A[l + _ia] * B[j + _lb];
			}
			C[i + _jc] *= alpha;
			}
			}
			#endif
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::aXplusY(UInt n, Real alpha, Real * x, Real * y) {
			#ifdef AKANTU_USE_BLAS
			/// y := alpha x + y
			int incx = 1, incy = 1;
			aka_axpy(&n, &alpha, x, &incx, y, &incy);
			#else
			for (UInt i = 0; i < n; ++i)
			(y++) += alpha *(x++);
			#endif
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::vectorDot(Real * v1, Real * v2, UInt in) {
			#ifdef AKANTU_USE_BLAS
			/// d := v1 . v2
			int incx = 1, incy = 1, n = in;
			Real d = aka_dot(&n, v1, &incx, v2, &incy);
			#else
			Real d = 0;
			for (UInt i = 0; i < in; ++i) {
			d += v1[i] * v2[i];
			}
			#endif
			return d;
			}

			/* -------------------------------------------------------------------------- */
			template <bool tr_A, bool tr_B>
			inline void Math::matMul(UInt m, UInt n, UInt k, Real alpha, Real * A, Real * B,
			__attribute__((unused)) Real beta, Real * C) {
			if (tr_A) {
			if (tr_B)
			matrixt_matrixt(m, n, k, A, B, C, alpha);
			else
			matrixt_matrix(m, n, k, A, B, C, alpha);
			} else {
			if (tr_B)
			matrix_matrixt(m, n, k, A, B, C, alpha);
			else
			matrix_matrix(m, n, k, A, B, C, alpha);
			}
			}

			/* -------------------------------------------------------------------------- */
			template <bool tr_A>
			inline void Math::matVectMul(UInt m, UInt n, Real alpha, Real * A, Real * x,
			__attribute__((unused)) Real beta, Real * y) {
			if (tr_A) {
			matrixt_vector(m, n, A, x, y, alpha);
			} else {
			matrix_vector(m, n, A, x, y, alpha);
			}
			}

			/* -------------------------------------------------------------------------- */
			template <typename T> inline void Math::matrixEig(UInt n, T * A, T * d, T * V) {

			// Matrix A is row major, so the lapack function in fortran will process
			// A^t. Asking for the left eigenvectors of A^t will give the transposed right
			// eigenvectors of A so in the C++ code the right eigenvectors.
			char jobvl;
			if (V != nullptr)
			jobvl = 'V'; // compute left eigenvectors
			else
			jobvl = 'N'; // compute left eigenvectors

			char jobvr('N'); // compute right eigenvectors

			auto * di = new T[n]; // imaginary part of the eigenvalues

			int info;
			int N = n;

			T wkopt;
			int lwork = -1;
			// query and allocate the optimal workspace
			aka_geev<T>(&jobvl, &jobvr, &N, A, &N, d, di, V, &N, nullptr, &N, &wkopt,
			&lwork, &info);

			lwork = int(wkopt);
			auto * work = new T[lwork];
			// solve the eigenproblem
			aka_geev<T>(&jobvl, &jobvr, &N, A, &N, d, di, V, &N, nullptr, &N, work,
			&lwork, &info);

			AKANTU_DEBUG_ASSERT(
			info == 0,
			"Problem computing eigenvalues/vectors. DGEEV exited with the value "
			<< info);

			delete[] work;
			delete[] di; // I hope for you that there was no complex eigenvalues !!!
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::matrix22_eigenvalues(Real * A, Real * Adiag) {
			/// d = determinant of Matrix A
			Real d = det2(A);
			/// b = trace of Matrix A
			Real b = A[0] + A[3];

			Real c = sqrt(b * b - 4 * d);
			Adiag[0] = .5 * (b + c);
			Adiag[1] = .5 * (b - c);
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::matrix33_eigenvalues(Real * A, Real * Adiag) {
			matrixEig(3, A, Adiag);
			}

			/* -------------------------------------------------------------------------- */
			template <UInt dim> inline void Math::eigenvalues(Real * A, Real * d) {
			if (dim == 1) {
			d[0] = A[0];
			} else if (dim == 2) {
			matrix22_eigenvalues(A, d);
			}
			// else if(dim == 3) { matrix33_eigenvalues(A, d); }
			else
			matrixEig(dim, A, d);
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::det2(const Real * mat) {
			return mat[0] * mat[3] - mat[1] * mat[2];
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::det3(const Real * mat) {
			return mat[0] * (mat[4] * mat[8] - mat[7] * mat[5]) -
			mat[3] * (mat[1] * mat[8] - mat[7] * mat[2]) +
			mat[6] * (mat[1] * mat[5] - mat[4] * mat[2]);
			}

			/* -------------------------------------------------------------------------- */
			template <UInt n> inline Real Math::det(const Real * mat) {
			if (n == 1)
			return *mat;
			else if (n == 2)
			return det2(mat);
			else if (n == 3)
			return det3(mat);
			else
			return det(n, mat);
			}

			/* -------------------------------------------------------------------------- */
			template <typename T> inline T Math::det(UInt n, const T * A) {
			int N = n;
			int info;
			auto * ipiv = new int[N + 1];

			auto * LU = new T[N * N];
			std::copy(A, A + N * N, LU);

			// LU factorization of A
			aka_getrf(&N, &N, LU, &N, ipiv, &info);
			if (info > 0) {
			AKANTU_DEBUG_ERROR("Singular matrix - cannot factorize it (info: " << info
			<< " )");
			}

			// det(A) = det(L) * det(U) = 1 * det(U) = product_i U_{ii}
			T det = 1.;
			for (int i = 0; i < N; ++i)
			det = (2 (ipiv[i] == i) - 1) * LU[i * n + i];

			delete[] ipiv;
			delete[] LU;
			return det;
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::normal2(const Real * vec, Real * normal) {
			normal[0] = vec[1];
			normal[1] = -vec[0];
			Math::normalize2(normal);
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::normal3(const Real * vec1, const Real * vec2, Real * normal) {
			Math::vectorProduct3(vec1, vec2, normal);
			Math::normalize3(normal);
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::normalize2(Real * vec) {
			Real norm = Math::norm2(vec);
			vec[0] /= norm;
			vec[1] /= norm;
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::normalize3(Real * vec) {
			Real norm = Math::norm3(vec);
			vec[0] /= norm;
			vec[1] /= norm;
			vec[2] /= norm;
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::norm2(const Real * vec) {
			return sqrt(vec[0] * vec[0] + vec[1] * vec[1]);
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::norm3(const Real * vec) {
			return sqrt(vec[0] * vec[0] + vec[1] * vec[1] + vec[2] * vec[2]);
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::norm(UInt n, const Real * vec) {
			Real norm = 0.;
			for (UInt i = 0; i < n; ++i) {
			norm += vec[i] * vec[i];
			}
			return sqrt(norm);
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::inv2(const Real * mat, Real * inv) {
			Real det_mat = det2(mat);

			inv[0] = mat[3] / det_mat;
			inv[1] = -mat[1] / det_mat;
			inv[2] = -mat[2] / det_mat;
			inv[3] = mat[0] / det_mat;
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::inv3(const Real * mat, Real * inv) {
			Real det_mat = det3(mat);

			inv[0] = (mat[4] * mat[8] - mat[7] * mat[5]) / det_mat;
			inv[1] = (mat[2] * mat[7] - mat[8] * mat[1]) / det_mat;
			inv[2] = (mat[1] * mat[5] - mat[4] * mat[2]) / det_mat;
			inv[3] = (mat[5] * mat[6] - mat[8] * mat[3]) / det_mat;
			inv[4] = (mat[0] * mat[8] - mat[6] * mat[2]) / det_mat;
			inv[5] = (mat[2] * mat[3] - mat[5] * mat[0]) / det_mat;
			inv[6] = (mat[3] * mat[7] - mat[6] * mat[4]) / det_mat;
			inv[7] = (mat[1] * mat[6] - mat[7] * mat[0]) / det_mat;
			inv[8] = (mat[0] * mat[4] - mat[3] * mat[1]) / det_mat;
			}

			/* -------------------------------------------------------------------------- */
			template <UInt n> inline void Math::inv(const Real * A, Real * Ainv) {
			if (n == 1)
			Ainv = 1. / A;
			else if (n == 2)
			inv2(A, Ainv);
			else if (n == 3)
			inv3(A, Ainv);
			else
			inv(n, A, Ainv);
			}

			/* -------------------------------------------------------------------------- */
			template <typename T> inline void Math::inv(UInt n, const T * A, T * invA) {
			int N = n;
			int info;
			auto * ipiv = new int[N + 1];
			int lwork = N * N;
			auto * work = new T[lwork];

			std::copy(A, A + n * n, invA);

			aka_getrf(&N, &N, invA, &N, ipiv, &info);
			if (info > 0) {
			AKANTU_DEBUG_ERROR("Singular matrix - cannot factorize it (info: " << info
			<< " )");
			}

			aka_getri(&N, invA, &N, ipiv, work, &lwork, &info);
			if (info != 0) {
			AKANTU_DEBUG_ERROR("Cannot invert the matrix (info: " << info << " )");
			}

			delete[] ipiv;
			delete[] work;
			}

			/* -------------------------------------------------------------------------- */
			template <typename T>
			inline void Math::solve(UInt n, const T * A, T * x, const T * b) {
			int N = n;
			int info;
			auto * ipiv = new int[N];
			auto * lu_A = new T[N * N];

			std::copy(A, A + N * N, lu_A);

			aka_getrf(&N, &N, lu_A, &N, ipiv, &info);
			if (info > 0) {
			AKANTU_DEBUG_ERROR("Singular matrix - cannot factorize it (info: " << info
			<< " )");
			}

			char trans = 'N';
			int nrhs = 1;

			std::copy(b, b + N, x);

			aka_getrs(&trans, &N, &nrhs, lu_A, &N, ipiv, x, &N, &info);
			if (info != 0) {
			AKANTU_DEBUG_ERROR("Cannot solve the system (info: " << info << " )");
			}

			delete[] ipiv;
			delete[] lu_A;
			}

			/* -------------------------------------------------------------------------- */
			/* -------------------------------------------------------------------------- */
			inline Real Math::matrixDoubleDot22(Real * A, Real * B) {
			Real d;
			d = A[0] * B[0] + A[1] * B[1] + A[2] * B[2] + A[3] * B[3];
			return d;
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::matrixDoubleDot33(Real * A, Real * B) {
			Real d;
			d = A[0] * B[0] + A[1] * B[1] + A[2] * B[2] + A[3] * B[3] + A[4] * B[4] +
			A[5] * B[5] + A[6] * B[6] + A[7] * B[7] + A[8] * B[8];
			return d;
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::matrixDoubleDot(UInt n, Real * A, Real * B) {
			Real d = 0.;
			for (UInt i = 0; i < n; ++i) {
			for (UInt j = 0; j < n; ++j) {
			d += A[i * n + j] * B[i * n + j];
			}
			}
			return d;
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::vectorProduct3(const Real * v1, const Real * v2, Real * res) {
			res[0] = v1[1] * v2[2] - v1[2] * v2[1];
			res[1] = v1[2] * v2[0] - v1[0] * v2[2];
			res[2] = v1[0] * v2[1] - v1[1] * v2[0];
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::vectorDot2(const Real * v1, const Real * v2) {
			return (v1[0] * v2[0] + v1[1] * v2[1]);
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::vectorDot3(const Real * v1, const Real * v2) {
			return (v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2]);
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::distance_2d(const Real * x, const Real * y) {
			return sqrt((y[0] - x[0]) * (y[0] - x[0]) + (y[1] - x[1]) * (y[1] - x[1]));
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::triangle_inradius(const Real * coord1, const Real * coord2,
			const Real * coord3) {
			/**
			* @f{eqnarray*}{
			* r &=& A / s \\
			* A &=& 1/4 * \sqrt{(a + b + c) * (a - b + c) * (a + b - c) (-a + b + c)} \\
			* s &=& \frac{a + b + c}{2}
			* @f}
			*/

			Real a, b, c;
			a = distance_2d(coord1, coord2);
			b = distance_2d(coord2, coord3);
			c = distance_2d(coord1, coord3);

			Real s;
			s = (a + b + c) * 0.5;

			return sqrt((s - a) * (s - b) * (s - c) / s);
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::distance_3d(const Real * x, const Real * y) {
			return sqrt((y[0] - x[0]) * (y[0] - x[0]) + (y[1] - x[1]) * (y[1] - x[1]) +
			(y[2] - x[2]) * (y[2] - x[2]));
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::tetrahedron_volume(const Real * coord1, const Real * coord2,
			const Real * coord3, const Real * coord4) {
			Real xx[9], vol;

			xx[0] = coord2[0];
			xx[1] = coord2[1];
			xx[2] = coord2[2];
			xx[3] = coord3[0];
			xx[4] = coord3[1];
			xx[5] = coord3[2];
			xx[6] = coord4[0];
			xx[7] = coord4[1];
			xx[8] = coord4[2];
			vol = det3(xx);

			xx[0] = coord1[0];
			xx[1] = coord1[1];
			xx[2] = coord1[2];
			xx[3] = coord3[0];
			xx[4] = coord3[1];
			xx[5] = coord3[2];
			xx[6] = coord4[0];
			xx[7] = coord4[1];
			xx[8] = coord4[2];
			vol -= det3(xx);

			xx[0] = coord1[0];
			xx[1] = coord1[1];
			xx[2] = coord1[2];
			xx[3] = coord2[0];
			xx[4] = coord2[1];
			xx[5] = coord2[2];
			xx[6] = coord4[0];
			xx[7] = coord4[1];
			xx[8] = coord4[2];
			vol += det3(xx);

			xx[0] = coord1[0];
			xx[1] = coord1[1];
			xx[2] = coord1[2];
			xx[3] = coord2[0];
			xx[4] = coord2[1];
			xx[5] = coord2[2];
			xx[6] = coord3[0];
			xx[7] = coord3[1];
			xx[8] = coord3[2];
			vol -= det3(xx);

			vol /= 6;

			return vol;
			}

			/* -------------------------------------------------------------------------- */
			inline Real Math::tetrahedron_inradius(const Real * coord1, const Real * coord2,
			const Real * coord3,
			const Real * coord4) {

			Real l12, l13, l14, l23, l24, l34;
			l12 = distance_3d(coord1, coord2);
			l13 = distance_3d(coord1, coord3);
			l14 = distance_3d(coord1, coord4);
			l23 = distance_3d(coord2, coord3);
			l24 = distance_3d(coord2, coord4);
			l34 = distance_3d(coord3, coord4);

			Real s1, s2, s3, s4;

			s1 = (l12 + l23 + l13) * 0.5;
			s1 = sqrt(s1 * (s1 - l12) * (s1 - l23) * (s1 - l13));

			s2 = (l12 + l24 + l14) * 0.5;
			s2 = sqrt(s2 * (s2 - l12) * (s2 - l24) * (s2 - l14));

			s3 = (l23 + l34 + l24) * 0.5;
			s3 = sqrt(s3 * (s3 - l23) * (s3 - l34) * (s3 - l24));

			s4 = (l13 + l34 + l14) * 0.5;
			s4 = sqrt(s4 * (s4 - l13) * (s4 - l34) * (s4 - l14));

			Real volume = Math::tetrahedron_volume(coord1, coord2, coord3, coord4);

			return 3 * volume / (s1 + s2 + s3 + s4);
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::barycenter(const Real * coord, UInt nb_points,
			UInt spatial_dimension, Real * barycenter) {
			memset(barycenter, 0, spatial_dimension * sizeof(Real));
			for (UInt n = 0; n < nb_points; ++n) {
			UInt offset = n * spatial_dimension;
			for (UInt i = 0; i < spatial_dimension; ++i) {
			barycenter[i] += coord[offset + i] / (Real)nb_points;
			}
			}
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::vector_2d(const Real * x, const Real * y, Real * res) {
			res[0] = y[0] - x[0];
			res[1] = y[1] - x[1];
			}

			/* -------------------------------------------------------------------------- */
			inline void Math::vector_3d(const Real * x, const Real * y, Real * res) {
			res[0] = y[0] - x[0];
			res[1] = y[1] - x[1];
			res[2] = y[2] - x[2];
			}

			/* -------------------------------------------------------------------------- */
			/// Combined absolute and relative tolerance test proposed in
			/// Real-time collision detection by C. Ericson (2004)
			inline bool Math::are_float_equal(const Real x, const Real y) {
			Real abs_max = std::max(std::abs(x), std::abs(y));
			abs_max = std::max(abs_max, Real(1.));
			return std::abs(x - y) <= (tolerance * abs_max);
			}

			/* -------------------------------------------------------------------------- */
			inline bool Math::isnan(Real x) {
			#if defined(__INTEL_COMPILER)
			#pragma warning(push)
			#pragma warning(disable : 1572)
			#endif // defined(__INTEL_COMPILER)

			// x = x return false means x = quiet_NaN
			return !(x == x);

			#if defined(__INTEL_COMPILER)
			#pragma warning(pop)
			#endif // defined(__INTEL_COMPILER)
			}

			/* -------------------------------------------------------------------------- */
			inline bool Math::are_vector_equal(UInt n, Real * x, Real * y) {
			bool test = true;
			for (UInt i = 0; i < n; ++i) {
			test &= are_float_equal(x[i], y[i]);
			}

			return test;
			}

			/* -------------------------------------------------------------------------- */
			inline bool Math::intersects(Real x_min, Real x_max, Real y_min, Real y_max) {
			return !((x_max <= y_min) \|\| (x_min >= y_max));
			}

			/* -------------------------------------------------------------------------- */
			inline bool Math::is_in_range(Real a, Real x_min, Real x_max) {
			return ((a >= x_min) && (a <= x_max));
			}

			/* -------------------------------------------------------------------------- */
			template <UInt p, typename T> inline T Math::pow(T x) {
			return (pow<p - 1, T>(x) * x);
			}
			template <> inline UInt Math::pow<0, UInt>(__attribute__((unused)) UInt x) {
			return (1);
			}
			template <> inline Real Math::pow<0, Real>(__attribute__((unused)) Real x) {
			return (1.);
			}

			/* -------------------------------------------------------------------------- */

			template <class Functor>
			Real Math::NewtonRaphson::solve(const Functor & funct, Real x_0) {
			Real x = x_0;
			Real f_x = funct.f(x);
			UInt iter = 0;
			while (std::abs(f_x) > this->tolerance && iter < this->max_iteration) {
			x -= f_x / funct.f_prime(x);
			f_x = funct.f(x);
			iter++;
			}

			AKANTU_DEBUG_ASSERT(iter < this->max_iteration,
			"Newton Raphson ("
			<< funct.name << ") solve did not converge in "
			<< this->max_iteration << " iterations (tolerance: "
			<< this->tolerance << ")");

			return x;
			}

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rAKA/src/common/aka_math_tmpl.hh
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