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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: Tue Feb 20 2018
*
* @brief Implementation of the inline functions of the math toolkit
*
* @section LICENSE
*
* Copyright (©) 2010-2018 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 = alpha*op(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 = alpha*op(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 := alpha*op(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 := alpha*op(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 := alpha*op(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 := alpha*op(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 jobvr, jobvl;
if (V != nullptr)
jobvr = 'V'; // compute left eigenvectors
else
jobvr = 'N'; // compute left eigenvectors
jobvl = '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, nullptr, &N, V, &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, nullptr, &N, V, &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_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_ERROR("Singular matrix - cannot factorize it (info: " << info
<< " )");
}
aka_getri(&N, invA, &N, ipiv, work, &lwork, &info);
if (info != 0) {
AKANTU_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_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_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 not ((x_max < y_min) or (x_min > y_max));
}
/* -------------------------------------------------------------------------- */
inline bool Math::is_in_range(Real a, Real x_min, Real x_max) {
return ((a >= x_min) and (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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