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/**
* @file aka_math_tmpl.hh
*
* @author Nicolas Richart <nicolas.richart@epfl.ch>
* @author Leonardo Snozzi <leonardo.snozzi@epfl.ch>
* @author Peter Spijker <peter.spijker@epfl.ch>
* @author Guillaume Anciaux <guillaume.anciaux@epfl.ch>
* @author Alejandro Marcos Aragon <alejandro.aragon@epfl.ch>
* @author Mathilde Radiguet <mathilde.radiguet@epfl.ch>
* @author Marco Vocialta <marco.vocialta@epfl.ch>
* @author David Simon Kammer <david.kammer@epfl.ch>
*
* @date Wed Aug 04 10:58:42 2010
*
* @brief Implementation of the inline functions of the math toolkit
*
* @section LICENSE
*
* Copyright (©) 2010-2011 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/>.
*
*/
__END_AKANTU__
#include <cmath>
#include <cstring>
#include <typeinfo>
#include "aka_blas_lapack.hh"
__BEGIN_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 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;
}
/* -------------------------------------------------------------------------- */
inline Real Math::matrixDoubleDot22(Real * A, Real * B) {
//#ifdef AKANTU_USE_BLAS
//#else
Real d = 0.0;
for (UInt i = 0; i < 2; ++i) {
for (UInt j = 0; j < 2; ++j) {
d = d + A[2*i+j] * B[2*i+j];
}
}
//#endif
return d;
}
/* -------------------------------------------------------------------------- */
inline Real Math::matrixDoubleDot33(Real * A, Real * B) {
//#ifdef AKANTU_USE_BLAS
//#else
Real d = 0.0;
for (UInt i = 0; i < 3; ++i) {
for (UInt j = 0; j < 3; ++j) {
d = d + A[3*i+j] * B[3*i+j];
}
}
//#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 != NULL)
jobvl = 'V'; // compute left eigenvectors
else
jobvl = 'N'; // compute left eigenvectors
char jobvr('N'); // compute right eigenvectors
T * 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, NULL, &N, &wkopt, &lwork, &info);
lwork = int(wkopt);
T * work = new T[lwork];
// solve the eigenproblem
aka_geev<T>(&jobvl, &jobvr, &N, A, &N, d, di, V, &N, NULL, &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;
int * ipiv = new int[N+1];
T * 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;
int * ipiv = new int[N+1];
int lwork = N*N;
T * 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;
int * ipiv = new int[N];
T * 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 <<" )");
exit (EXIT_FAILURE);
}
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 <<" )");
exit (EXIT_FAILURE);
}
delete [] ipiv;
delete [] lu_A;
}
/* -------------------------------------------------------------------------- */
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];
}
/* -------------------------------------------------------------------------- */
inline bool Math::are_float_equal(const Real x, const Real y){
Real abs_max = std::max(std::abs(x), std::abs(y));
if (abs_max < tolerance) return true;
else return ( std::abs(x - y) / abs_max < tolerance);
}
/* -------------------------------------------------------------------------- */
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> inline UInt Math::pow(UInt x) { return (pow<p-1>(x)*x); }
template<> inline UInt Math::pow<0>(__attribute__((unused)) UInt x) { return (1); }
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