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/**
* @file
* @section LICENSE
*
* Copyright (©) 2016-19 EPFL (École Polytechnique Fédérale de Lausanne),
* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
*
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU Affero General Public License as published
* by the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* This program 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 Affero General Public License for more details.
*
* You should have received a copy of the GNU Affero General Public License
* along with this program. If not, see <https://www.gnu.org/licenses/>.
*
*/
/* -------------------------------------------------------------------------- */
#ifndef __STATIC_TYPES_HH__
#define __STATIC_TYPES_HH__
/* -------------------------------------------------------------------------- */
#include "tamaas.hh"
#include <type_traits>
namespace tamaas {
/* -------------------------------------------------------------------------- */
namespace detail {
template <UInt acc, UInt n, UInt... ns>
struct product_tail_rec : product_tail_rec<acc * n, ns...> {};
template <UInt acc, UInt n>
struct product_tail_rec<acc, n> : std::integral_constant<UInt, acc * n> {};
template <UInt N, UInt n, UInt... ns>
struct get_rec : get_rec<N - 1, ns...> {};
template <UInt n, UInt... ns>
struct get_rec<0, n, ns...> : std::integral_constant<UInt, n> {};
} // namespace detail
template <UInt... ns>
struct product : detail::product_tail_rec<1, ns...> {};
template <UInt N, UInt... ns>
struct get : detail::get_rec<N, ns...> {};
template <typename T>
struct is_arithmetic : std::is_arithmetic<T> {};
template <typename T>
struct is_arithmetic<thrust::complex<T>> : std::true_type {};
template <UInt dim>
struct voigt_size;
template <>
struct voigt_size<3> : std::integral_constant<UInt, 6> {};
template <>
struct voigt_size<2> : std::integral_constant<UInt, 3> {};
template <>
struct voigt_size<1> : std::integral_constant<UInt, 1> {};
/* -------------------------------------------------------------------------- */
/**
* @brief Static Array
*
* This class is meant to be a small and fast object for intermediate
* calculations, possibly on wrapped memory belonging to a grid. Support type
* show be either a pointer or a C array. It should not contain any virtual
* method.
*/
template <typename DataType, typename SupportType, UInt _size>
class StaticArray {
static_assert(std::is_array<SupportType>::value ||
std::is_pointer<SupportType>::value,
"the support type of StaticArray should be either a pointer or "
"a C-array");
using T = DataType;
using T_bare = typename std::remove_cv_t<T>;
public:
using value_type = T;
static constexpr UInt size = _size;
public:
/// Access operator
__device__ __host__ T& operator()(UInt i) {
// TAMAAS_ASSERT(i < n, "Access out of bounds");
return _mem[i];
}
/// Access operator
__device__ __host__ const T& operator()(UInt i) const {
// TAMAAS_ASSERT(i < n, "Access out of bounds");
return _mem[i];
}
/// Scalar product
template <typename DT, typename ST>
__device__ __host__ T_bare dot(const StaticArray<DT, ST, size>& o) const {
decltype(T_bare(0) * DT(0)) res = 0;
for (UInt i = 0; i < size; ++i)
res += (*this)(i)*o(i);
return res;
}
/// L2 norm squared
__device__ __host__ T_bare l2squared() const { return this->dot(*this); }
/// L2 norm
__device__ __host__ T_bare l2norm() const { return std::sqrt(l2squared()); }
/// Sum of all elements
__device__ __host__ T_bare sum() const {
T_bare res = 0;
for (UInt i = 0; i < size; ++i)
res += (*this)(i);
return res;
}
#define VECTOR_OP(op) \
template <typename DT, typename ST> \
__device__ __host__ void operator op(const StaticArray<DT, ST, size>& o) { \
for (UInt i = 0; i < size; ++i) \
(*this)(i) op o(i); \
}
VECTOR_OP(+=)
VECTOR_OP(-=)
VECTOR_OP(*=)
VECTOR_OP(/=)
#undef VECTOR_OP
#define SCALAR_OP(op) \
template <typename T1> \
__device__ __host__ std::enable_if_t<is_arithmetic<T1>::value, StaticArray&> \
operator op(const T1& x) { \
for (UInt i = 0; i < size; ++i) \
(*this)(i) op x; \
return *this; \
}
SCALAR_OP(+=)
SCALAR_OP(-=)
SCALAR_OP(*=)
SCALAR_OP(/=)
SCALAR_OP(=)
#undef SCALAR_OP
/// Overriding the implicit copy operator
__device__ __host__ StaticArray& operator=(const StaticArray& o) {
return this->copy(o);
}
template <typename DT, typename ST>
__device__ __host__ void operator=(const StaticArray<DT, ST, size>& o) {
this->copy(o);
}
template <typename DT, typename ST>
__device__ __host__ StaticArray& copy(const StaticArray<DT, ST, size>& o) {
for (UInt i = 0; i < size; ++i)
(*this)(i) = o(i);
return *this;
}
T* begin() { return _mem; }
const T* begin() const { return _mem; }
T* end() { return _mem + size; }
const T* end() const { return _mem + size; }
private:
template <typename U>
using valid_size_t = std::enable_if_t<(size > 0), U>;
public:
valid_size_t<T&> front() { return *_mem; }
valid_size_t<const T&> front() const { return *_mem; }
valid_size_t<T&> back() { return _mem[size - 1]; }
valid_size_t<const T&> back() const { return _mem[size - 1]; }
protected:
SupportType _mem;
};
/**
* @brief Static Tensor
*
* This class implements a multi-dimensional tensor behavior.
*/
template <typename DataType, typename SupportType = DataType*, UInt... dims>
class StaticTensor
: public StaticArray<DataType, SupportType, product<dims...>::value> {
using parent = StaticArray<DataType, SupportType, product<dims...>::value>;
using T = DataType;
public:
static constexpr UInt dim = sizeof...(dims);
using parent::operator=;
private:
template <typename... Idx>
__device__ __host__ static UInt unpackOffset(UInt offset, UInt index,
Idx... rest) {
constexpr UInt size = sizeof...(rest);
offset += index;
offset *= get<dim - size, dims...>::value;
return unpackOffset(offset, rest...);
}
template <typename... Idx>
__device__ __host__ static UInt unpackOffset(UInt offset, UInt index) {
return offset + index;
}
public:
template <typename... Idx>
__device__ __host__ const T& operator()(Idx... idx) const {
return parent::operator()(unpackOffset(0, idx...));
}
template <typename... Idx>
__device__ __host__ T& operator()(Idx... idx) {
return parent::operator()(unpackOffset(0, idx...));
}
};
/* -------------------------------------------------------------------------- */
/* Common Static Types */
/* -------------------------------------------------------------------------- */
// Forward declaration
template <typename DataType, typename SupportType, UInt n>
class StaticVector;
template <typename DataType, typename SupportType, UInt n>
class StaticSymMatrix;
/* -------------------------------------------------------------------------- */
template <typename DataType, typename SupportType, UInt n, UInt m>
class StaticMatrix : public StaticTensor<DataType, SupportType, n, m> {
using T = DataType;
using T_bare = typename std::remove_cv_t<T>;
public:
using StaticTensor<DataType, SupportType, n, m>::operator=;
// /// Initialize from a symmetric matrix
template <typename DT, typename ST>
__device__ __host__ std::enable_if_t<n == m>
fromSymmetric(const StaticSymMatrix<DT, ST, n>& o);
/// Outer product of two vectors
template <typename DT1, typename ST1, typename DT2, typename ST2>
__device__ __host__ void outer(const StaticVector<DT1, ST1, n>& a,
const StaticVector<DT2, ST2, m>& b);
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt l>
__device__ __host__ void mul(const StaticMatrix<DT1, ST1, n, l>& a,
const StaticMatrix<DT2, ST2, l, m>& b) {
(*this) = T(0);
for (UInt i = 0; i < n; ++i)
for (UInt j = 0; j < m; ++j)
for (UInt k = 0; k < l; ++k)
(*this)(i, j) += a(i, k) * b(k, j);
}
__device__ __host__ std::enable_if_t<n == m, T_bare> trace() const {
T_bare res{0};
for (UInt i = 0; i < n; ++i)
res += (*this)(i, i);
return res;
}
template <typename DT1, typename ST1>
__device__ __host__ std::enable_if_t<n == m>
deviatoric(const StaticMatrix<DT1, ST1, n, m>& mat, Real factor = n) {
auto norm_trace = mat.trace() / factor;
for (UInt i = 0; i < n; ++i)
for (UInt j = 0; j < m; ++j)
(*this)(i, j) = mat(i, j) - (i == j) * norm_trace;
}
};
/* -------------------------------------------------------------------------- */
/// Vector class with size determined at compile-time
template <typename DataType, typename SupportType, UInt n>
class StaticVector : public StaticTensor<DataType, SupportType, n> {
using T = std::remove_cv_t<DataType>;
public:
using StaticTensor<DataType, SupportType, n>::operator=;
/// Matrix-vector product
template <bool transpose, typename DT1, typename ST1, typename DT2,
typename ST2, UInt m>
__device__ __host__ void mul(const StaticMatrix<DT1, ST1, n, m>& mat,
const StaticVector<DT2, ST2, m>& vec) {
*this = T(0);
for (UInt i = 0; i < n; ++i)
for (UInt j = 0; j < m; ++j)
(*this)(i) +=
((transpose) ? mat(j, i) : mat(i, j)) * vec(j); // can be optimized
}
};
/* -------------------------------------------------------------------------- */
/// Symmetric matrix in Voigt notation
template <typename DataType, typename SupportType, UInt n>
class StaticSymMatrix
: public StaticVector<DataType, SupportType, voigt_size<n>::value> {
using parent = StaticVector<DataType, SupportType, voigt_size<n>::value>;
using T = std::remove_cv_t<DataType>;
private:
template <typename DT, typename ST, typename BinOp>
__device__ __host__ void sym_binary(const StaticMatrix<DT, ST, n, n>& m,
BinOp&& op) {
for (UInt i = 0; i < n; ++i)
op((*this)(i), m(i, i));
const auto a = 0.5 * std::sqrt(2);
for (UInt j = n - 1, b = n; j > 0; --j)
for (int i = j - 1; i >= 0; --i)
op((*this)(b++), a * (m(i, j) + m(j, i)));
}
public:
/// Copy values from matrix and symmetrize
template <typename DT, typename ST>
__device__ __host__ void symmetrize(const StaticMatrix<DT, ST, n, n>& m) {
sym_binary(m, [](auto&& v, auto&& w) { v = w; });
}
/// Add values from symmetrized matrix
template <typename DT, typename ST>
__device__ __host__ void operator+=(const StaticMatrix<DT, ST, n, n>& m) {
sym_binary(m, [](auto&& v, auto&& w) { v += w; });
}
__device__ __host__ auto trace() const {
std::remove_cv_t<DataType> res = 0;
for (UInt i = 0; i < n; ++i)
res += (*this)(i);
return res;
}
template <typename DT, typename ST>
__device__ __host__ void deviatoric(const StaticSymMatrix<DT, ST, n>& m,
Real factor = n) {
auto tr = m.trace() / factor;
for (UInt i = 0; i < n; ++i)
(*this)(i) = m(i) - tr;
for (UInt i = n; i < voigt_size<n>::value; ++i)
(*this)(i) = m(i);
}
using parent::operator+=;
using parent::operator=;
};
/* -------------------------------------------------------------------------- */
// Implementation of constructor from symmetric matrix
template <typename DataType, typename SupportType, UInt n, UInt m>
template <typename DT, typename ST>
__device__ __host__ std::enable_if_t<n == m>
StaticMatrix<DataType, SupportType, n, m>::fromSymmetric(
const StaticSymMatrix<DT, ST, n>& o) {
for (UInt i = 0; i < n; ++i)
(*this)(i, i) = o(i);
// We use Mendel notation for the vector representation
const auto a = 1. / std::sqrt(2);
for (UInt j = n - 1, b = n; j > 0; --j)
for (int i = j - 1; i >= 0; --i)
(*this)(i, j) = (*this)(j, i) = a * o(b++);
}
// Implementation of outer product
template <typename DataType, typename SupportType, UInt n, UInt m>
template <typename DT1, typename ST1, typename DT2, typename ST2>
__device__ __host__ void StaticMatrix<DataType, SupportType, n, m>::outer(
const StaticVector<DT1, ST1, n>& a, const StaticVector<DT2, ST2, m>& b) {
for (UInt i = 0; i < n; ++i)
for (UInt j = 0; j < m; ++j)
(*this)(i, j) = a(i) * b(j);
}
/* -------------------------------------------------------------------------- */
/* On the stack static types */
/* -------------------------------------------------------------------------- */
template <template <typename, typename, UInt...> class StaticParent,
UInt... dims>
struct static_size_helper : product<dims...> {};
template <UInt n>
struct static_size_helper<StaticSymMatrix, n> : voigt_size<n> {};
template <template <typename, typename, UInt...> class StaticParent, typename T,
UInt... dims>
class Tensor
: public StaticParent<
T, T[static_size_helper<StaticParent, dims...>::value], dims...> {
static constexpr UInt size = static_size_helper<StaticParent, dims...>::value;
using parent = StaticParent<T, T[size], dims...>;
public:
using parent::operator=;
using parent::copy;
/// Default constructor
__device__ __host__ Tensor() = default;
/// Construct with default value
__device__ __host__ Tensor(T val) { *this = val; }
/// Construct from array
__device__ __host__ Tensor(const std::array<T, size>& arr) {
// we use size to ensure static loop unrolling
for (UInt i = 0; i < size; ++i)
this->_mem[i] = arr[i];
}
/// Copy from array
__device__ __host__ Tensor& operator=(const std::array<T, size>& arr) {
// we use size to ensure static loop unrolling
for (UInt i = 0; i < size; ++i)
(*this)(i) = arr[i];
}
/// Construct by copy from static tensor
template <typename DT, typename ST>
__device__ __host__ Tensor(const StaticParent<DT, ST, dims...>& o) {
this->copy(o);
}
};
template <typename T, UInt n, UInt m>
using Matrix = Tensor<StaticMatrix, T, n, m>;
template <typename T, UInt n>
using SymMatrix = Tensor<StaticSymMatrix, T, n>;
template <typename T, UInt n>
using Vector = Tensor<StaticVector, T, n>;
/* -------------------------------------------------------------------------- */
/* Proxy Static Types */
/* -------------------------------------------------------------------------- */
/// Proxy type for tensor
template <template <typename, typename, UInt...> class StaticParent, typename T,
UInt... dims>
class TensorProxy : public StaticParent<T, T*, dims...> {
using parent = StaticParent<T, T*, dims...>;
public:
/// Explicit construction from data location
__device__ __host__ explicit TensorProxy(T* spot) { this->_mem = spot; }
/// Explicit construction from lvalue-reference
__device__ __host__ explicit TensorProxy(T& spot) : TensorProxy(&spot) {}
/// Construction from static tensor
template <typename DataType, typename SupportType>
__device__ __host__
TensorProxy(StaticParent<DataType, SupportType, dims...>& o)
: TensorProxy(o.begin()) {}
using parent::operator=;
public:
using stack_type = Tensor<StaticParent, T, dims...>;
};
template <typename T, UInt n, UInt m>
using MatrixProxy = TensorProxy<StaticMatrix, T, n, m>;
template <typename T, UInt n>
using SymMatrixProxy = TensorProxy<StaticSymMatrix, T, n>;
template <typename T, UInt n>
using VectorProxy = TensorProxy<StaticVector, T, n>;
/* -------------------------------------------------------------------------- */
/* -------------------------------------------------------------------------- */
/* Arithmetic operators creating temporaries */
/* -------------------------------------------------------------------------- */
/* -------------------------------------------------------------------------- */
/* -------------------------------------------------------------------------- */
/* Simple operators */
/* -------------------------------------------------------------------------- */
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt dim>
__device__ __host__ Vector<decltype(DT1(0) + DT2(0)), dim>
operator+(const StaticVector<DT1, ST1, dim>& a,
const StaticVector<DT2, ST2, dim>& b) {
Vector<decltype(DT1(0) + DT2(0)), dim> res(a);
res += b;
return res;
}
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt dim>
__device__ __host__ Vector<decltype(DT1(0) - DT2(0)), dim>
operator-(const StaticVector<DT1, ST1, dim>& a,
const StaticVector<DT2, ST2, dim>& b) {
Vector<decltype(DT1(0) - DT2(0)), dim> res(a);
res -= b;
return res;
}
template <typename DT1, typename ST1, UInt dim>
__device__ __host__ Vector<decltype(DT1(0)), dim>
operator-(const StaticVector<DT1, ST1, dim>& a) {
Vector<decltype(DT1(0)), dim> res(a);
res *= -1;
return res;
}
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt n,
UInt m>
__device__ __host__ Matrix<decltype(DT1(0) + DT2(0)), n, m>
operator+(const StaticMatrix<DT1, ST1, n, m>& a,
const StaticMatrix<DT2, ST2, n, m>& b) {
Matrix<decltype(DT1(0) + DT2(0)), n, m> res(a);
res += b;
return res;
}
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt n,
UInt m>
__device__ __host__ Matrix<decltype(DT1(0) - DT2(0)), n, m>
operator-(const StaticMatrix<DT1, ST1, n, m>& a,
const StaticMatrix<DT2, ST2, n, m>& b) {
Matrix<decltype(DT1(0) - DT2(0)), n, m> res(a);
res -= b;
return res;
}
template <typename DT1, typename ST1, UInt n, UInt m>
__device__ __host__ Matrix<decltype(DT1(0)), n, m>
operator-(const StaticMatrix<DT1, ST1, n, m>& a) {
Matrix<decltype(DT1(0)), n, m> res(a);
res *= -1;
return res;
}
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt n>
__device__ __host__ SymMatrix<decltype(DT1(0) + DT2(0)), n>
operator+(const StaticSymMatrix<DT1, ST1, n>& a,
const StaticSymMatrix<DT2, ST2, n>& b) {
SymMatrix<decltype(DT1(0) + DT2(0)), n> res(a);
res += b;
return res;
}
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt n>
__device__ __host__ SymMatrix<decltype(DT1(0) - DT2(0)), n>
operator-(const StaticSymMatrix<DT1, ST1, n>& a,
const StaticSymMatrix<DT2, ST2, n>& b) {
SymMatrix<decltype(DT1(0) - DT2(0)), n> res(a);
res -= b;
return res;
}
template <typename DT1, typename ST1, UInt dim>
__device__ __host__ SymMatrix<decltype(DT1(0)), dim>
operator-(const StaticSymMatrix<DT1, ST1, dim>& a) {
SymMatrix<decltype(DT1(0)), dim> res(a);
res *= -1;
return res;
}
template <typename DT, typename ST, typename T, UInt n,
typename = std::enable_if_t<is_arithmetic<T>::value>>
Vector<decltype(DT(0) * T(0)), n> operator*(const StaticVector<DT, ST, n>& a,
const T& b) {
Vector<decltype(DT(0) * T(0)), n> res{a};
res *= b;
return res;
}
// symmetry
template <typename DT, typename ST, typename T, UInt n,
typename = std::enable_if_t<is_arithmetic<T>::value>>
Vector<decltype(DT(0) * T(0)), n> operator*(const T& b,
const StaticVector<DT, ST, n>& a) {
return a * b;
}
template <typename DT, typename ST, typename T, UInt n, UInt m,
typename = std::enable_if_t<is_arithmetic<T>::value>>
Matrix<decltype(DT(0) * T(0)), n, m>
operator*(const StaticMatrix<DT, ST, n, m>& a, const T& b) {
Matrix<decltype(DT(0) * T(0)), n, m> res{a};
res *= b;
return res;
}
// symmetry
template <typename DT, typename ST, typename T, UInt n, UInt m,
typename = std::enable_if_t<is_arithmetic<T>::value, void>>
Matrix<decltype(DT(0) * T(0)), n, m>
operator*(const T& b, const StaticMatrix<DT, ST, n, m>& a) {
return a * b;
}
template <typename DT, typename ST, typename T, UInt n,
typename = std::enable_if_t<is_arithmetic<T>::value>>
SymMatrix<decltype(DT(0) * T(0)), n>
operator*(const StaticSymMatrix<DT, ST, n>& a, const T& b) {
SymMatrix<decltype(DT(0) * T(0)), n> res{a};
res *= b;
return res;
}
// symmetry
template <typename DT, typename ST, typename T, UInt n,
typename = std::enable_if_t<is_arithmetic<T>::value>>
SymMatrix<decltype(DT(0) * T(0)), n>
operator*(const T& b, const StaticSymMatrix<DT, ST, n>& a) {
return a * b;
}
/* -------------------------------------------------------------------------- */
/* Linear algebra operators */
/* -------------------------------------------------------------------------- */
/// Matrix-vector multiplication
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt n,
UInt m>
__device__ __host__ Vector<decltype(DT1(0) * DT2(0)), n>
operator*(const StaticMatrix<DT1, ST1, n, m>& a,
const StaticVector<DT2, ST2, m>& b) {
Vector<decltype(DT1(0) * DT2(0)), n> res;
res.template mul<false>(a, b);
return res;
}
/// Matrix-matrix multiplication
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt n,
UInt m, UInt l>
__device__ __host__ Matrix<decltype(DT1(0) * DT2(0)), n, m>
operator*(const StaticMatrix<DT1, ST1, n, l>& a,
const StaticMatrix<DT2, ST2, l, m>& b) {
Matrix<decltype(DT1(0) * DT2(0)), n, m> res;
res.mul(a, b);
return res;
}
template <typename DT1, typename ST1, typename DT2, typename ST2, UInt n,
UInt m>
__device__ __host__ Matrix<decltype(DT1(0) * DT2(0)), n, m>
outer(const StaticVector<DT1, ST1, n>& a, const StaticVector<DT2, ST2, m>& b) {
Matrix<decltype(DT1(0) * DT2(0)), n, m> res;
res.outer(a, b);
return res;
}
/* -------------------------------------------------------------------------- */
/* Dense/Sparse */
/* -------------------------------------------------------------------------- */
template <typename DT, typename ST, UInt n>
__device__ __host__ Matrix<std::remove_cv_t<DT>, n, n>
dense(const StaticSymMatrix<DT, ST, n>& m) {
Matrix<std::remove_cv_t<DT>, n, n> res;
res.fromSymmetric(m);
return res;
}
template <typename DT, typename ST, UInt n>
__device__ __host__ auto dense(const StaticVector<DT, ST, n>& v) {
return v;
}
template <typename DT, typename ST, UInt n>
__device__ __host__ SymMatrix<std::remove_cv_t<DT>, n>
symmetrize(const StaticMatrix<DT, ST, n, n>& m) {
SymMatrix<std::remove_cv_t<DT>, n> res;
res.symmetrize(m);
return res;
}
/* -------------------------------------------------------------------------- */
template <typename DT, typename ST>
__device__ __host__ std::array<std::remove_cv_t<DT>, 3>
invariants(const StaticSymMatrix<DT, ST, 3>& m) {
return {// I1 = tr(A)
m.trace(),
// I2 = 1/2 * (tr(A)^2 - tr(A^2))
m(0) * m(1) + m(1) * m(2) + m(0) * m(2) - m(3) * m(3) * 0.5 -
m(4) * m(4) * 0.5 - m(5) * m(5) * 0.5,
// I3 = det(A)
m(0) * m(1) * m(2) + m(5) * m(3) * m(4) / std::sqrt(2) -
m(4) * m(4) * m(1) * 0.5 - m(3) * m(3) * m(0) * 0.5 -
m(5) * m(5) * m(2) * 0.5};
}
template <typename DT, typename ST>
__device__ __host__ std::array<std::remove_cv_t<DT>, 3>
eigenvalues(const StaticSymMatrix<DT, ST, 3>& m) {
constexpr UInt n = 3;
std::array<std::remove_cv_t<DT>, n> eigenv;
auto inv = invariants(m);
auto a = 1., b = -inv[0], c = inv[1], d = inv[2];
auto p = (3 * a * c - b * b) / (3 * a * a);
auto q = (2 * b * b * b - 9 * a * b * c + 27 * a * a * d) / (27 * a * a * a);
for (UInt k = 0; k < n; ++k)
eigenv[k] =
2. * std::sqrt(-p / 3.) *
std::cos(1. / 3. *
std::acos(3. * q / (2. * p) * std::sqrt(-3. / p)) -
2. * M_PI * k / 3.) -
b / (3. * a);
return eigenv;
}
/* -------------------------------------------------------------------------- */
/* Type traits */
/* -------------------------------------------------------------------------- */
template <class Type>
struct is_proxy : std::false_type {};
template <template <typename, typename, UInt...> class StaticParent, typename T,
UInt... dims>
struct is_proxy<TensorProxy<StaticParent, T, dims...>> : std::true_type {};
template <typename T, UInt n, UInt m>
struct is_proxy<MatrixProxy<T, n, m>> : std::true_type {};
template <typename T, UInt n>
struct is_proxy<SymMatrixProxy<T, n>> : std::true_type {};
template <typename T, UInt n>
struct is_proxy<VectorProxy<T, n>> : std::true_type {};
} // namespace tamaas
#endif // __STATIC_TYPES_HH__
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