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tensor

d-dimensional physical tensor

# Description

The tensor class defines a d*d matrix with floating coefficients. This class is suitable for defining tensors, i.e. field with d*d matrix values at each physical position.

It is represented as a bidimensional array of coordinates. The coordinate indexes start at zero and finishes at d-1, e.g. a(0,0), a(0,1), ..., a(2,2).

The default constructor set all components to zero:

        tensor a;

and this default could be overridden:

        tensor a = {{1, 2, 3.14},
                    {2, 6, 6.2 },
                    {5, 8, 9.0 }};

The standard linear algebra with scalars, vectors of R^3 (see the point class) and tensor is supported.

The computation of eigenvalues and eigenvectors, together with the SVD decomposition are also provided for convenience.

# Implementation

This documentation has been generated from file fem/geo_element/tensor.h

The tensor class is simply an alias to the tensor_basic class

typedef tensor_basic<Float> tensor;

The tensor_basic class is a template class with the floating type as parameter:

template<class T>
class tensor_basic {
public:
typedef size_t size_type;
typedef T element_type;
typedef T float_type;
// allocators:
tensor_basic (const T& init_val = 0);
tensor_basic (T x[3][3]);
tensor_basic (const tensor_basic<T>& a);
static tensor_basic<T> eye (size_type d = 3);
tensor_basic (const std::initializer_list<std::initializer_list<T> >& il);
// affectation:
tensor_basic<T>& operator= (const tensor_basic<T>& a);
tensor_basic<T>& operator= (const T& val);
// modifiers:
void fill (const T& init_val);
void reset ();
void set_row (const point_basic<T>& r, size_t i, size_t d = 3);
void set_column (const point_basic<T>& c, size_t j, size_t d = 3);
// accessors:
T& operator()(size_type i, size_type j);
const T& operator()(size_type i, size_type j) const;
point_basic<T> row(size_type i) const;
point_basic<T> col(size_type i) const;
size_t nrow() const; // = 3, for template matrix compatibility
size_t ncol() const;
// inputs/outputs:
std::ostream& put (std::ostream& s, size_type d = 3) const;
std::istream& get (std::istream&);
// algebra:
bool operator== (const tensor_basic<T>&) const;
bool operator!= (const tensor_basic<T>& b) const { return ! operator== (b); }
const tensor_basic<T>& operator+ () const { return *this; }
tensor_basic<T> operator- () const;
tensor_basic<T> operator+ (const tensor_basic<T>& b) const;
tensor_basic<T> operator- (const tensor_basic<T>& b) const;
tensor_basic<T> operator* (const tensor_basic<T>& b) const;
tensor_basic<T> operator* (const T& k) const;
tensor_basic<T> operator/ (const T& k) const;
point_basic<T> operator* (const point_basic<T>&) const;
point_basic<T> trans_mult (const point_basic<T>& x) const;
tensor_basic<T>& operator+= (const tensor_basic<T>&);
tensor_basic<T>& operator-= (const tensor_basic<T>&);
tensor_basic<T>& operator*= (const T& k);
tensor_basic<T>& operator/= (const T& k);
T determinant (size_type d = 3) const;
bool is_symmetric (size_type d = 3) const;
// eigenvalues & eigenvectors:
// a = q*d*q^T
// a may be symmetric
// where q=(q1,q2,q3) are eigenvectors in rows (othonormal matrix)
// and d=(d1,d2,d3) are eigenvalues, sorted in decreasing order d1 >= d2 >= d3
// return d
point_basic<T> eig (tensor_basic<T>& q, size_t dim = 3) const;
point_basic<T> eig (size_t dim = 3) const;
// singular value decomposition:
// a = u*s*v^T
// a can be unsymmetric
// where u=(u1,u2,u3) are left pseudo-eigenvectors in rows (othonormal matrix)
// v=(v1,v2,v3) are right pseudo-eigenvectors in rows (othonormal matrix)
// and s=(s1,s2,s3) are eigenvalues, sorted in decreasing order s1 >= s2 >= s3
// return s
point_basic<T> svd (tensor_basic<T>& u, tensor_basic<T>& v, size_t dim = 3) const;
};

The linear algebra is completed by some classical operators and the matrix exponential:

template <class U>
point_basic<U> operator* (const point_basic<U>& yt, const tensor_basic<U>& a);
template <class U>
tensor_basic<U> trans (const tensor_basic<U>& a, size_t d = 3);
template <class U>
void prod (const tensor_basic<U>& a, const tensor_basic<U>& b, tensor_basic<U>& result,
size_t di=3, size_t dj=3, size_t dk=3);
// tr(a) = a00 + a11 + a22
template <class U>
U tr (const tensor_basic<U>& a, size_t d=3);
template <class U>
U ddot (const tensor_basic<U>&, const tensor_basic<U>&);
// a = u otimes v <==> aij = ui*vj
template <class U>
tensor_basic<U> otimes (const point_basic<U>& u, const point_basic<U>& v, size_t d=3);
template <class U>
tensor_basic<U> inv (const tensor_basic<U>& a, size_t d = 3);
template <class U>
tensor_basic<U> diag (const point_basic<U>& d);
template <class U>
point_basic<U> diag (const tensor_basic<U>& a);
template <class U>
U determinant (const tensor_basic<U>& A, size_t d = 3);
template <class U>
bool invert_3x3 (const tensor_basic<U>& A, tensor_basic<U>& result);
// matrix exponential:
template<class T>
tensor_basic<T> exp (const tensor_basic<T>& a, size_t d = 3);
// inputs/outputs:
template<class T>
inline
std::istream& operator>> (std::istream& in, tensor_basic<T>& a) {
return a.get (in); }
template<class T>
inline
std::ostream& operator<< (std::ostream& out, const tensor_basic<T>& a) {
return a.put (out); }
// t += a otimes b
template<class T>
void cumul_otimes (tensor_basic<T>& t, const point_basic<T>& a, const point_basic<T>& b, size_t na = 3);
template<class T>
void cumul_otimes (tensor_basic<T>& t, const point_basic<T>& a, const point_basic<T>& b, size_t na, size_t nb);