Reference documentation for deal.II version Git 70d02132d0 2020-07-06 10:39:37 -0400
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Public Types | Public Member Functions | Static Public Member Functions | Static Public Attributes | Private Member Functions | Private Attributes | Friends | Related Functions | List of all members
Tensor< rank_, dim, Number > Class Template Reference

#include <deal.II/base/tensor.h>

Inheritance diagram for Tensor< rank_, dim, Number >:
[legend]

Public Types

using value_type = typename Tensor< rank_ - 1, dim, Number >::tensor_type
 
using array_type = typename Tensor< rank_ - 1, dim, Number >::array_type[(dim !=0) ? dim :1]
 
using tensor_type = Tensor< rank_, dim, Number >
 

Public Member Functions

constexpr Tensor ()=default
 
constexpr Tensor (const array_type &initializer)
 
template<typename ElementType , typename MemorySpace >
constexpr Tensor (const ArrayView< ElementType, MemorySpace > &initializer)
 
template<typename OtherNumber >
constexpr Tensor (const Tensor< rank_, dim, OtherNumber > &initializer)
 
template<typename OtherNumber >
constexpr Tensor (const Tensor< 1, dim, Tensor< rank_ - 1, dim, OtherNumber >> &initializer)
 
template<typename OtherNumber >
constexpr operator Tensor< 1, dim, Tensor< rank_ - 1, dim, OtherNumber >> () const
 
constexpr value_typeoperator[] (const unsigned int i)
 
constexpr const value_typeoperator[] (const unsigned int i) const
 
constexpr const Number & operator[] (const TableIndices< rank_ > &indices) const
 
constexpr Number & operator[] (const TableIndices< rank_ > &indices)
 
Number * begin_raw ()
 
const Number * begin_raw () const
 
Number * end_raw ()
 
const Number * end_raw () const
 
template<typename OtherNumber >
constexpr Tensoroperator= (const Tensor< rank_, dim, OtherNumber > &rhs)
 
constexpr Tensoroperator= (const Number &d)
 
template<typename OtherNumber >
constexpr bool operator== (const Tensor< rank_, dim, OtherNumber > &) const
 
template<typename OtherNumber >
constexpr bool operator!= (const Tensor< rank_, dim, OtherNumber > &) const
 
template<typename OtherNumber >
constexpr Tensoroperator+= (const Tensor< rank_, dim, OtherNumber > &)
 
template<typename OtherNumber >
constexpr Tensoroperator-= (const Tensor< rank_, dim, OtherNumber > &)
 
template<typename OtherNumber >
constexpr Tensoroperator*= (const OtherNumber &factor)
 
template<typename OtherNumber >
constexpr Tensoroperator/= (const OtherNumber &factor)
 
constexpr Tensor operator- () const
 
constexpr void clear ()
 
numbers::NumberTraits< Number >::real_type norm () const
 
constexpr numbers::NumberTraits< Number >::real_type norm_square () const
 
template<typename OtherNumber >
void unroll (Vector< OtherNumber > &result) const
 
template<class Archive >
void serialize (Archive &ar, const unsigned int version)
 

Static Public Member Functions

static constexpr unsigned int component_to_unrolled_index (const TableIndices< rank_ > &indices)
 
static constexpr TableIndices< rank_ > unrolled_to_component_indices (const unsigned int i)
 
static constexpr std::size_t memory_consumption ()
 

Static Public Attributes

static constexpr unsigned int dimension = dim
 
static constexpr unsigned int rank = rank_
 
static constexpr unsigned int n_independent_components
 

Private Member Functions

template<typename OtherNumber >
void unroll_recursion (Vector< OtherNumber > &result, unsigned int &start_index) const
 
template<typename ArrayLike , std::size_t... Indices>
constexpr Tensor (const ArrayLike &initializer, std::index_sequence< Indices... >)
 

Private Attributes

Tensor< rank_ - 1, dim, Number > values [(dim !=0) ? dim :1]
 

Friends

template<int , int , typename >
class Tensor
 
class Point< dim, Number >
 

Related Functions

(Note that these are not member functions.)

template<int rank, int dim, typename Number >
Tensor< rank, dim, Number > sum (const Tensor< rank, dim, Number > &local, const MPI_Comm &mpi_communicator)
 
Output functions for Tensor objects
template<int rank_, int dim, typename Number >
std::ostream & operator<< (std::ostream &out, const Tensor< rank_, dim, Number > &p)
 
template<int dim, typename Number >
std::ostream & operator<< (std::ostream &out, const Tensor< 0, dim, Number > &p)
 
Vector space operations on Tensor objects:
template<int dim, typename Number , typename Other >
constexpr ProductType< Other, Number >::type operator* (const Other &object, const Tensor< 0, dim, Number > &t)
 
template<int dim, typename Number , typename Other >
constexpr ProductType< Number, Other >::type operator* (const Tensor< 0, dim, Number > &t, const Other &object)
 
template<int dim, typename Number , typename OtherNumber >
constexpr ProductType< Number, OtherNumber >::type operator* (const Tensor< 0, dim, Number > &src1, const Tensor< 0, dim, OtherNumber > &src2)
 
template<int dim, typename Number , typename OtherNumber >
constexpr Tensor< 0, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > operator/ (const Tensor< 0, dim, Number > &t, const OtherNumber &factor)
 
template<int dim, typename Number , typename OtherNumber >
constexpr Tensor< 0, dim, typename ProductType< Number, OtherNumber >::type > operator+ (const Tensor< 0, dim, Number > &p, const Tensor< 0, dim, OtherNumber > &q)
 
template<int dim, typename Number , typename OtherNumber >
constexpr Tensor< 0, dim, typename ProductType< Number, OtherNumber >::type > operator- (const Tensor< 0, dim, Number > &p, const Tensor< 0, dim, OtherNumber > &q)
 
template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > operator* (const Tensor< rank, dim, Number > &t, const OtherNumber &factor)
 
template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< typename EnableIfScalar< Number >::type, OtherNumber >::type > operator* (const Number &factor, const Tensor< rank, dim, OtherNumber > &t)
 
template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > operator/ (const Tensor< rank, dim, Number > &t, const OtherNumber &factor)
 
template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type > operator+ (const Tensor< rank, dim, Number > &p, const Tensor< rank, dim, OtherNumber > &q)
 
template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type > operator- (const Tensor< rank, dim, Number > &p, const Tensor< rank, dim, OtherNumber > &q)
 
template<int dim, typename Number , typename OtherNumber >
constexpr Tensor< 0, dim, typename ProductType< Number, OtherNumber >::type > schur_product (const Tensor< 0, dim, Number > &src1, const Tensor< 0, dim, OtherNumber > &src2)
 
template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type > schur_product (const Tensor< rank, dim, Number > &src1, const Tensor< rank, dim, OtherNumber > &src2)
 
Contraction operations and the outer product for tensor objects
template<int rank_1, int rank_2, int dim, typename Number , typename OtherNumber , typename = typename std::enable_if<rank_1 >= 1 && rank_2>
OtherNumber::type::tensor_type operator* (const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, OtherNumber > &src2)
 
template<int index_1, int index_2, int rank_1, int rank_2, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank_1+rank_2 - 2, dim, typename ProductType< Number, OtherNumber >::type >::tensor_type contract (const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, OtherNumber > &src2)
 
template<int index_1, int index_2, int index_3, int index_4, int rank_1, int rank_2, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank_1+rank_2 - 4, dim, typename ProductType< Number, OtherNumber >::type >::tensor_type double_contract (const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, OtherNumber > &src2)
 
template<int rank, int dim, typename Number , typename OtherNumber >
constexpr ProductType< Number, OtherNumber >::type scalar_product (const Tensor< rank, dim, Number > &left, const Tensor< rank, dim, OtherNumber > &right)
 
template<template< int, int, typename > class TensorT1, template< int, int, typename > class TensorT2, template< int, int, typename > class TensorT3, int rank_1, int rank_2, int dim, typename T1 , typename T2 , typename T3 >
constexpr ProductType< T1, typename ProductType< T2, T3 >::type >::type contract3 (const TensorT1< rank_1, dim, T1 > &left, const TensorT2< rank_1+rank_2, dim, T2 > &middle, const TensorT3< rank_2, dim, T3 > &right)
 
template<int rank_1, int rank_2, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank_1+rank_2, dim, typename ProductType< Number, OtherNumber >::type > outer_product (const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, OtherNumber > &src2)
 
Special operations on tensors of rank 1
template<int dim, typename Number >
constexpr Tensor< 1, dim, Number > cross_product_2d (const Tensor< 1, dim, Number > &src)
 
template<int dim, typename Number1 , typename Number2 >
constexpr Tensor< 1, dim, typename ProductType< Number1, Number2 >::type > cross_product_3d (const Tensor< 1, dim, Number1 > &src1, const Tensor< 1, dim, Number2 > &src2)
 
Special operations on tensors of rank 2
template<int dim, typename Number >
constexpr Number determinant (const Tensor< 2, dim, Number > &t)
 
template<typename Number >
constexpr Number determinant (const Tensor< 2, 1, Number > &t)
 
template<int dim, typename Number >
constexpr Number trace (const Tensor< 2, dim, Number > &d)
 
template<int dim, typename Number >
constexpr Tensor< 2, dim, Number > invert (const Tensor< 2, dim, Number > &)
 
template<int dim, typename Number >
constexpr Tensor< 2, dim, Number > transpose (const Tensor< 2, dim, Number > &t)
 
template<int dim, typename Number >
constexpr Tensor< 2, dim, Number > adjugate (const Tensor< 2, dim, Number > &t)
 
template<int dim, typename Number >
constexpr Tensor< 2, dim, Number > cofactor (const Tensor< 2, dim, Number > &t)
 
template<int dim, typename Number >
Tensor< 2, dim, Number > project_onto_orthogonal_tensors (const Tensor< 2, dim, Number > &A)
 
template<int dim, typename Number >
Number l1_norm (const Tensor< 2, dim, Number > &t)
 
template<int dim, typename Number >
Number linfty_norm (const Tensor< 2, dim, Number > &t)
 

Detailed Description

template<int rank_, int dim, typename Number>
class Tensor< rank_, dim, Number >

A general tensor class with an arbitrary rank, i.e. with an arbitrary number of indices. The Tensor class provides an indexing operator and a bit of infrastructure, but most functionality is recursively handed down to tensors of rank 1 or put into external templated functions, e.g. the contract family.

The rank of a tensor specifies which types of physical quantities it can represent:

Using this tensor class for objects of rank 2 has advantages over matrices in many cases since the dimension is known to the compiler as well as the location of the data. It is therefore possible to produce far more efficient code than for matrices with runtime-dependent dimension. It also makes the code easier to read because of the semantic difference between a tensor (an object that relates to a coordinate system and has transformation properties with regard to coordinate rotations and transforms) and matrices (which we consider as operators on arbitrary vector spaces related to linear algebra things).

Template Parameters
rank_An integer that denotes the rank of this tensor. A specialization of this class exists for rank-0 tensors.
dimAn integer that denotes the dimension of the space in which this tensor operates. This of course equals the number of coordinates that identify a point and rank-1 tensor.
NumberThe data type in which the tensor elements are to be stored. This will, in almost all cases, simply be the default double, but there are cases where one may want to store elements in a different (and always scalar) type. It can be used to base tensors on float or complex numbers or any other data type that implements basic arithmetic operations. Another example would be a type that allows for Automatic Differentiation (see, for example, the Sacado type used in step-33) and thereby can generate analytic (spatial) derivatives of a function that takes a tensor as argument.

Definition at line 448 of file tensor.h.

Member Typedef Documentation

◆ value_type

template<int rank_, int dim, typename Number>
using Tensor< rank_, dim, Number >::value_type = typename Tensor<rank_ - 1, dim, Number>::tensor_type

Type of objects encapsulated by this container and returned by operator[](). This is a tensor of lower rank for a general tensor, and a scalar number type for Tensor<1,dim,Number>.

Definition at line 482 of file tensor.h.

◆ array_type

template<int rank_, int dim, typename Number>
using Tensor< rank_, dim, Number >::array_type = typename Tensor<rank_ - 1, dim, Number>::array_type[(dim != 0) ? dim : 1]

Declare an array type which can be used to initialize an object of this type statically. For dim == 0, its size is 1. Otherwise, it is dim.

Definition at line 489 of file tensor.h.

◆ tensor_type

template<int rank_, int dim, typename Number>
using Tensor< rank_, dim, Number >::tensor_type = Tensor<rank_, dim, Number>

Internal type declaration that is used to specialize the return type of operator[]() for Tensor<1,dim,Number>

Definition at line 766 of file tensor.h.

Constructor & Destructor Documentation

◆ Tensor() [1/6]

template<int rank_, int dim, typename Number>
constexpr Tensor< rank_, dim, Number >::Tensor ( )
default

Constructor. Initialize all entries to zero.

Note
This function can also be used in CUDA device code.

◆ Tensor() [2/6]

template<int rank_, int dim, typename Number>
constexpr Tensor< rank_, dim, Number >::Tensor ( const array_type initializer)
explicit

A constructor where the data is copied from a C-style array.

Note
This function can also be used in CUDA device code.

◆ Tensor() [3/6]

template<int rank_, int dim, typename Number>
template<typename ElementType , typename MemorySpace >
constexpr Tensor< rank_, dim, Number >::Tensor ( const ArrayView< ElementType, MemorySpace > &  initializer)
explicit

A constructor where the data is copied from an ArrayView object. Obviously, the ArrayView object must represent a stretch of data of size dimrank. The sequentially ordered elements of the argument initializer are interpreted as described by unrolled_to_component_index().

This constructor obviously requires that the ElementType type is either equal to Number, or is convertible to Number. Number.

Note
This function can also be used in CUDA device code.

◆ Tensor() [4/6]

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr Tensor< rank_, dim, Number >::Tensor ( const Tensor< rank_, dim, OtherNumber > &  initializer)

Constructor from tensors with different underlying scalar type. This obviously requires that the OtherNumber type is convertible to Number.

Note
This function can also be used in CUDA device code.

◆ Tensor() [5/6]

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr Tensor< rank_, dim, Number >::Tensor ( const Tensor< 1, dim, Tensor< rank_ - 1, dim, OtherNumber >> &  initializer)

Constructor that converts from a "tensor of tensors".

◆ Tensor() [6/6]

template<int rank_, int dim, typename Number>
template<typename ArrayLike , std::size_t... Indices>
constexpr Tensor< rank_, dim, Number >::Tensor ( const ArrayLike &  initializer,
std::index_sequence< Indices... >   
)
private

This constructor is for internal use. It provides a way to create constexpr constructors for Tensor<rank, dim, Number>

Note
This function can also be used in CUDA device code.

Member Function Documentation

◆ operator Tensor< 1, dim, Tensor< rank_ - 1, dim, OtherNumber >>()

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr Tensor< rank_, dim, Number >::operator Tensor< 1, dim, Tensor< rank_ - 1, dim, OtherNumber >> ( ) const

Conversion operator to tensor of tensors.

◆ operator[]() [1/4]

template<int rank_, int dim, typename Number>
constexpr value_type& Tensor< rank_, dim, Number >::operator[] ( const unsigned int  i)

Read-Write access operator.

Note
This function can also be used in CUDA device code.

◆ operator[]() [2/4]

template<int rank_, int dim, typename Number>
constexpr const value_type& Tensor< rank_, dim, Number >::operator[] ( const unsigned int  i) const

Read-only access operator.

Note
This function can also be used in CUDA device code.

◆ operator[]() [3/4]

template<int rank_, int dim, typename Number>
constexpr const Number& Tensor< rank_, dim, Number >::operator[] ( const TableIndices< rank_ > &  indices) const

Read access using TableIndices indices

◆ operator[]() [4/4]

template<int rank_, int dim, typename Number>
constexpr Number& Tensor< rank_, dim, Number >::operator[] ( const TableIndices< rank_ > &  indices)

Read and write access using TableIndices indices

◆ begin_raw() [1/2]

template<int rank_, int dim, typename Number>
Number* Tensor< rank_, dim, Number >::begin_raw ( )

Return a pointer to the first element of the underlying storage.

◆ begin_raw() [2/2]

template<int rank_, int dim, typename Number>
const Number* Tensor< rank_, dim, Number >::begin_raw ( ) const

Return a const pointer to the first element of the underlying storage.

◆ end_raw() [1/2]

template<int rank_, int dim, typename Number>
Number* Tensor< rank_, dim, Number >::end_raw ( )

Return a pointer to the element past the end of the underlying storage.

◆ end_raw() [2/2]

template<int rank_, int dim, typename Number>
const Number* Tensor< rank_, dim, Number >::end_raw ( ) const

Return a pointer to the element past the end of the underlying storage.

◆ operator=() [1/2]

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr Tensor& Tensor< rank_, dim, Number >::operator= ( const Tensor< rank_, dim, OtherNumber > &  rhs)

Assignment operator from tensors with different underlying scalar type. This obviously requires that the OtherNumber type is convertible to Number.

Note
This function can also be used in CUDA device code.

◆ operator=() [2/2]

template<int rank_, int dim, typename Number>
constexpr Tensor& Tensor< rank_, dim, Number >::operator= ( const Number &  d)

This operator assigns a scalar to a tensor. To avoid confusion with what exactly it means to assign a scalar value to a tensor, zero is the only value allowed for d, allowing the intuitive notation t=0 to reset all elements of the tensor to zero.

◆ operator==()

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr bool Tensor< rank_, dim, Number >::operator== ( const Tensor< rank_, dim, OtherNumber > &  ) const

Test for equality of two tensors.

◆ operator!=()

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr bool Tensor< rank_, dim, Number >::operator!= ( const Tensor< rank_, dim, OtherNumber > &  ) const

Test for inequality of two tensors.

◆ operator+=()

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr Tensor& Tensor< rank_, dim, Number >::operator+= ( const Tensor< rank_, dim, OtherNumber > &  )

Add another tensor.

Note
This function can also be used in CUDA device code.

◆ operator-=()

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr Tensor& Tensor< rank_, dim, Number >::operator-= ( const Tensor< rank_, dim, OtherNumber > &  )

Subtract another tensor.

Note
This function can also be used in CUDA device code.

◆ operator*=()

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr Tensor& Tensor< rank_, dim, Number >::operator*= ( const OtherNumber &  factor)

Scale the tensor by factor, i.e. multiply all components by factor.

Note
This function can also be used in CUDA device code.

◆ operator/=()

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
constexpr Tensor& Tensor< rank_, dim, Number >::operator/= ( const OtherNumber &  factor)

Scale the vector by 1/factor.

Note
This function can also be used in CUDA device code.

◆ operator-()

template<int rank_, int dim, typename Number>
constexpr Tensor Tensor< rank_, dim, Number >::operator- ( ) const

Unary minus operator. Negate all entries of a tensor.

Note
This function can also be used in CUDA device code.

◆ clear()

template<int rank_, int dim, typename Number>
constexpr void Tensor< rank_, dim, Number >::clear ( )

Reset all values to zero.

Note that this is partly inconsistent with the semantics of the clear() member functions of the standard library containers and of several other classes within deal.II, which not only reset the values of stored elements to zero, but release all memory and return the object into a virginial state. However, since the size of objects of the present type is determined by its template parameters, resizing is not an option, and indeed the state where all elements have a zero value is the state right after construction of such an object.

◆ norm()

template<int rank_, int dim, typename Number>
numbers::NumberTraits<Number>::real_type Tensor< rank_, dim, Number >::norm ( ) const

Return the Frobenius-norm of a tensor, i.e. the square root of the sum of the absolute squares of all entries. For the present case of rank-1 tensors, this equals the usual l2 norm of the vector.

Note
This function can also be used in CUDA device code.

◆ norm_square()

template<int rank_, int dim, typename Number>
constexpr numbers::NumberTraits<Number>::real_type Tensor< rank_, dim, Number >::norm_square ( ) const

Return the square of the Frobenius-norm of a tensor, i.e. the sum of the absolute squares of all entries.

Note
This function can also be used in CUDA device code.

◆ unroll()

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
void Tensor< rank_, dim, Number >::unroll ( Vector< OtherNumber > &  result) const

Fill a vector with all tensor elements.

This function unrolls all tensor entries into a single, linearly numbered vector. As usual in C++, the rightmost index of the tensor marches fastest.

◆ component_to_unrolled_index()

template<int rank_, int dim, typename Number>
static constexpr unsigned int Tensor< rank_, dim, Number >::component_to_unrolled_index ( const TableIndices< rank_ > &  indices)
static

Return an unrolled index in the range \([0,\text{dim}^{\text{rank}}-1]\) for the element of the tensor indexed by the argument to the function.

◆ unrolled_to_component_indices()

template<int rank_, int dim, typename Number>
static constexpr TableIndices<rank_> Tensor< rank_, dim, Number >::unrolled_to_component_indices ( const unsigned int  i)
static

Opposite of component_to_unrolled_index: For an index in the range \([0, \text{dim}^{\text{rank}}-1]\), return which set of indices it would correspond to.

◆ memory_consumption()

template<int rank_, int dim, typename Number>
static constexpr std::size_t Tensor< rank_, dim, Number >::memory_consumption ( )
static

Determine an estimate for the memory consumption (in bytes) of this object.

◆ serialize()

template<int rank_, int dim, typename Number>
template<class Archive >
void Tensor< rank_, dim, Number >::serialize ( Archive &  ar,
const unsigned int  version 
)

Read or write the data of this object to or from a stream for the purpose of serialization

◆ unroll_recursion()

template<int rank_, int dim, typename Number>
template<typename OtherNumber >
void Tensor< rank_, dim, Number >::unroll_recursion ( Vector< OtherNumber > &  result,
unsigned int start_index 
) const
private

Internal helper function for unroll.

Friends And Related Function Documentation

◆ Tensor

template<int rank_, int dim, typename Number>
template<int , int , typename >
friend class Tensor
friend

Definition at line 796 of file tensor.h.

◆ Point< dim, Number >

template<int rank_, int dim, typename Number>
friend class Point< dim, Number >
friend

Definition at line 800 of file tensor.h.

◆ sum()

template<int rank, int dim, typename Number >
Tensor< rank, dim, Number > sum ( const Tensor< rank, dim, Number > &  local,
const MPI_Comm &  mpi_communicator 
)
related

Perform an MPI sum of the entries of a tensor.

◆ operator<<() [1/2]

template<int rank_, int dim, typename Number >
std::ostream & operator<< ( std::ostream &  out,
const Tensor< rank_, dim, Number > &  p 
)
related

Output operator for tensors. Print the elements consecutively, with a space in between, two spaces between rank 1 subtensors, three between rank 2 and so on.

Definition at line 1692 of file tensor.h.

◆ operator<<() [2/2]

template<int dim, typename Number >
std::ostream & operator<< ( std::ostream &  out,
const Tensor< 0, dim, Number > &  p 
)
related

Output operator for tensors of rank 0. Since such tensors are scalars, we simply print this one value.

Definition at line 1713 of file tensor.h.

◆ operator*() [1/6]

template<int dim, typename Number , typename Other >
constexpr ProductType< Other, Number >::type operator* ( const Other &  object,
const Tensor< 0, dim, Number > &  t 
)
related

Scalar multiplication of a tensor of rank 0 with an object from the left.

This function unwraps the underlying Number stored in the Tensor and multiplies object with it.

Note
This function can also be used in CUDA device code.

Definition at line 1740 of file tensor.h.

◆ operator*() [2/6]

template<int dim, typename Number , typename Other >
constexpr ProductType< Number, Other >::type operator* ( const Tensor< 0, dim, Number > &  t,
const Other &  object 
)
related

Scalar multiplication of a tensor of rank 0 with an object from the right.

This function unwraps the underlying Number stored in the Tensor and multiplies object with it.

Note
This function can also be used in CUDA device code.

Definition at line 1760 of file tensor.h.

◆ operator*() [3/6]

template<int dim, typename Number , typename OtherNumber >
constexpr ProductType< Number, OtherNumber >::type operator* ( const Tensor< 0, dim, Number > &  src1,
const Tensor< 0, dim, OtherNumber > &  src2 
)
related

Scalar multiplication of two tensors of rank 0.

This function unwraps the underlying objects of type Number and OtherNumber that are stored within the Tensor and multiplies them. It returns an unwrapped number of product type.

Note
This function can also be used in CUDA device code.

Definition at line 1780 of file tensor.h.

◆ operator/() [1/2]

template<int dim, typename Number , typename OtherNumber >
constexpr Tensor< 0, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > operator/ ( const Tensor< 0, dim, Number > &  t,
const OtherNumber &  factor 
)
related

Division of a tensor of rank 0 by a scalar number.

Note
This function can also be used in CUDA device code.

Definition at line 1801 of file tensor.h.

◆ operator+() [1/2]

template<int dim, typename Number , typename OtherNumber >
constexpr Tensor< 0, dim, typename ProductType< Number, OtherNumber >::type > operator+ ( const Tensor< 0, dim, Number > &  p,
const Tensor< 0, dim, OtherNumber > &  q 
)
related

Add two tensors of rank 0.

Note
This function can also be used in CUDA device code.

Definition at line 1817 of file tensor.h.

◆ operator-() [1/2]

template<int dim, typename Number , typename OtherNumber >
constexpr Tensor< 0, dim, typename ProductType< Number, OtherNumber >::type > operator- ( const Tensor< 0, dim, Number > &  p,
const Tensor< 0, dim, OtherNumber > &  q 
)
related

Subtract two tensors of rank 0.

Note
This function can also be used in CUDA device code.

Definition at line 1834 of file tensor.h.

◆ operator*() [4/6]

template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > operator* ( const Tensor< rank, dim, Number > &  t,
const OtherNumber &  factor 
)
related

Multiplication of a tensor of general rank with a scalar number from the right.

Only multiplication with a scalar number type (i.e., a floating point number, a complex floating point number, etc.) is allowed, see the documentation of EnableIfScalar for details.

Note
This function can also be used in CUDA device code.

Definition at line 1859 of file tensor.h.

◆ operator*() [5/6]

template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< typename EnableIfScalar< Number >::type, OtherNumber >::type > operator* ( const Number &  factor,
const Tensor< rank, dim, OtherNumber > &  t 
)
related

Multiplication of a tensor of general rank with a scalar number from the left.

Only multiplication with a scalar number type (i.e., a floating point number, a complex floating point number, etc.) is allowed, see the documentation of EnableIfScalar for details.

Note
This function can also be used in CUDA device code.

Definition at line 1887 of file tensor.h.

◆ operator/() [2/2]

template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > operator/ ( const Tensor< rank, dim, Number > &  t,
const OtherNumber &  factor 
)
related

Division of a tensor of general rank with a scalar number. See the discussion on operator*() above for more information about template arguments and the return type.

Note
This function can also be used in CUDA device code.

Definition at line 1958 of file tensor.h.

◆ operator+() [2/2]

template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type > operator+ ( const Tensor< rank, dim, Number > &  p,
const Tensor< rank, dim, OtherNumber > &  q 
)
related

Addition of two tensors of general rank.

Template Parameters
rankThe rank of both tensors.
Note
This function can also be used in CUDA device code.

Definition at line 1976 of file tensor.h.

◆ operator-() [2/2]

template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type > operator- ( const Tensor< rank, dim, Number > &  p,
const Tensor< rank, dim, OtherNumber > &  q 
)
related

Subtraction of two tensors of general rank.

Template Parameters
rankThe rank of both tensors.
Note
This function can also be used in CUDA device code.

Definition at line 2000 of file tensor.h.

◆ schur_product() [1/2]

template<int dim, typename Number , typename OtherNumber >
constexpr Tensor< 0, dim, typename ProductType< Number, OtherNumber >::type > schur_product ( const Tensor< 0, dim, Number > &  src1,
const Tensor< 0, dim, OtherNumber > &  src2 
)
related

Entrywise multiplication of two tensor objects of rank 0 (i.e. the multiplication of two scalar values).

Definition at line 2020 of file tensor.h.

◆ schur_product() [2/2]

template<int rank, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type > schur_product ( const Tensor< rank, dim, Number > &  src1,
const Tensor< rank, dim, OtherNumber > &  src2 
)
related

Entrywise multiplication of two tensor objects of general rank.

This multiplication is also called "Hadamard-product" (c.f. https://en.wikipedia.org/wiki/Hadamard_product_(matrices)), and generates a new tensor of size <rank, dim>:

\[ \text{result}_{i, j} = \text{left}_{i, j}\circ \text{right}_{i, j} \]

Template Parameters
rankThe rank of both tensors.

Definition at line 2049 of file tensor.h.

◆ operator*() [6/6]

template<int rank_1, int rank_2, int dim, typename Number , typename OtherNumber , typename = typename std::enable_if<rank_1 >= 1 && rank_2>
OtherNumber::type::tensor_type operator* ( const Tensor< rank_1, dim, Number > &  src1,
const Tensor< rank_2, dim, OtherNumber > &  src2 
)
related

Definition at line 2100 of file tensor.h.

◆ contract()

template<int index_1, int index_2, int rank_1, int rank_2, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank_1+rank_2 - 2, dim, typename ProductType< Number, OtherNumber >::type >::tensor_type contract ( const Tensor< rank_1, dim, Number > &  src1,
const Tensor< rank_2, dim, OtherNumber > &  src2 
)
related

Generic contraction of a pair of indices of two tensors of arbitrary rank: Return a tensor of rank \((\text{rank}_1 + \text{rank}_2 - 2)\) that is the contraction of index index_1 of a tensor src1 of rank rank_1 with the index index_2 of a tensor src2 of rank rank_2:

\[ \text{result}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} = \sum_{k} \text{left}_{i_1,\ldots,k,\ldots,i_{r1}} \text{right}_{j_1,\ldots,k,\ldots,j_{r2}} \]

If for example the first index (index_1==0) of a tensor t1 shall be contracted with the third index (index_2==2) of a tensor t2, this function should be invoked as

contract<0, 2>(t1, t2);
Note
The position of the index is counted from 0, i.e., \(0\le\text{index}_i<\text{range}_i\).
In case the contraction yields a tensor of rank 0 the scalar number is returned as an unwrapped number type.

Definition at line 2156 of file tensor.h.

◆ double_contract()

template<int index_1, int index_2, int index_3, int index_4, int rank_1, int rank_2, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank_1+rank_2 - 4, dim, typename ProductType< Number, OtherNumber >::type >::tensor_type double_contract ( const Tensor< rank_1, dim, Number > &  src1,
const Tensor< rank_2, dim, OtherNumber > &  src2 
)
related

Generic contraction of two pairs of indices of two tensors of arbitrary rank: Return a tensor of rank \((\text{rank}_1 + \text{rank}_2 - 4)\) that is the contraction of index index_1 with index index_2, and index index_3 with index index_4 of a tensor src1 of rank rank_1 and a tensor src2 of rank rank_2:

\[ \text{result}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} = \sum_{k, l} \text{left}_{i_1,\ldots,k,\ldots,l,\ldots,i_{r1}} \text{right}_{j_1,\ldots,k,\ldots,l\ldots,j_{r2}} \]

If for example the first index (index_1==0) shall be contracted with the third index (index_2==2), and the second index (index_3==1) with the first index (index_4==0) of a tensor t2, this function should be invoked as

double_contract<0, 2, 1, 0>(t1, t2);
Note
The position of the index is counted from 0, i.e., \(0\le\text{index}_i<\text{range}_i\).
In case the contraction yields a tensor of rank 0 the scalar number is returned as an unwrapped number type.

Definition at line 2229 of file tensor.h.

◆ scalar_product()

template<int rank, int dim, typename Number , typename OtherNumber >
constexpr ProductType< Number, OtherNumber >::type scalar_product ( const Tensor< rank, dim, Number > &  left,
const Tensor< rank, dim, OtherNumber > &  right 
)
related

The scalar product, or (generalized) Frobenius inner product of two tensors of equal rank: Return a scalar number that is the result of a full contraction of a tensor left and right:

\[ \sum_{i_1,\ldots,i_r} \text{left}_{i_1,\ldots,i_r} \text{right}_{i_1,\ldots,i_r} \]

Definition at line 2308 of file tensor.h.

◆ contract3()

template<template< int, int, typename > class TensorT1, template< int, int, typename > class TensorT2, template< int, int, typename > class TensorT3, int rank_1, int rank_2, int dim, typename T1 , typename T2 , typename T3 >
constexpr ProductType< T1, typename ProductType< T2, T3 >::type >::type contract3 ( const TensorT1< rank_1, dim, T1 > &  left,
const TensorT2< rank_1+rank_2, dim, T2 > &  middle,
const TensorT3< rank_2, dim, T3 > &  right 
)
related

Full contraction of three tensors: Return a scalar number that is the result of a full contraction of a tensor left of rank rank_1, a tensor middle of rank \((\text{rank}_1+\text{rank}_2)\) and a tensor right of rank rank_2:

\[ \sum_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} \text{left}_{i_1,\ldots,i_{r1}} \text{middle}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} \text{right}_{j_1,\ldots,j_{r2}} \]

Note
Each of the three input tensors can be either a Tensor or SymmetricTensor.

Definition at line 2345 of file tensor.h.

◆ outer_product()

template<int rank_1, int rank_2, int dim, typename Number , typename OtherNumber >
constexpr Tensor< rank_1+rank_2, dim, typename ProductType< Number, OtherNumber >::type > outer_product ( const Tensor< rank_1, dim, Number > &  src1,
const Tensor< rank_2, dim, OtherNumber > &  src2 
)
related

The outer product of two tensors of rank_1 and rank_2: Returns a tensor of rank \((\text{rank}_1 + \text{rank}_2)\):

\[ \text{result}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} = \text{left}_{i_1,\ldots,i_{r1}}\,\text{right}_{j_1,\ldots,j_{r2}.} \]

Definition at line 2374 of file tensor.h.

◆ cross_product_2d()

template<int dim, typename Number >
constexpr Tensor< 1, dim, Number > cross_product_2d ( const Tensor< 1, dim, Number > &  src)
related

Return the cross product in 2d. This is just a rotation by 90 degrees clockwise to compute the outer normal from a tangential vector. This function is defined for all space dimensions to allow for dimension independent programming (e.g. within switches over the space dimension), but may only be called if the actual dimension of the arguments is two (e.g. from the dim==2 case in the switch).

Definition at line 2405 of file tensor.h.

◆ cross_product_3d()

template<int dim, typename Number1 , typename Number2 >
constexpr Tensor< 1, dim, typename ProductType< Number1, Number2 >::type > cross_product_3d ( const Tensor< 1, dim, Number1 > &  src1,
const Tensor< 1, dim, Number2 > &  src2 
)
related

Return the cross product of 2 vectors in 3d. This function is defined for all space dimensions to allow for dimension independent programming (e.g. within switches over the space dimension), but may only be called if the actual dimension of the arguments is three (e.g. from the dim==3 case in the switch).

Definition at line 2430 of file tensor.h.

◆ determinant() [1/2]

template<int dim, typename Number >
constexpr Number determinant ( const Tensor< 2, dim, Number > &  t)
related

Compute the determinant of a tensor or rank 2.

Definition at line 2464 of file tensor.h.

◆ determinant() [2/2]

template<typename Number >
constexpr Number determinant ( const Tensor< 2, 1, Number > &  t)
related

Specialization for dim==1.

Definition at line 2492 of file tensor.h.

◆ trace()

template<int dim, typename Number >
constexpr Number trace ( const Tensor< 2, dim, Number > &  d)
related

Compute and return the trace of a tensor of rank 2, i.e. the sum of its diagonal entries.

Definition at line 2506 of file tensor.h.

◆ invert()

template<int dim, typename Number >
constexpr Tensor< 2, dim, Number > invert ( const Tensor< 2, dim, Number > &  )
related

Compute and return the inverse of the given tensor. Since the compiler can perform the return value optimization, and since the size of the return object is known, it is acceptable to return the result by value, rather than by reference as a parameter.

Definition at line 2525 of file tensor.h.

◆ transpose()

template<int dim, typename Number >
constexpr Tensor< 2, dim, Number > transpose ( const Tensor< 2, dim, Number > &  t)
related

Return the transpose of the given tensor.

Definition at line 2618 of file tensor.h.

◆ adjugate()

template<int dim, typename Number >
constexpr Tensor< 2, dim, Number > adjugate ( const Tensor< 2, dim, Number > &  t)
related

Return the adjugate of the given tensor of rank 2. The adjugate of a tensor \(\mathbf A\) is defined as

\[ \textrm{adj}\mathbf A \dealcoloneq \textrm{det}\mathbf A \; \mathbf{A}^{-1} \; . \]

Note
This requires that the tensor is invertible.

Definition at line 2649 of file tensor.h.

◆ cofactor()

template<int dim, typename Number >
constexpr Tensor< 2, dim, Number > cofactor ( const Tensor< 2, dim, Number > &  t)
related

Return the cofactor of the given tensor of rank 2. The cofactor of a tensor \(\mathbf A\) is defined as

\[ \textrm{cof}\mathbf A \dealcoloneq \textrm{det}\mathbf A \; \mathbf{A}^{-T} = \left[ \textrm{adj}\mathbf A \right]^{T} \; . \]

Note
This requires that the tensor is invertible.

Definition at line 2670 of file tensor.h.

◆ project_onto_orthogonal_tensors()

template<int dim, typename Number >
Tensor< 2, dim, Number > project_onto_orthogonal_tensors ( const Tensor< 2, dim, Number > &  A)
related

Return the nearest orthogonal matrix \(\hat {\mathbf A}=\mathbf U \mathbf{V}^T\) by combining the products of the singular value decomposition (SVD) \({\mathbf A}=\mathbf U \mathbf S \mathbf V^T\) for a given input \({\mathbf A}\), effectively replacing \(\mathbf S\) with the identity matrix.

This is a (nonlinear) projection operation since when applied twice, we have \(\hat{\hat{\mathbf A}}=\hat{\mathbf A}\) as is easy to see. (That is because the SVD of \(\hat {\mathbf A}\) is simply \(\mathbf U \mathbf I \mathbf{V}^T\).) Furthermore, \(\hat {\mathbf A}\) is really an orthogonal matrix because orthogonal matrices have to satisfy \({\hat {\mathbf A}}^T \hat {\mathbf A}={\mathbf I}\), which here implies that

\begin{align*} {\hat {\mathbf A}}^T \hat {\mathbf A} &= \left(\mathbf U \mathbf{V}^T\right)^T\left(\mathbf U \mathbf{V}^T\right) \\ &= \mathbf V \mathbf{U}^T \mathbf U \mathbf{V}^T \\ &= \mathbf V \left(\mathbf{U}^T \mathbf U\right) \mathbf{V}^T \\ &= \mathbf V \mathbf I \mathbf{V}^T \\ &= \mathbf V \mathbf{V}^T \\ &= \mathbf I \end{align*}

due to the fact that the \(\mathbf U\) and \(\mathbf V\) factors that come out of the SVD are themselves orthogonal matrices.

Parameters
AThe tensor for which to find the closest orthogonal tensor.
Template Parameters
NumberThe type used to store the entries of the tensor. Must be either float or double.
Precondition
In order to use this function, this program must be linked with the LAPACK library.
A must not be singular. This is because, conceptually, the problem to be solved here is trying to find a matrix \(\hat{\mathbf A}\) that minimizes some kind of distance from \(\mathbf A\) while satisfying the quadratic constraint \({\hat {\mathbf A}}^T \hat {\mathbf A}={\mathbf I}\). This is not so dissimilar to the kind of problem where one wants to find a vector \(\hat{\mathbf x}\in{\mathbb R}^n\) that minimizes the quadratic objective function \(\|\hat {\mathbf x} - \mathbf x\|^2\) for a given \(\mathbf x\) subject to the constraint \(\|\mathbf x\|^2=1\) – in other words, we are seeking the point \(\hat{\mathbf x}\) on the unit sphere that is closest to \(\mathbf x\). This problem has a solution for all \(\mathbf x\) except if \(\mathbf x=0\). The corresponding condition for the problem we are considering here is that \(\mathbf A\) must not have a zero eigenvalue.

◆ l1_norm()

template<int dim, typename Number >
Number l1_norm ( const Tensor< 2, dim, Number > &  t)
related

Return the \(l_1\) norm of the given rank-2 tensor, where \(\|\mathbf T\|_1 = \max_j \sum_i |T_{ij}|\) (maximum of the sums over columns).

Definition at line 2753 of file tensor.h.

◆ linfty_norm()

template<int dim, typename Number >
Number linfty_norm ( const Tensor< 2, dim, Number > &  t)
related

Return the \(l_\infty\) norm of the given rank-2 tensor, where \(\|\mathbf T\|_\infty = \max_i \sum_j |T_{ij}|\) (maximum of the sums over rows).

Definition at line 2779 of file tensor.h.

Member Data Documentation

◆ dimension

template<int rank_, int dim, typename Number>
constexpr unsigned int Tensor< rank_, dim, Number >::dimension = dim
static

Provide a way to get the dimension of an object without explicit knowledge of it's data type. Implementation is this way instead of providing a function dimension() because now it is possible to get the dimension at compile time without the expansion and preevaluation of an inlined function; the compiler may therefore produce more efficient code and you may use this value to declare other data types.

Definition at line 463 of file tensor.h.

◆ rank

template<int rank_, int dim, typename Number>
constexpr unsigned int Tensor< rank_, dim, Number >::rank = rank_
static

Publish the rank of this tensor to the outside world.

Definition at line 468 of file tensor.h.

◆ n_independent_components

template<int rank_, int dim, typename Number>
constexpr unsigned int Tensor< rank_, dim, Number >::n_independent_components
static
Initial value:
=
Tensor<rank_ - 1, dim>::n_independent_components * dim

Number of independent components of a tensor of current rank. This is dim times the number of independent components of each sub-tensor.

Definition at line 474 of file tensor.h.

◆ values

template<int rank_, int dim, typename Number>
Tensor<rank_ - 1, dim, Number> Tensor< rank_, dim, Number >::values[(dim !=0) ? dim :1]
private

Array of tensors holding the subelements.

Definition at line 772 of file tensor.h.


The documentation for this class was generated from the following files: