Reference documentation for deal.II version GIT eef19498cf 2022-10-05 14:45:02+00:00
Point< dim, Number > Class Template Reference

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

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

## Public Types

using value_type = typename Tensor< rank_ - 1, dim, double >::tensor_type

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

using tensor_type = Tensor< rank_, dim, double >

## Public Member Functions

Point ()

Point (const Tensor< 1, dim, Number > &)

Point (const Number x)

Point (const Number x, const Number y)

Point (const Number x, const Number y, const Number z)

template<std::size_t dummy_dim, std::enable_if_t<(dim==dummy_dim) &&(dummy_dim !=0), int > = 0>
Point (const boost::geometry::model::point< Number, dummy_dim, boost::geometry::cs::cartesian > &boost_pt)

Number operator() (const unsigned int index) const

Number & operator() (const unsigned int index)

template<typename OtherNumber >
Point< dim, Number > & operator= (const Tensor< 1, dim, OtherNumber > &p)

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

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 doubleoperator[] (const TableIndices< rank_ > &indices) const

constexpr doubleoperator[] (const TableIndices< rank_ > &indices)

doublebegin_raw ()

const doublebegin_raw () const

doubleend_raw ()

const doubleend_raw () const

constexpr bool operator== (const Tensor< rank_, dim, OtherNumber > &) const

constexpr bool operator!= (const Tensor< rank_, dim, OtherNumber > &) const

constexpr Tensoroperator+= (const Tensor< rank_, dim, OtherNumber > &)

constexpr Tensoroperator-= (const Tensor< rank_, dim, OtherNumber > &)

constexpr Tensoroperator*= (const OtherNumber &factor)

constexpr Tensoroperator/= (const OtherNumber &factor)

constexpr void clear ()

numbers::NumberTraits< double >::real_type norm () const

constexpr numbers::NumberTraits< double >::real_type norm_square () const

void unroll (Vector< OtherNumber > &result) const

void unroll (const Iterator begin, const Iterator end) const

Addition and subtraction of points.
Point< dim, Number > operator+ (const Tensor< 1, dim, Number > &) const

Tensor< 1, dim, Number > operator- (const Point< dim, Number > &) const

Point< dim, Number > operator- (const Tensor< 1, dim, Number > &) const

Point< dim, Number > operator- () const

## Static Public Member Functions

static Point< dim, Number > unit_vector (const unsigned int i)

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

static constexpr unsigned int rank

static constexpr unsigned int n_independent_components

## Private Member Functions

Iterator unroll_recursion (const Iterator current, const Iterator end) const

## Private Attributes

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

## Related Functions

(Note that these are not member functions.)

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

template<int dim, typename Number >
std::istream & operator>> (std::istream &in, Point< dim, Number > &p)

Tensor< rank, dim, doublesum (const Tensor< rank, dim, double > &local, const MPI_Comm &mpi_communicator)

double l1_norm (const Tensor< 2, dim, double > &t)

double linfty_norm (const Tensor< 2, dim, double > &t)

Vector space operations on Tensor objects:
constexpr Tensor< 0, dim, typename ProductType< double, OtherNumber >::type > operator- (const Tensor< 0, dim, double > &p, const Tensor< 0, dim, OtherNumber > &q)

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

constexpr ProductType< Other, double >::type operator* (const Other &object, const Tensor< 0, dim, double > &t)

constexpr ProductType< double, Other >::type operator* (const Tensor< 0, dim, double > &t, const Other &object)

constexpr ProductType< double, OtherNumber >::type operator* (const Tensor< 0, dim, double > &src1, const Tensor< 0, dim, OtherNumber > &src2)

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

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

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

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

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

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

constexpr Tensor< 0, dim, typename ProductType< double, OtherNumber >::type > schur_product (const Tensor< 0, dim, double > &src1, const Tensor< 0, dim, OtherNumber > &src2)

constexpr Tensor< rank, dim, typename ProductType< double, OtherNumber >::type > schur_product (const Tensor< rank, dim, double > &src1, const Tensor< rank, dim, OtherNumber > &src2)

Output functions for Tensor objects
std::ostream & operator<< (std::ostream &out, const Tensor< rank_, dim, double > &p)

std::ostream & operator<< (std::ostream &out, const Tensor< 0, dim, double > &p)

## Multiplication and scaling of points. Dot products. Norms.

template<typename OtherNumber >
Point< dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > operator/ (const OtherNumber) const

Number operator* (const Tensor< 1, dim, Number > &p) const

numbers::NumberTraits< Number >::real_type square () const

numbers::NumberTraits< Number >::real_type distance (const Point< dim, Number > &p) const

numbers::NumberTraits< Number >::real_type distance_square (const Point< dim, Number > &p) const

template<typename OtherNumber >
Point< dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > operator* (const OtherNumber) const

## Detailed Description

### template<int dim, typename Number = double> class Point< dim, Number >

A class that represents a point in a Cartesian space of dimension dim .

Objects of this class are used to represent points (i.e., vectors anchored at the origin) of a vector space equipped with a Cartesian coordinate system. They are, among other uses, passed to functions that operate on points in spaces of a priori fixed dimension: rather than using functions like double f(const double x) and double f(const double x, const double y), you can use double f(const Point<dim> &p) instead as it allows writing dimension independent code.

deal.II specifically uses Point objects as indicating points that are represented by Cartesian coordinates, i.e., where a point in dim space dimensions is characterized by signed distances along the axes of a coordinate system spanned by dim mutually orthogonal unit vectors (called the "coordinate axes"). This choice of representing a vector makes addition and scaling of vectors particularly simple: one only has to add or multiply each coordinate value. On the other hand, adding or scaling vectors is not nearly as simple when a vector is represented in other kinds of coordinate systems (e.g., spherical coordinate systems).

### What's a Point<dim> and what is a Tensor<1,dim>?

The Point class is derived from Tensor<1,dim> and consequently shares the latter's member functions and other attributes. In fact, it has relatively few additional functions itself (the most notable exception being the distance() function to compute the Euclidean distance between two points in space), and these two classes can therefore often be used interchangeably.

Nonetheless, there are semantic differences that make us use these classes in different and well-defined contexts. Within deal.II, we use the Point class to denote points in space, i.e., for vectors (rank-1 tensors) that are anchored at the origin. On the other hand, vectors that are anchored elsewhere (and consequently do not represent points in the common usage of the word) are represented by objects of type Tensor<1,dim>. In particular, this is the case for direction vectors, normal vectors, gradients, and the differences between two points (i.e., what you get when you subtract one point from another): all of these are represented by Tensor<1,dim> objects rather than Point<dim>.

Furthermore, the Point class is only used where the coordinates of an object can be thought to possess the dimension of a length. An object that represents the weight, height, and cost of an object is neither a point nor a tensor (because it lacks the transformation properties under rotation of the coordinate system) and should consequently not be represented by either of these classes. Use an array of size 3 in this case, or the std::array class. Alternatively, as in the case of vector-valued functions, you can use objects of type Vector or std::vector.

Template Parameters
 dim An integer that denotes the dimension of the space in which a point lies. This of course equals the number of coordinates that identify a point. Number The data type in which the coordinates values 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 coordinates in a different (and always scalar) type. An example would be an interval type that can store the value of a coordinate as well as its uncertainty. 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 when passed a Point object whose coordinates are stored in such a type.

Definition at line 110 of file point.h.

## ◆ value_type

 using Tensor< rank_, dim, double >::value_type = typename Tensor::tensor_type
inherited

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 536 of file tensor.h.

## ◆ array_type

 using Tensor< rank_, dim, double >::array_type = typename Tensor::array_type[(dim != 0) ? dim : 1]
inherited

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 542 of file tensor.h.

## ◆ tensor_type

 using Tensor< rank_, dim, double >::tensor_type = Tensor
inherited

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

Definition at line 859 of file tensor.h.

## ◆ Point() [1/6]

template<int dim, typename Number = double>
 Point< dim, Number >::Point ( )

Standard constructor. Creates an object that corresponds to the origin, i.e., all coordinates are set to zero.

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

## ◆ Point() [2/6]

template<int dim, typename Number = double>
 Point< dim, Number >::Point ( const Tensor< 1, dim, Number > & )
explicit

Convert a tensor to a point.

## ◆ Point() [3/6]

template<int dim, typename Number = double>
 Point< dim, Number >::Point ( const Number x )
explicit

Constructor for one dimensional points. This function is only implemented for dim==1 since the usage is considered unsafe for points with dim!=1 as it would leave some components of the point coordinates uninitialized.

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

## ◆ Point() [4/6]

template<int dim, typename Number = double>
 Point< dim, Number >::Point ( const Number x, const Number y )

Constructor for two dimensional points. This function is only implemented for dim==2 since the usage is considered unsafe for points with dim!=2 as it would leave some components of the point coordinates uninitialized (if dim>2) or would not use some arguments (if dim<2).

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

## ◆ Point() [5/6]

template<int dim, typename Number = double>
 Point< dim, Number >::Point ( const Number x, const Number y, const Number z )

Constructor for three dimensional points. This function is only implemented for dim==3 since the usage is considered unsafe for points with dim!=3 as it would leave some components of the point coordinates uninitialized (if dim>3) or would not use some arguments (if dim<3).

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

## ◆ Point() [6/6]

template<int dim, typename Number = double>
template<std::size_t dummy_dim, std::enable_if_t<(dim==dummy_dim) &&(dummy_dim !=0), int > = 0>
 Point< dim, Number >::Point ( const boost::geometry::model::point< Number, dummy_dim, boost::geometry::cs::cartesian > & boost_pt )

Convert a boost::geometry::point to a Point.

## ◆ unit_vector()

template<int dim, typename Number = double>
 static Point Point< dim, Number >::unit_vector ( const unsigned int i )
static

Return a unit vector in coordinate direction i, i.e., a vector that is zero in all coordinates except for a single 1 in the ith coordinate.

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

## ◆ operator()() [1/2]

template<int dim, typename Number = double>
 Number Point< dim, Number >::operator() ( const unsigned int index ) const

Read access to the indexth coordinate.

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

## ◆ operator()() [2/2]

template<int dim, typename Number = double>
 Number& Point< dim, Number >::operator() ( const unsigned int index )

Read and write access to the indexth coordinate.

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

## ◆ operator=()

template<int dim, typename Number = double>
template<typename OtherNumber >
 Point& Point< dim, Number >::operator= ( const Tensor< 1, dim, OtherNumber > & p )

Assignment operator from Tensor<1, dim, Number> with different underlying scalar type. This obviously requires that the OtherNumber type is convertible to Number.

## ◆ operator+()

template<int dim, typename Number = double>
 Point Point< dim, Number >::operator+ ( const Tensor< 1, dim, Number > & ) const

Add an offset given as Tensor<1,dim,Number> to a point.

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

## ◆ operator-() [1/3]

template<int dim, typename Number = double>
 Tensor<1, dim, Number> Point< dim, Number >::operator- ( const Point< dim, Number > & ) const

Subtract two points, i.e., obtain the vector that connects the two. As discussed in the documentation of this class, subtracting two points results in a vector anchored at one of the two points (rather than at the origin) and, consequently, the result is returned as a Tensor<1,dim> rather than as a Point<dim>.

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

## ◆ operator-() [2/3]

template<int dim, typename Number = double>
 Point Point< dim, Number >::operator- ( const Tensor< 1, dim, Number > & ) const

Subtract a difference vector (represented by a Tensor<1,dim>) from the current point. This results in another point and, as discussed in the documentation of this class, the result is then naturally returned as a Point<dim> object rather than as a Tensor<1,dim>.

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

## ◆ operator-() [3/3]

template<int dim, typename Number = double>
 Point Point< dim, Number >::operator- ( ) const

The opposite vector.

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

## ◆ operator/()

template<int dim, typename Number = double>
template<typename OtherNumber >
 Point< dim, typename ProductType::type>::type> Point< dim, Number >::operator/ ( const OtherNumber ) const

Divide the current point by a factor.

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

## ◆ operator*()

template<int dim, typename Number = double>
 Number Point< dim, Number >::operator* ( const Tensor< 1, dim, Number > & p ) const

Return the scalar product of the vectors representing two points.

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

## ◆ square()

template<int dim, typename Number = double>
 numbers::NumberTraits::real_type Point< dim, Number >::square ( ) const

Return the scalar product of this point vector with itself, i.e. the square, or the square of the norm. In case of a complex number type it is equivalent to the contraction of this point vector with a complex conjugate of itself.

Note
This function is equivalent to Tensor<rank,dim,Number>::norm_square() which returns the square of the Frobenius norm.
This function can also be used in CUDA device code.

## ◆ distance()

template<int dim, typename Number = double>
 numbers::NumberTraits::real_type Point< dim, Number >::distance ( const Point< dim, Number > & p ) const

Return the Euclidean distance of this point to the point p, i.e. the $$l_2$$ norm of the difference between the vectors representing the two points.

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

## ◆ distance_square()

template<int dim, typename Number = double>
 numbers::NumberTraits::real_type Point< dim, Number >::distance_square ( const Point< dim, Number > & p ) const

Return the squared Euclidean distance of this point to the point p.

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

## ◆ serialize()

template<int dim, typename Number = double>
template<class Archive >
 void Point< 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 using the BOOST serialization library.

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

 constexpr Tensor< rank_, dim, double >::operator Tensor< 1, dim, Tensor< rank_ - 1, dim, OtherNumber >> ( ) const
constexprinherited

Conversion operator to tensor of tensors.

## ◆ operator[]() [1/4]

 constexpr value_type& Tensor< rank_, dim, double >::operator[] ( const unsigned int i )
constexprinherited

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

## ◆ operator[]() [2/4]

 constexpr const value_type& Tensor< rank_, dim, double >::operator[] ( const unsigned int i ) const
constexprinherited

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

## ◆ operator[]() [3/4]

 constexpr const double & Tensor< rank_, dim, double >::operator[] ( const TableIndices< rank_ > & indices ) const
constexprinherited

Read access using TableIndices indices

## ◆ operator[]() [4/4]

 constexpr double & Tensor< rank_, dim, double >::operator[] ( const TableIndices< rank_ > & indices )
constexprinherited

Read and write access using TableIndices indices

## ◆ begin_raw() [1/2]

 double * Tensor< rank_, dim, double >::begin_raw ( )
inherited

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

## ◆ begin_raw() [2/2]

 const double * Tensor< rank_, dim, double >::begin_raw ( ) const
inherited

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

## ◆ end_raw() [1/2]

 double * Tensor< rank_, dim, double >::end_raw ( )
inherited

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

## ◆ end_raw() [2/2]

 const double * Tensor< rank_, dim, double >::end_raw ( ) const
inherited

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

## ◆ operator==()

 constexpr bool Tensor< rank_, dim, double >::operator== ( const Tensor< rank_, dim, OtherNumber > & ) const
constexprinherited

Test for equality of two tensors.

## ◆ operator!=()

 constexpr bool Tensor< rank_, dim, double >::operator!= ( const Tensor< rank_, dim, OtherNumber > & ) const
constexprinherited

Test for inequality of two tensors.

## ◆ operator+=()

 constexpr Tensor& Tensor< rank_, dim, double >::operator+= ( const Tensor< rank_, dim, OtherNumber > & )
constexprinherited

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

## ◆ operator-=()

 constexpr Tensor& Tensor< rank_, dim, double >::operator-= ( const Tensor< rank_, dim, OtherNumber > & )
constexprinherited

Subtract another tensor.

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

## ◆ operator*=()

 constexpr Tensor& Tensor< rank_, dim, double >::operator*= ( const OtherNumber & factor )
constexprinherited

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

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

## ◆ operator/=()

 constexpr Tensor& Tensor< rank_, dim, double >::operator/= ( const OtherNumber & factor )
constexprinherited

Scale the vector by 1/factor.

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

## ◆ clear()

 constexpr void Tensor< rank_, dim, double >::clear ( )
constexprinherited

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()

 numbers::NumberTraits::real_type Tensor< rank_, dim, double >::norm ( ) const
inherited

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()

 constexpr numbers::NumberTraits::real_type Tensor< rank_, dim, double >::norm_square ( ) const
constexprinherited

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() [1/2]

 void Tensor< rank_, dim, double >::unroll ( Vector< OtherNumber > & result ) const
inherited

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.

Deprecated:
Use the more general function that takes a pair of iterators instead.

## ◆ unroll() [2/2]

 void Tensor< rank_, dim, double >::unroll ( const Iterator begin, const Iterator end ) const
inherited

Fill a range with all tensor elements.

This function unrolls all tensor entries into a single, linearly numbered sequence. The order of the elements is the one given by component_to_unrolled_index().

The template type Number must be convertible to the type of *begin.

## ◆ component_to_unrolled_index()

 static constexpr unsigned int Tensor< rank_, dim, double >::component_to_unrolled_index ( const TableIndices< rank_ > & indices )
staticconstexprinherited

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()

 static constexpr TableIndices Tensor< rank_, dim, double >::unrolled_to_component_indices ( const unsigned int i )
staticconstexprinherited

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()

 static constexpr std::size_t Tensor< rank_, dim, double >::memory_consumption ( )
staticconstexprinherited

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

## ◆ unroll_recursion()

 Iterator Tensor< rank_, dim, double >::unroll_recursion ( const Iterator current, const Iterator end ) const
privateinherited

Internal helper function for unroll.

## ◆ operator*() [1/6]

template<int dim, typename Number = double>
template<typename OtherNumber >
 Point< dim, typename ProductType::type>::type> operator* ( const OtherNumber ) const
related

Multiply the current point by a factor.

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

## ◆ operator<<() [1/3]

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

Output operator for points. Print the elements consecutively, with a space in between.

Definition at line 668 of file point.h.

## ◆ operator>>()

template<int dim, typename Number >
 std::istream & operator>> ( std::istream & in, Point< dim, Number > & p )
related

Input operator for points. Inputs the elements consecutively.

Definition at line 686 of file point.h.

## ◆ operator-() [1/2]

 constexpr Tensor< 0, dim, typename ProductType< double , OtherNumber >::type > operator- ( const Tensor< 0, dim, double > & 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 2047 of file tensor.h.

## ◆ operator-() [2/2]

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

Subtraction of two tensors of general rank.

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

Definition at line 2213 of file tensor.h.

## ◆ sum()

 Tensor< rank, dim, double > sum ( const Tensor< rank, dim, double > & local, const MPI_Comm & mpi_communicator )
related

Perform an MPI sum of the entries of a tensor.

## ◆ operator<<() [2/3]

 std::ostream & operator<< ( std::ostream & out, const Tensor< rank_, dim, double > & 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 1905 of file tensor.h.

## ◆ operator<<() [3/3]

 std::ostream & operator<< ( std::ostream & out, const Tensor< 0, dim, double > & p )
related

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

Definition at line 1926 of file tensor.h.

## ◆ operator*() [2/6]

 constexpr ProductType< Other, double >::type operator* ( const Other & object, const Tensor< 0, dim, double > & 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 1953 of file tensor.h.

## ◆ operator*() [3/6]

 constexpr ProductType< double , Other >::type operator* ( const Tensor< 0, dim, double > & 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 1973 of file tensor.h.

## ◆ operator*() [4/6]

 constexpr ProductType< double , OtherNumber >::type operator* ( const Tensor< 0, dim, double > & 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 1993 of file tensor.h.

## ◆ operator*() [5/6]

 constexpr Tensor< rank, dim, typename ProductType< double , typename EnableIfScalar< OtherNumber >::type >::type > operator* ( const Tensor< rank, dim, double > & 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 2072 of file tensor.h.

## ◆ operator*() [6/6]

 constexpr Tensor< rank, dim, typename ProductType< typename EnableIfScalar< double >::type, OtherNumber >::type > operator* ( const double & 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 2100 of file tensor.h.

## ◆ operator/() [1/2]

 constexpr Tensor< 0, dim, typename ProductType< double , typename EnableIfScalar< OtherNumber >::type >::type > operator/ ( const Tensor< 0, dim, double > & 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 2014 of file tensor.h.

## ◆ operator/() [2/2]

 constexpr Tensor< rank, dim, typename ProductType< double , typename EnableIfScalar< OtherNumber >::type >::type > operator/ ( const Tensor< rank, dim, double > & 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 2171 of file tensor.h.

## ◆ operator+() [1/2]

 constexpr Tensor< 0, dim, typename ProductType< double , OtherNumber >::type > operator+ ( const Tensor< 0, dim, double > & 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 2030 of file tensor.h.

## ◆ operator+() [2/2]

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

Addition of two tensors of general rank.

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

Definition at line 2189 of file tensor.h.

## ◆ schur_product() [1/2]

 constexpr Tensor< 0, dim, typename ProductType< double , OtherNumber >::type > schur_product ( const Tensor< 0, dim, double > & 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 2233 of file tensor.h.

## ◆ schur_product() [2/2]

 constexpr Tensor< rank, dim, typename ProductType< double , OtherNumber >::type > schur_product ( const Tensor< rank, dim, double > & 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
 rank The rank of both tensors.

Definition at line 2262 of file tensor.h.

## ◆ l1_norm()

 double l1_norm ( const Tensor< 2, dim, double > & t )
related

The dot product (single contraction) for tensors. This function return a tensor of rank $$(\text{rank}_1 + \text{rank}_2 - 2)$$ that is the contraction of the last index of a tensor src1 of rank rank_1 with the first index of a tensor src2 of rank rank_2:

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

Note
For the Tensor class, the multiplication operator only performs a contraction over a single pair of indices. This is in contrast to the multiplication operator for SymmetricTensor, for which the corresponding operator*() performs a double contraction. The origin of the difference in how operator*() is implemented between Tensor and SymmetricTensor is that for the former, the product between two Tensor objects of same rank and dimension results in another Tensor object – that it, operator*() corresponds to the multiplicative group action within the group of tensors. On the other hand, there is no corresponding multiplicative group action with the set of symmetric tensors because, in general, the product of two symmetric tensors is a nonsymmetric tensor. As a consequence, for a mathematician, it is clear that operator*() for symmetric tensors must have a different meaning: namely the dot or scalar product that maps two symmetric tensors of rank 2 to a scalar. This corresponds to the double-dot (colon) operator whose meaning is then extended to the product of any two even-ranked symmetric tensors.
In case the contraction yields a tensor of rank 0, that is, if rank_1==rank_2==1, then a scalar number is returned as an unwrapped number type. 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 3007 of file tensor.h.

## ◆ linfty_norm()

 double linfty_norm ( const Tensor< 2, dim, double > & 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 3033 of file tensor.h.

## ◆ dimension

 constexpr unsigned int Tensor< rank_, dim, double >::dimension
staticconstexprinherited

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 517 of file tensor.h.

## ◆ rank

 constexpr unsigned int Tensor< rank_, dim, double >::rank
staticconstexprinherited

Publish the rank of this tensor to the outside world.

Definition at line 522 of file tensor.h.

## ◆ n_independent_components

 constexpr unsigned int Tensor< rank_, dim, double >::n_independent_components
staticconstexprinherited

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 528 of file tensor.h.

## ◆ values

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

Array of tensors holding the subelements.

Definition at line 865 of file tensor.h.

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