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Reference documentation for deal.II version 9.3.3
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Reference documentation for deal.II version 9.3.3
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#include <deal.II/base/point.h>
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, typename std::enable_if<(dim==dummy_dim) &&(dummy_dim !=0), int >::type = 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_type & | operator[] (const unsigned int i) |
| constexpr const value_type & | operator[] (const unsigned int i) const |
| constexpr const double & | operator[] (const TableIndices< rank_ > &indices) const |
| constexpr double & | operator[] (const TableIndices< rank_ > &indices) |
| double * | begin_raw () |
| const double * | begin_raw () const |
| double * | end_raw () |
| const double * | end_raw () const |
| constexpr bool | operator== (const Tensor< rank_, dim, OtherNumber > &) const |
| constexpr bool | operator!= (const Tensor< rank_, dim, OtherNumber > &) const |
| constexpr Tensor & | operator+= (const Tensor< rank_, dim, OtherNumber > &) |
| constexpr Tensor & | operator-= (const Tensor< rank_, dim, OtherNumber > &) |
| constexpr Tensor & | operator*= (const OtherNumber &factor) |
| constexpr Tensor & | operator/= (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 |
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 | |
| void | unroll_recursion (Vector< OtherNumber > &result, unsigned int &start_index) const |
Private Attributes | |
| Tensor< rank_ - 1, dim, double > | values [(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, double > | sum (const Tensor< rank, dim, double > &local, const MPI_Comm &mpi_communicator) |
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) |
Contraction operations and the outer product for tensor objects | |
| OtherNumber::type::tensor_type | operator* (const Tensor< rank_1, dim, double > &src1, const Tensor< rank_2, dim, OtherNumber > &src2) |
| constexpr Tensor< rank_1+rank_2-2, dim, typenameProductType< double, OtherNumber >::type >::tensor_type | contract (const Tensor< rank_1, dim, double > &src1, const Tensor< rank_2, dim, OtherNumber > &src2) |
| constexpr Tensor< rank_1+rank_2-4, dim, typenameProductType< double, OtherNumber >::type >::tensor_type | double_contract (const Tensor< rank_1, dim, double > &src1, const Tensor< rank_2, dim, OtherNumber > &src2) |
| constexpr ProductType< double, OtherNumber >::type | scalar_product (const Tensor< rank, dim, double > &left, const Tensor< rank, dim, OtherNumber > &right) |
| constexpr ProductType< T1, typenameProductType< 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) |
| constexpr Tensor< rank_1+rank_2, dim, typename ProductType< double, OtherNumber >::type > | outer_product (const Tensor< rank_1, dim, double > &src1, const Tensor< rank_2, dim, OtherNumber > &src2) |
Special operations on tensors of rank 1 | |
| constexpr Tensor< 1, dim, double > | cross_product_2d (const Tensor< 1, dim, double > &src) |
| 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 | |
| constexpr double | determinant (const Tensor< 2, dim, double > &t) |
| constexpr double | determinant (const Tensor< 2, 1, double > &t) |
| constexpr double | determinant (const Tensor< 2, 2, double > &t) |
| constexpr double | determinant (const Tensor< 2, 3, double > &t) |
| constexpr double | trace (const Tensor< 2, dim, double > &d) |
| constexpr Tensor< 2, dim, double > | invert (const Tensor< 2, dim, double > &) |
| constexpr Tensor< 2, dim, double > | transpose (const Tensor< 2, dim, double > &t) |
| constexpr Tensor< 2, dim, double > | adjugate (const Tensor< 2, dim, double > &t) |
| constexpr Tensor< 2, dim, double > | cofactor (const Tensor< 2, dim, double > &t) |
| Tensor< 2, dim, double > | project_onto_orthogonal_tensors (const Tensor< 2, dim, double > &A) |
| double | l1_norm (const Tensor< 2, dim, double > &t) |
| double | linfty_norm (const Tensor< 2, dim, double > &t) |
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 |
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).
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.
| 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. |
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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>.
Internal type declaration that is used to specialize the return type of operator[]() for Tensor<1,dim,Number>
Standard constructor. Creates an object that corresponds to the origin, i.e., all coordinates are set to zero.
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explicit |
Convert a tensor to a point.
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.
| 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).
| 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).
| 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.
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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.
| Number Point< dim, Number >::operator() | ( | const unsigned int | index | ) | const |
Read access to the indexth coordinate.
| Number & Point< dim, Number >::operator() | ( | const unsigned int | index | ) |
Read and write access to the indexth coordinate.
| Point< dim, Number > & 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.
| Point< dim, Number > Point< dim, Number >::operator+ | ( | const Tensor< 1, dim, Number > & | ) | const |
Add an offset given as Tensor<1,dim,Number> to a point.
| 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>.
| Point< dim, Number > 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>.
| Point< dim, Number > Point< dim, Number >::operator- | ( | ) | const |
The opposite vector.
| Point< dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type > Point< dim, Number >::operator/ | ( | const | OtherNumber | ) | const |
Divide the current point by a factor.
| Number Point< dim, Number >::operator* | ( | const Tensor< 1, dim, Number > & | p | ) | const |
Return the scalar product of the vectors representing two points.
| numbers::NumberTraits< Number >::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.
| numbers::NumberTraits< Number >::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.
| numbers::NumberTraits< Number >::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.
| 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.
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constexprinherited |
Conversion operator to tensor of tensors.
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constexprinherited |
Read-Write access operator.
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constexprinherited |
Read-only access operator.
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constexprinherited |
Read access using TableIndices indices
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constexprinherited |
Read and write access using TableIndices indices
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inherited |
Return a pointer to the first element of the underlying storage.
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inherited |
Return a const pointer to the first element of the underlying storage.
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inherited |
Return a pointer to the element past the end of the underlying storage.
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inherited |
Return a pointer to the element past the end of the underlying storage.
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constexprinherited |
Test for equality of two tensors.
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constexprinherited |
Test for inequality of two tensors.
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constexprinherited |
Add another tensor.
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constexprinherited |
Subtract another tensor.
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constexprinherited |
Scale the tensor by factor, i.e. multiply all components by factor.
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constexprinherited |
Scale the vector by 1/factor.
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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.
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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.
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Return the square of the Frobenius-norm of a tensor, i.e. the sum of the absolute squares of all entries.
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.
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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.
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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.
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Determine an estimate for the memory consumption (in bytes) of this object.
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privateinherited |
Internal helper function for unroll.
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Multiply the current point by a factor.
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Perform an MPI sum of the entries of a tensor.
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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.
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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.
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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.
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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.
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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>:
\[ \text{result}_{i, j} = \text{left}_{i, j}\circ \text{right}_{i, j} \]
| rank | The rank of both tensors. |
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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
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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
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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:
\[ \sum_{i_1,\ldots,i_r} \text{left}_{i_1,\ldots,i_r} \text{right}_{i_1,\ldots,i_r} \]
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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:
\[ \sum_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} \text{left}_{i_1,\ldots,i_{r1}} \text{middle}_{i_1,\ldots,i_{r1},j_1,\ldots,j_{r2}} \text{right}_{j_1,\ldots,j_{r2}} \]
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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).
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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).
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Return the cofactor of the given tensor of rank 2. The cofactor of a tensor \(\mathbf A\) is defined as
\[ \textrm{cof}\mathbf A \dealcoloneq \textrm{det}\mathbf A \; \mathbf{A}^{-T} = \left[ \textrm{adj}\mathbf A \right]^{T} \; . \]
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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.
| A | The tensor for which to find the closest orthogonal tensor. |
| Number | The type used to store the entries of the tensor. Must be either float or double. |
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.
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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.
1.9.4