#include <representations.h>
Public Types | |
| using | base_type = hila::arithmetic_type< radix > |
| The underlying arithmetic type of the matrix. | |
| using | sun = SU< N, radix > |
| The SU(N) type the adjoint matrix is constructed of. | |
Public Member Functions | |
| auto | abs () const |
| Returns absolute value of Matrix. | |
| Rtype | adjoint () const |
| Adjoint of matrix. | |
| antisymmetric ()=default | |
| default constructor | |
| template<typename scalart , std::enable_if_t< hila::is_arithmetic< scalart >::value, int > = 0> | |
| antisymmetric (const antisymmetric< N, scalart > m) | |
| Copy constructor. | |
| template<typename scalart , std::enable_if_t< hila::is_arithmetic< scalart >::value, int > = 0> | |
| antisymmetric (const scalart m) | |
| Square matrix constructor from scalar. | |
| antisymmetric (const SquareMatrix< size, Complex< radix > > m) | |
| automatic conversion from SquareMatrix – needed for methods! | |
| const Array< n, m, T > & | asArray () const |
| Cast Matrix to Array. | |
| const DiagonalMatrix< n, T > & | asDiagonalMatrix () const |
| Cast Vector to DiagonalMatrix. | |
| Vector< n, T > | column (int c) const |
| Returns column vector as value at index c. | |
| Matrix< n, m, T > | conj () const |
| Returns complex conjugate of Matrix. | |
| Rtype | dagger () const |
| Hermitian conjugate of matrix. | |
| T | det () const |
| determinant function - if matrix size is < 5, use Laplace, otherwise LU | |
| T | det_laplace () const |
| Determinant of matrix, using Laplace method. | |
| T | det_lu () const |
| following calculate the determinant of a square matrix det() is the generic interface, using laplace for small matrices and LU for large | |
| DiagonalMatrix< n, T > | diagonal () |
| Return diagonal of square matrix. | |
| R | dot (const Matrix< p, q, S > &rhs) const |
| Dot product. | |
| T | e (const int i) const |
| Standard array indexing operation for vectors. | |
| T | e (const int i, const int j) const |
| Standard array indexing operation for matrices. | |
| int | eigen_hermitean (DiagonalMatrix< n, Et > &eigenvalues, Matrix_t< n, n, Mt, MT > &eigenvectors, enum hila::sort sorted=hila::sort::unsorted) const |
| Calculate eigenvalues and -vectors of an hermitean (or symmetric) matrix. | |
| eigen_result< Matrix< n, m, T > > | eigen_hermitean (enum hila::sort sorted=hila::sort::unsorted) const |
| eigenvalues and -vectors of hermitean/symmetric matrix, alternative interface | |
| template<int n, int m, typename T , typename MT > | |
| Matrix_t< n, m, T, MT > | exp (const Matrix_t< n, m, T, MT > &mat, const int order=20) |
| Calculate exp of a square matrix. | |
| const Matrix< n, m, T > & | fill (const S rhs) |
| Matrix fill. | |
| Matrix< n, m, T > & | gaussian_random (double width=1.0) |
| Fills Matrix with gaussian random elements from gaussian distribution. | |
| Matrix< n, m, hila::arithmetic_type< T > > | imag () const |
| Returns imaginary part of Matrix or Vector. | |
| T | max () const |
| Find max of Matrix only for arithmetic types. | |
| T | min () const |
| Find min of Matrix only for arithmetic types. | |
| hila::type_mul< T, S > | mul_trace (const Matrix_t< p, q, S, MT > &rm) const |
| Multiply with given matrix and compute trace of result. | |
| void | mult_by_2x2_left (int p, int q, const Matrix_t< 2, 2, R, Mt > &M) |
| Multiply \( n \times m \)-matrix from the left by \( n \times m \) matrix defined by \( 2 \times 2 \) sub matrix. | |
| void | mult_by_2x2_right (int p, int q, const Matrix_t< 2, 2, R, Mt > &M) |
| Multiply \( n \times m \)-matrix from the right by \( n \times m \) matrix defined by \( 2 \times 2 \) sub matrix. | |
| hila::arithmetic_type< T > | norm () const |
| Calculate vector norm - sqrt of squarenorm. | |
| bool | operator!= (const Matrix< n, m, S > &rhs) const |
| Boolean operator != to check if matrices are exactly different. | |
| Rt | operator* (const Matrix< 1, m, T1 > &A, const Matrix< n, 1, T2 > &B) |
| Multiplication operator RowVector * Vector. | |
| Rt | operator* (const Mt &mat, const S &rhs) |
| Multiplication operator Matrix * scalar. | |
| Matrix< n, m, R > | operator* (const Mt1 &a, const Mt2 &b) |
| Multiplication operator. | |
| Rt | operator* (const S &rhs, const Mt &mat) |
| Multiplication operator Scalar * Matrix. | |
| Matrix< n, m, T > & | operator*= (const DiagonalMatrix< m, S > &rhs) |
| Multiply assign operator for DiagonalMatrix to Matrix. | |
| Matrix< n, m, T > & | operator*= (const Matrix_t< m, p, S, MT > &rhs) |
| Multiply assign scalar or matrix. | |
| Matrix< n, m, T > & | operator*= (const S rhs) |
| Multiply assign operator scalar to Matrix. | |
| const Matrix< n, m, T > & | operator+ () const |
| Addition operator. | |
| Rtype | operator+ (const Matrix< n, m, T > &a, const S &b) |
| Addition operator Matrix + scalar. | |
| Rtype | operator+ (const Mtype1 &a, const Mtype2 &b) |
| Addition operator Matrix + Matrix. | |
| Rtype | operator+ (const S &b, const Matrix< n, m, T > &a) |
| Addition operator scalar + Matrix. | |
| Matrix< n, m, T > & | operator+= (const DiagonalMatrix< n, S > &rhs) |
| Addition assign operator for DiagonalMatrix to Matrix. | |
| Matrix< n, m, T > & | operator+= (const Matrix_t< n, m, S, MT > &rhs) |
| Add assign operator Matrix to Matrix. | |
| Matrix< n, m, T > & | operator+= (const S &rhs) |
| Add assign operator scalar to Matrix. | |
| Matrix< n, m, T > | operator- () const |
| Unary operator. | |
| Rtype | operator- (const Matrix< n, m, T > &a, const S &b) |
| Subtraction operator Matrix - scalar. | |
| Rtype | operator- (const Mtype1 &a, const Mtype2 &b) |
| Subtraction operator Matrix - Matrix. | |
| Rtype | operator- (const S &b, const Matrix< n, m, T > &a) |
| Subtraction operator Scalar - Matrix. | |
| Matrix< n, m, T > & | operator-= (const DiagonalMatrix< n, S > &rhs) |
| Subtract assign operator for DiagonalMatrix to Matrix. | |
| Matrix< n, m, T > & | operator-= (const Matrix_t< n, m, S, MT > &rhs) |
| Subtract assign operator Matrix to MAtrix. | |
| Matrix< n, m, T > & | operator-= (const S rhs) |
| Subtract assign operator scalar to Matrix. | |
| Rt | operator/ (const Mt &mat, const S &rhs) |
| Division operator. | |
| Matrix< n, m, T > & | operator/= (const DiagonalMatrix< m, S > &rhs) |
| Divide assign operator for DiagonalMatrix to Matrix. | |
| Matrix< n, m, T > & | operator/= (const S rhs) |
| Divide assign oprator scalar with matrix. | |
| template<int n, int m, typename T , typename MT > | |
| std::ostream & | operator<< (std::ostream &strm, const Matrix_t< n, m, T, MT > &A) |
| Stream operator. | |
| template<typename scalart , std::enable_if_t< hila::is_arithmetic< scalart >::value, int > = 0> | |
| antisymmetric & | operator= (const antisymmetric< N, scalart > m) |
| Needs assignment as well. | |
| bool | operator== (const Matrix< n, m, S > &rhs) const |
| Boolean operator == to determine if two matrices are exactly the same. | |
| T | operator[] (const int i) const |
| Indexing operation [] defined only for vectors. | |
| Matrix< n, p, R > | outer_product (const Matrix< p, q, S > &rhs) const |
| Outer product. | |
| Matrix< n, m, T > | permute (const Vector< N, int > &permutation) const |
| Permute elements of vector. | |
| Matrix< n, m, T > | permute_columns (const Vector< m, int > &permutation) const |
| permute columns of Matrix | |
| Matrix< n, m, T > | permute_rows (const Vector< n, int > &permutation) const |
| permute rows of Matrix | |
| Matrix< n, m, T > & | random () |
| dot with matrix - matrix | |
| Matrix< n, m, hila::arithmetic_type< T > > | real () const |
| Returns real part of Matrix or Vector. | |
| void | represent (sun &m) |
| Project a matrix into the antisymmetric representation. | |
| const RowVector< m, T > & | row (int r) const |
| Return reference to row in a matrix. | |
| void | set_column (int c, const Vector< n, S > &v) |
| get column of a matrix | |
| void | set_diagonal (const Vector< n, S > &v) |
| Set diagonal of square matrix to Vector which is passed to the method. | |
| void | set_row (int r, const RowVector< m, S > &v) |
| Set row of Matrix with RowVector if types are assignable. | |
| Matrix< n, m, T > | sort (hila::sort order=hila::sort::ascending) const |
| Sort method for Vector. | |
| Matrix< n, m, T > | sort (Vector< N, int > &permutation, hila::sort order=hila::sort::ascending) const |
| Sort method for Vector which returns permutation order. | |
| hila::arithmetic_type< T > | squarenorm () const |
| Calculate square norm - sum of squared elements. | |
| std::string | str (int prec=8, char separator=' ') const |
| svd_result< Matrix< n, m, T > > | svd (enum hila::sort sorted=hila::sort::unsorted) const |
| svd and svd_pivot - alternative interface | |
| int | svd (Matrix_t< n, n, Mt, MT > &_U, DiagonalMatrix< n, Et > &_D, Matrix_t< n, n, Mt, MT > &_V, enum hila::sort sorted=hila::sort::unsorted) const |
| Singular value decomposition: divide matrix A = U S V*, where U,V unitary and S diagonal matrix of real singular values. Unpivoted Jacobi rotations. | |
| int | svd_pivot (Matrix_t< n, n, Mt, MT > &_U, DiagonalMatrix< n, Et > &_D, Matrix_t< n, n, Mt, MT > &_V, enum hila::sort sorted=hila::sort::unsorted) const |
| Singular value decomposition: divide matrix A = U S V*, where U,V unitary and S diagonal matrix of real singular values. Fully pivoted Jacobi rotations. | |
| T | trace () const |
| Computes Trace for Matrix. | |
| Rtype | transpose () const |
| Transpose of matrix. | |
| const RowVector< n, T > & | transpose () const |
| Transpose of vector. | |
| const Vector< n, T > & | transpose () const |
| Transpose of RowVector. | |
Static Public Member Functions | |
| static constexpr int | columns () |
| Returns column length. | |
| static sun | generator (int ng) |
| The antisymmetric generator. | |
| static constexpr bool | is_square () |
| Returns true if matrix is a square matrix. | |
| static constexpr bool | is_vector () |
| Returns true if Matrix is a vector. | |
| static SquareMatrix< N, Complex< radix > > | project_force (SquareMatrix< size, Complex< radix > > rforce) |
| static antisymmetric | represented_generator_I (int i) |
| Return a SU(N) generator (times I) in the antisymmetric representation. | |
| static constexpr int | rows () |
| Define constant methods rows(), columns() - may be useful in template code. | |
Public Attributes | |
| T | c [n *m] |
| The data as a one dimensional array. | |
Static Public Attributes | |
| static constexpr int | size = N * (N - 1) / 2 |
| Matrix size. | |
Detailed Description
class antisymmetric< N, radix >
A matrix in the antisymmetric representation of the SU(N) group
Members antisymmetric.represent(sun m): projects the sU(N) matrix to the antisymmetric representation and replaces this
Class functions: antisymmetric::generator(int i): returns antisymmetric matrices in the fundamental representation antisymmetric::represented_generator_I(int i): returns antisymmetric SU(N) matrices (times I) in the antisymmetric representation antisymmetric::project_force(squarematrix): projects a square matrix the size of an antisymmetric matrix to the SU(N) algebra. This is used to calculate the force of an antisymmetric action term to a derivative of the underlying su(N) group
Definition at line 126 of file representations.h.
Member Function Documentation
◆ abs()
◆ adjoint()
◆ asArray()
◆ asDiagonalMatrix()
|
inlineinherited |
◆ column()
Returns column vector as value at index c.
- Parameters
-
c index of column vector to be returned
- Returns
- const Vector<n, T>
◆ columns()
◆ conj()
◆ dagger()
◆ det_laplace()
Determinant of matrix, using Laplace method.
Defined only for square matrices
For perfomance overloads exist for \( 1 \times 1 \), \( 2 \times 2 \) and \( 3 \times 3 \) matrices.
- Parameters
-
mat matrix to compute determinant for
- Returns
- T result determinant
Definition at line 1589 of file matrix_linalg.h.
◆ det_lu()
following calculate the determinant of a square matrix det() is the generic interface, using laplace for small matrices and LU for large
Matrix determinant with LU decomposition.
Algorithm: numerical Recipes, 2nd ed. p. 47 ff Works for Real and Complex matrices Defined only for square matrices
- Returns
- Complex<radix> Determinant
Definition at line 1587 of file matrix_linalg.h.
◆ diagonal()
|
inlineinherited |
Return diagonal of square matrix.
If called for non square matrix the program will throw an error.
- Returns
- Vector<n, T> returned vector.
◆ dot()
|
inlineinherited |
Dot product.
Only works between two Vector objects
- Template Parameters
-
p Row length for rhs q Column length for rhs S Type for rhs R Gives resulting type of lhs and rhs multiplication
- Parameters
-
rhs Vector to compute dot product with
- Returns
- R Value of dot product
◆ e() [1/2]
Standard array indexing operation for vectors.
Accessing singular elements is insufficient, but Vector elements are often quite small.
- Template Parameters
-
q Number of rows p Number of columns
- Parameters
-
i Index
- Returns
- T
◆ e() [2/2]
◆ eigen_hermitean() [1/2]
|
inherited |
Calculate eigenvalues and -vectors of an hermitean (or symmetric) matrix.
Calculate eigenvalues and vectors of an hermitean or real symmetric matrix.
Returns the number of Jacobi iterations, or -1 if did not converge. Arguments will contain eigenvalues and eigenvectors in columns of the "eigenvectors" matrix. Computation is done in double precision despite the input matrix types.
- Parameters
-
eigenvaluevec Vector of computed eigenvalue eigenvectors Eigenvectors as columns of \( n \times n \) Matrix
Algorithm uses fully pivoted Jacobi rotations.
Two interfaces:
H.eigen_hermitean(E, U, [optional: sort]); E: output is DiagnoalMatrix containing real eigenvalues U: nxn unitary matrix, columns are normalized eigenvectors
auto res = H.eigen_hermitean([optional: sort]); This saves the trouble of defining E and U (see below)
Thus, H = U E U^* and H U = U E
Both interfaces allow optional sorting according to eigenvalues: hila::sort::unsorted [default] hila::sort::ascending / descending
Example:
- Note
- The matrix has to be hermitean/symmetric
- Parameters
-
E diagonal matrix of real eigenvalues U Unitary nxn matrix of eigenvectors sorted sorting of eigenvalues (default:unsorted)
- Returns
- int number of jacobi rotations
Definition at line 1605 of file matrix_linalg.h.
◆ eigen_hermitean() [2/2]
|
inherited |
eigenvalues and -vectors of hermitean/symmetric matrix, alternative interface
result = eigen_hermitean(hila::sort [optional]) returns struct eigen_result<Mtype>, with fields eigenvalues: DiagonalMatrix of eigenvalues eigenvectors: nxn unitary matrix of eigenvectors
Example:
This interface saves the trouble of defining the eigenvalue and -vector variables.
Definition at line 1609 of file matrix_linalg.h.
◆ exp()
|
inlineinherited |
Calculate exp of a square matrix.
Computes up to certain order given as argument
Evaluation is done as:
\begin{align} \exp(H) &= 1 + H + \frac{H^2}{2!} + \frac{H^2}{2!} + \frac{H^3}{3!} \\ &= 1 + H\cdot(1 + (\frac{H}{2})\cdot (1 + (\frac{H}{3})\cdot(...))) \end{align}
Done backwards in order to reduce accumulation of errors
- Parameters
-
mat Matrix to compute exponential for order order to compute exponential to
- Returns
- Matrix_t<n, m, T, MT>
◆ fill()
|
inlineinherited |
◆ gaussian_random()
|
inlineinherited |
Fills Matrix with gaussian random elements from gaussian distribution.
- Parameters
-
width
- Returns
- Mtype&
◆ imag()
|
inlineinherited |
◆ is_square()
◆ is_vector()
◆ mul_trace()
|
inlineinherited |
Multiply with given matrix and compute trace of result.
Slightly cheaper operation than
- Template Parameters
-
p q S MT
- Parameters
-
rm
- Returns
- hila::type_mul<T, S>
◆ mult_by_2x2_left()
|
inlineinherited |
Multiply \( n \times m \)-matrix from the left by \( n \times m \) matrix defined by \( 2 \times 2 \) sub matrix.
The multiplication is defined as follows, let \(M\) as the \( 2 \times 2 \) input matrix and \(B\) be (this) matrix, being the matrix stored in the object this method is called for. Let \(A = I\) be a \( n \times m \) unit matrix. We then set the values of A to be:
\begin{align} A_{p,p} = M_{0,0}, \hspace{5px} A_{p,q} = M_{0,1}, \hspace{5px} A_{q,p} = M_{1,0}, \hspace{5px} A_{q,q} = M_{1,1}. \end{align}
Then the resulting matrix will be:
\begin{align} B = A \cdot B \end{align}
- Template Parameters
-
R Element type of M Mt Matrix type of M
- Parameters
-
p First row and column q Second row and column M \( 2 \times 2\) Matrix to multiply with
◆ mult_by_2x2_right()
|
inlineinherited |
Multiply \( n \times m \)-matrix from the right by \( n \times m \) matrix defined by \( 2 \times 2 \) sub matrix.
See Matrix::mult_by_2x2_left, only difference being that the multiplication is from the right.
- Template Parameters
-
R Element type of M Mt Matrix type of M
- Parameters
-
p First row and column q Second row and column M \( 2 \times 2\) Matrix to multiply with
◆ norm()
◆ operator!=()
|
inlineinherited |
Boolean operator != to check if matrices are exactly different.
if matrices are exactly the same then this will return false
- Template Parameters
-
S Type for MAtrix which is being compared to
- Parameters
-
rhs right hand side Matrix which we are comparing
- Returns
- true
- false
◆ operator*() [1/4]
Multiplication operator RowVector * Vector.
Defined as standard dot product between vectors as long as vectors are of same length
- Template Parameters
-
m Dimensions of RowVector n Dimension of Vector T1 Element type of RowVector T2 Element type of Vector Rt Return type of T1 \( \cdot \) T2 product
- Parameters
-
A RowVector to multiply B Vector to multiply
- Returns
- Rt
◆ operator*() [2/4]
|
inlineinherited |
◆ operator*() [3/4]
|
inlineinherited |
Multiplication operator.
Multiplication type depends on the original types of the multiplied matrices. Defined for the following operations.
Standard matrix multiplication where LHS columns must match RHS rows
Vector * RowVector is same as outer product which is equivalent to a matrix multiplication
- Returns
- Matrix<n, m, R>
◆ operator*() [4/4]
|
inlineinherited |
◆ operator*=() [1/3]
|
inlineinherited |
Multiply assign operator for DiagonalMatrix to Matrix.
Simply defined as matrix multiplication, but since rhs is guaranteed to be diagonal the method is optimized to skip most of the elements.
- Template Parameters
-
S Element type of rhs
- Parameters
-
rhs DiagonalMatrix to multiply
- Returns
- Mtype&
◆ operator*=() [2/3]
|
inlineinherited |
Multiply assign scalar or matrix.
Multiplication works as defined for matrix multiplication
Matrix multiply assign only defined for square matrices, since the matrix dimensions would change otherwise.
- Parameters
-
rhs Matrix to multiply with
- Returns
- template <int p, typename S, typename MT, std::enable_if_t<hila::is_assignable<T &, hila::type_mul<T, S>>::value, int> = 0>&
◆ operator*=() [3/3]
|
inlineinherited |
◆ operator+() [1/4]
|
inlineinherited |
◆ operator+() [2/4]
|
inlineinherited |
Addition operator Matrix + scalar.
Addition operator between matrix and scalar is defined in the usual way, where the scalar is added to the diagonal elements.
- \( M + 2 = M + 2\cdot\mathbb{1} \)
- Template Parameters
-
Mtype Matrix type of b S Type of scalar
- Parameters
-
a Matrix to add to b Scalar to add
- Returns
- Rtype
◆ operator+() [3/4]
|
inlineinherited |
◆ operator+() [4/4]
|
inlineinherited |
Addition operator scalar + Matrix.
Addition operator between Scalar and Matrix is defined in the usual way, where the scalar is treated as diagonal matrix which is then added to. \( 2 + M = 2\cdot\mathbb{1} + M \)
- Template Parameters
-
Mtype Matrix type of a S scalar type of b
- Parameters
-
b Matrix to add a Scalar to add to
- Returns
- Rtype
◆ operator+=() [1/3]
|
inlineinherited |
Addition assign operator for DiagonalMatrix to Matrix.
Works as long as Matrix to assign to is square
- Template Parameters
-
S Element type of rhs
- Parameters
-
rhs DiagonalMatrix to add
- Returns
- Mtype&
◆ operator+=() [2/3]
|
inlineinherited |
◆ operator+=() [3/3]
|
inlineinherited |
Add assign operator scalar to Matrix.
Adds scalar \( a \) to square matrix as \( M + a\cdot\mathbb{1} \)
- Template Parameters
-
S Element type of scalar rhs
- Parameters
-
rhs Sclar to add
- Returns
- Mtype&
◆ operator-() [1/4]
◆ operator-() [2/4]
|
inlineinherited |
Subtraction operator Matrix - scalar.
Subtraction operator between matrix and scalar is defined in the usual way, where the scalar is subtracted from the diagonal elements.
\( M - 2 = M - 2\cdot\mathbb{1} \)
- Template Parameters
-
Mtype Matrix type of a S Scalar type of b
- Parameters
-
a Matrix to subtract from b Scalar to subtract
- Returns
- Rtype
◆ operator-() [3/4]
|
inlineinherited |
◆ operator-() [4/4]
|
inlineinherited |
Subtraction operator Scalar - Matrix.
Subtraction operator between Scalar and Matrix is defined in the usual way, where the scalar is treated as diagonal matrix which is then subtracted from.
\( 2 - M = 2\cdot\mathbb{1} - M \)
- Template Parameters
-
Mtype Matrix type a S Scalar type of b
- Parameters
-
b Scalar to subtract from a Matrix to subtract
- Returns
- Rtype
◆ operator-=() [1/3]
|
inlineinherited |
Subtract assign operator for DiagonalMatrix to Matrix.
Works as long as Matrix to assign to is square
- Template Parameters
-
S Element type of rhs
- Parameters
-
rhs DiagonalMatrix to add
- Returns
- Mtype&
◆ operator-=() [2/3]
◆ operator-=() [3/3]
|
inlineinherited |
Subtract assign operator scalar to Matrix.
Subtract scalar \( a \) to square matrix as \( M - a\cdot\mathbb{1} \)
- Template Parameters
-
S Element type of scalar rhs
- Parameters
-
rhs scalar to add
- Returns
- Mtype&
◆ operator/()
|
inlineinherited |
◆ operator/=() [1/2]
|
inlineinherited |
Divide assign operator for DiagonalMatrix to Matrix.
Well defined since rhs is guaranteed to be Diagonal.
Let M be the Matrix which we divide and D be the DiagonalMatrix which we divide with then the operation is defined as \( M \cdot D^{-1} \).
- Template Parameters
-
S Element type of rhs
- Parameters
-
rhs DiagonalMatrix to divide
- Returns
- Mtype&
◆ operator/=() [2/2]
|
inlineinherited |
Divide assign oprator scalar with matrix.
Divide works as defined for scalar matrix division.
- Template Parameters
-
S Scalar type of rhs
- Parameters
-
rhs Scalar to divide with
- Returns
◆ operator<<()
|
inherited |
◆ operator==()
|
inlineinherited |
◆ operator[]()
Indexing operation [] defined only for vectors.
Example:
- Template Parameters
-
q row size n p column size m
- Parameters
-
i row or vector index depending on which is being indexed
- Returns
- T
◆ outer_product()
|
inlineinherited |
Outer product.
Only works between two Vector objects
- Template Parameters
-
p Row length for rhs q Column length for rhs S Element type for rhs R Type between lhs and rhs multiplication
- Parameters
-
rhs Vector to compute outer product with
- Returns
- Matrix<n, p, R>
◆ permute()
◆ permute_columns()
◆ permute_rows()
◆ project_force()
|
inlinestatic |
Project a complex antisymmetric matrix into the algebra and represent as a complex NxN (momentum) matrix
Definition at line 227 of file representations.h.
◆ random()
dot with matrix - matrix
Generate random elements
Fills Matrix with random elements from uniform distribution
- Returns
- Mtype&
◆ real()
|
inlineinherited |
◆ row()
|
inlineinherited |
Return reference to row in a matrix.
Since the Matrix data is ordered in a row major order accessing a row returns a reference to the row.
- Parameters
-
r index of row to be referenced
- Returns
- const RowVector<m, T>&
◆ rows()
◆ set_column()
|
inlineinherited |
get column of a matrix
Set column of Matrix with Vector if types are assignable
- Template Parameters
-
S Vector type
- Parameters
-
c Index of column to be set v Vector to be set
◆ set_diagonal()
|
inlineinherited |
Set diagonal of square matrix to Vector which is passed to the method.
If called for non square matrix the program will throw an error.
Example:
- Template Parameters
-
S type vector to assign values to
- Parameters
-
v Vector to assign to diagonal
◆ set_row()
|
inlineinherited |
◆ sort() [1/2]
|
inlineinherited |
Sort method for Vector.
Order can be past as argument as either ascending (default) or descending
Ascending
Descending
- Parameters
-
order Order to sort in
- Returns
- Mtype
◆ sort() [2/2]
|
inlineinherited |
Sort method for Vector which returns permutation order.
Order can be past as argument as either ascending (default) or descending
Ascending
Descending
- Template Parameters
-
N
- Parameters
-
permutation order
- Returns
- Mtype
◆ squarenorm()
◆ str()
◆ svd() [1/2]
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inherited |
svd and svd_pivot - alternative interface
svd(hila::sort [optional]) returns struct svd_result<Mtype>, with fields U: nxn unitary matrix singularvalues: DiagonalMatrix of singular values (real, > 0) V: nxn unitary matrix
auto res = M.svd(); now res.U : nxn unitary res.singularvalues : DiagonalMatrix of singular values res.V : nxn unitary
result satifies M = res.U res.singularvalues res.V.dagger()
Definition at line 1630 of file matrix_linalg.h.
◆ svd() [2/2]
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inherited |
Singular value decomposition: divide matrix A = U S V*, where U,V unitary and S diagonal matrix of real singular values. Unpivoted Jacobi rotations.
Unpivoted rotation is generally faster than pivoted (esp. on gpu), but a bit less accurate
Use: M.svd(U, S, V, [sorted]);
Result satisfies M = U S V.dagger()
The algorithm works by one-sided Jacobi rotations.
- U = V = 1
- Loop
- for i=0..(n-2); j=(i+1)..(n-1)
- calculate B_ii, B_ij, B_jj, (B_ij = (A^* A)_ij)
- If B_ij > 0
- diagonalize J^* [ B_ii B_ij ] J = diag(D_ii, D_jj) [ B_ji B_jj ]
- A <- A J , Columns i and j change
- V <- V J
- endfor
- if nothing changed, break Loop
- for i=0..(n-2); j=(i+1)..(n-1)
- Endloop Now A = U S, A^* A = S* S = S^2
- S_ii = A_ii / |A_i| (norm of column i)
- U = A / S
Definition at line 1626 of file matrix_linalg.h.
◆ svd_pivot()
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inherited |
Singular value decomposition: divide matrix A = U S V*, where U,V unitary and S diagonal matrix of real singular values. Fully pivoted Jacobi rotations.
Use: M.svd_pivot(U, S, V, [sorted]);
The algorithm works by one-sided Jacobi rotations.
- U = V = 1
- Compute Gram matrix B = A^* A. B is Hermitean, with B_ii >= 0.
- Loop:
- Find largest B_ij, j > i - full pivoting if |B_ij|^2 <= tol * B_ii B_jj we're done
- find J to diagonalize J^* [ B_ii B_ij ] J = diag(D_ii, D_jj) [ B_ji B_jj ]
- A <- A J , Columns i and j change
- V <- V J
- Recompute B_ik and B_jk, k /= i,j (B_ii = D_ii, B_jj = D_jj, B_ij = 0) (n^2)
- Endloop Now A = U S, A^* A = S* S = S^2
- S_ii = A_ii / |A_i| (norm of column i)
- U = A / S
- Parameters
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U
Definition at line 1635 of file matrix_linalg.h.
◆ trace()
◆ transpose() [1/3]
◆ transpose() [2/3]
◆ transpose() [3/3]
The documentation for this class was generated from the following file:
