- Generated on Wed Nov 20 2024 16:28:30 for Theoretica by
1.9.1
|
Theoretica
A C++ numerical and automatic mathematical library
|
Linear algebra routines. More...
Go to the source code of this file.
Namespaces | |
| theoretica | |
| Main namespace of the library which contains all functions and objects. | |
| theoretica::algebra | |
| Linear algebra routines. | |
Functions | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::mat_error (Matrix &m) |
| Overwrite the given matrix with the error matrix with NaN values on the diagonal and zeroes everywhere else. More... | |
| template<typename Vector > | |
| Vector & | theoretica::algebra::vec_error (Vector &v) |
| Overwrite the given vector with the error vector with NaN values. More... | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::make_identity (Matrix &m) |
| Overwrite a matrix with the identity matrix. More... | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::mat_zeroes (Matrix &m) |
| Overwrite a matrix with all zeroes. More... | |
| template<typename Vector > | |
| Vector & | theoretica::algebra::vec_zeroes (Vector &v) |
| Overwrite a vector with all zeroes. More... | |
| template<typename Matrix1 , typename Matrix2 > | |
| Matrix1 & | theoretica::algebra::mat_copy (Matrix1 &dest, const Matrix2 &src) |
| Copy a matrix by overwriting another. More... | |
| template<typename Vector1 , typename Vector2 > | |
| Vector1 & | theoretica::algebra::vec_copy (Vector1 &dest, const Vector2 &src) |
| Copy a vector by overwriting another. More... | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::mat_swap_rows (Matrix &A, unsigned int row1, unsigned int row2) |
| Swap two rows of a matrix, given the matrix and the two indices of the rows. More... | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::mat_swap_cols (Matrix &A, unsigned int col1, unsigned int col2) |
| Swap two columns of a matrix, given the matrix and the two indices of the columns. More... | |
| template<typename Matrix , typename Type = matrix_element_t<Matrix>> | |
| Matrix & | theoretica::algebra::mat_shift_diagonal (Matrix &A, const Type &sigma) |
| Shift the diagonal of a matrix by the given amount, overwriting the matrix itself, as \((A + \sigma I)\). More... | |
| template<typename Type > | |
| auto | theoretica::algebra::pair_inner_product (const Type &v_i, const Type &w_i) |
| Compute the contribution of the inner product between a pair of elements of two vectors, automatically selecting whether to compute the conjugate or not. More... | |
| template<typename Type > | |
| auto | theoretica::algebra::pair_inner_product (const complex< Type > &v_i, const complex< Type > &w_i) |
| Compute the contribution of the inner product between a pair of elements of two vectors, automatically selecting whether to compute the conjugate or not. More... | |
| template<typename Vector > | |
| auto | theoretica::algebra::sqr_norm (const Vector &v) |
| Returns the square of the Euclidean/Hermitian norm of the given vector. More... | |
| template<typename Vector > | |
| auto | theoretica::algebra::norm (const Vector &v) |
| Returns the Euclidean/Hermitian norm of the given vector. More... | |
| template<typename Vector > | |
| Vector | theoretica::algebra::normalize (const Vector &v) |
| Returns the normalized vector. More... | |
| template<typename Vector > | |
| Vector & | theoretica::algebra::make_normalized (Vector &v) |
| Normalize a given vector overwriting it. More... | |
| template<typename Vector1 , typename Vector2 > | |
| auto | theoretica::algebra::dot (const Vector1 &v, const Vector2 &w) |
| Computes the dot product between two vectors, using the Hermitian form if needed. More... | |
| template<typename Vector1 , typename Vector2 > | |
| Vector1 | theoretica::algebra::cross (const Vector1 &v1, const Vector2 &v2) |
| Compute the cross product between two tridimensional vectors. More... | |
| template<typename Matrix , typename MatrixT = Matrix> | |
| MatrixT | theoretica::algebra::transpose (const Matrix &m) |
| Returns the transpose of the given matrix. More... | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::make_transposed (Matrix &m) |
| Transpose the given matrix. More... | |
| template<typename Matrix1 , typename Matrix2 > | |
| Matrix1 & | theoretica::algebra::transpose (Matrix1 &dest, const Matrix2 &src) |
| Compute the transpose matrix and write the result to another matrix. More... | |
| template<typename Matrix , typename MatrixT = Matrix> | |
| MatrixT | theoretica::algebra::hermitian (const Matrix &m) |
| Returns the hermitian of the given matrix. More... | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::make_hermitian (Matrix &m) |
| Compute the hermitian of a given matrix and overwrite it. More... | |
| template<typename Matrix1 , typename Matrix2 > | |
| Matrix1 & | theoretica::algebra::hermitian (Matrix1 &dest, const Matrix2 &src) |
| Hermitian (conjugate transpose) of a matrix. More... | |
| template<typename Matrix > | |
| auto | theoretica::algebra::trace (const Matrix &m) |
| Compute the trace of the given matrix. More... | |
| template<typename Matrix > | |
| auto | theoretica::algebra::diagonal_product (const Matrix &m) |
| Compute the product of the elements of the main diagonal of a generic matrix. More... | |
| template<typename Field , typename Matrix > | |
| Matrix & | theoretica::algebra::mat_scalmul (Field a, Matrix &m) |
| Multiply a matrix by a scalar of any compatible type. More... | |
| template<typename Field , typename Matrix1 , typename Matrix2 > | |
| Matrix1 & | theoretica::algebra::mat_scalmul (Matrix1 &dest, Field a, const Matrix2 &src) |
| Multiply a matrix by a scalar of any compatible type which can be cast to the type of element of the output matrix. More... | |
| template<typename Field , typename Vector > | |
| Vector & | theoretica::algebra::vec_scalmul (Field a, Vector &v) |
| Multiply a vector by a scalar of any compatible type. More... | |
| template<typename Field , typename Vector1 , typename Vector2 > | |
| Vector1 & | theoretica::algebra::vec_scalmul (Vector1 &dest, Field a, const Vector2 &src) |
| Multiply a vector by a scalar of any compatible type which can be cast to the type of element of the output vector. More... | |
| template<typename Matrix , typename Vector > | |
| Vector & | theoretica::algebra::apply_transform (const Matrix &A, Vector &v) |
| Apply a matrix transformation to a vector and store the result in the vector. More... | |
| template<typename Matrix , typename Vector > | |
| Vector | theoretica::algebra::transform (const Matrix &A, const Vector &v) |
| Returns the matrix transformation of a vector. More... | |
| template<typename Matrix , typename Vector1 , typename Vector2 > | |
| Vector1 & | theoretica::algebra::transform (Vector1 &res, const Matrix &A, const Vector2 &v) |
| Apply a matrix transformation to a vector and store the result in the vector. More... | |
| template<typename Matrix1 , typename Matrix2 > | |
| Matrix2 & | theoretica::algebra::mat_sum (Matrix1 &A, const Matrix2 &B) |
| Sum two matrices and store the result in the first matrix. More... | |
| template<typename Matrix1 , typename Matrix2 , typename Matrix3 > | |
| Matrix1 & | theoretica::algebra::mat_sum (Matrix1 &res, const Matrix2 &A, const Matrix3 &B) |
| Sum two matrices and store the result in another matrix Equivalent to the operation res = A + B. More... | |
| template<typename Matrix1 , typename Matrix2 > | |
| Matrix2 & | theoretica::algebra::mat_diff (Matrix1 &A, const Matrix2 &B) |
| Subtract two matrices and store the result in the first matrix. More... | |
| template<typename Matrix1 , typename Matrix2 , typename Matrix3 > | |
| Matrix1 & | theoretica::algebra::mat_diff (Matrix1 &res, const Matrix2 &A, const Matrix3 &B) |
| Subtract two matrices and store the result in another matrix Equivalent to the operation res = A - B. More... | |
| template<typename Field1 , typename Matrix1 , typename Field2 , typename Matrix2 > | |
| Matrix2 & | theoretica::algebra::mat_lincomb (Field1 alpha, Matrix1 &A, Field2 beta, const Matrix2 &B) |
| Compute the linear combination of two matrices and store the result in the first matrix. More... | |
| template<typename Matrix1 , typename Field1 , typename Matrix2 , typename Field2 , typename Matrix3 > | |
| Matrix1 & | theoretica::algebra::mat_lincomb (Matrix1 &res, Field1 alpha, const Matrix2 &A, Field2 beta, const Matrix3 &B) |
| Compute the linear combination of two matrices and store the result in the first matrix. More... | |
| template<typename Matrix1 , typename Matrix2 , typename Matrix3 = Matrix1> | |
| Matrix3 | theoretica::algebra::mat_mul (const Matrix1 &A, const Matrix2 &B) |
| Multiply two matrices and store the result in the first matrix, equivalent to the operation \(R = A B\). More... | |
| template<typename Matrix1 , typename Matrix2 , typename Matrix3 > | |
| Matrix1 & | theoretica::algebra::mat_mul (Matrix1 &R, const Matrix2 &A, const Matrix3 &B) |
| Multiply two matrices and store the result in another matrix, equivalent to the operation \(R = A B\). More... | |
| template<typename Matrix1 , typename Matrix2 , typename Matrix3 = Matrix1> | |
| Matrix3 | theoretica::algebra::mat_transpose_mul (const Matrix1 &A, const Matrix2 &B) |
| Multiply the transpose of a matrix by another matrix, equivalent to the operation \(R = A^T B\). More... | |
| template<typename Matrix1 , typename Matrix2 , typename Matrix3 = Matrix1> | |
| Matrix3 | theoretica::algebra::mat_mul_transpose (const Matrix1 &A, const Matrix2 &B) |
| Multiply a matrix by the transpose of another matrix, equivalent to the operation \(R = A B^T\). More... | |
| template<typename Matrix1 , typename Matrix2 > | |
| bool | theoretica::algebra::mat_equals (const Matrix1 &A, const Matrix2 &B, real tolerance=10 *MACH_EPSILON) |
| Checks whether two matrices are equal. More... | |
| template<typename Vector1 , typename Vector2 > | |
| Vector2 & | theoretica::algebra::vec_sum (Vector1 &v1, const Vector2 &v2) |
| Sum two vectors and store the result in the first vector. More... | |
| template<typename Vector1 , typename Vector2 , typename Vector3 > | |
| Vector1 & | theoretica::algebra::vec_sum (Vector1 &res, const Vector2 &v1, const Vector3 &v2) |
| Sum two vectors and store the result in another vector Equivalent to the operation res = v1 + v2. More... | |
| template<typename Vector1 , typename Vector2 > | |
| Vector2 & | theoretica::algebra::vec_diff (Vector1 &v1, const Vector2 &v2) |
| Subtract two vectors and store the result in the first vector. More... | |
| template<typename Vector1 , typename Vector2 , typename Vector3 > | |
| Vector1 & | theoretica::algebra::vec_diff (Vector1 &res, const Vector2 &v1, const Vector3 &v2) |
| Subtract two vectors and store the result in another vector Equivalent to the operation res = v1 - v2. More... | |
| template<typename Matrix > | |
| bool | theoretica::algebra::is_square (const Matrix &m) |
| Returns whether the matrix is square. More... | |
| template<typename Matrix > | |
| bool | theoretica::algebra::is_diagonal (const Matrix &m, real tolerance=10 *MACH_EPSILON) |
| Returns whether the matrix is diagonal. More... | |
| template<typename Matrix > | |
| bool | theoretica::algebra::is_symmetric (const Matrix &m, real tolerance=ALGEBRA_ELEMENT_TOL) |
| Returns whether the matrix is symmetric. More... | |
| template<typename Matrix > | |
| bool | theoretica::algebra::is_lower_triangular (const Matrix &m, real tolerance=ALGEBRA_ELEMENT_TOL) |
| Returns whether the matrix is lower triangular. More... | |
| template<typename Matrix > | |
| bool | theoretica::algebra::is_upper_triangular (const Matrix &m, real tolerance=ALGEBRA_ELEMENT_TOL) |
| Returns whether the matrix is upper triangular. More... | |
| template<typename Matrix , enable_matrix< Matrix > = true> | |
| real | theoretica::algebra::density (const Matrix &A, real tolerance=1E-12) |
| Compute the density of a matrix, counting the proportion of non-zero (bigger in module than the given tolerance) elements with respect to the total number of elements. More... | |
| template<typename Matrix , enable_matrix< Matrix > = true> | |
| real | theoretica::algebra::sparsity (const Matrix &A, real tolerance=1E-12) |
| Compute the sparsity of a matrix, counting the proportion of zero (smaller in module than the given tolerance) elements with respect to the total number of elements. More... | |
| template<typename Matrix1 , typename Matrix2 , typename Matrix3 > | |
| void | theoretica::algebra::decompose_lu (const Matrix1 &A, Matrix2 &L, Matrix3 &U) |
| Decompose a square matrix to two triangular matrices, L and U where L is lower and U is upper, so that \(A = LU\). More... | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::decompose_lu_inplace (Matrix &A) |
| Decompose a square matrix to two triangular matrices, L and U where L is lower and U is upper, so that \(A = LU\) overwriting the matrix A with the elements of both matrices, omitting the diagonal of L (equal to all ones). More... | |
| template<typename Matrix > | |
| Matrix | theoretica::algebra::decompose_cholesky (const Matrix &A) |
| Decompose a symmetric positive definite matrix into a triangular matrix so that \(A = L L^T\) using Cholesky decomposition. More... | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::decompose_cholesky_inplace (Matrix &A) |
| Decompose a symmetric positive definite matrix in-place, overwriting the starting matrix, without using additional space. More... | |
| template<typename Matrix , typename Vector > | |
| Vector | theoretica::algebra::solve_triangular_lower (const Matrix &L, const Vector &b) |
| Solve the linear system \(L \vec x = b\) for lower triangular \(L\). More... | |
| template<typename Matrix , typename Vector > | |
| Vector | theoretica::algebra::solve_triangular_upper (const Matrix &U, const Vector &b) |
| Solve the linear system \(U \vec x = b\) for upper triangular \(U\). More... | |
| template<typename Matrix , typename Vector > | |
| Vector | theoretica::algebra::solve_triangular (const Matrix &T, const Vector &b) |
| Solve the linear system \(T \vec x = b\) for triangular \(T\). More... | |
| template<typename Matrix , typename Vector > | |
| Vector & | theoretica::algebra::solve_lu_inplace (const Matrix &A, Vector &b) |
| Solve the linear system \(A \vec x = \vec b\), finding \(\vec x\), where the matrix A has already undergone in-place LU decomposition. More... | |
| template<typename Matrix , typename Vector > | |
| Vector | theoretica::algebra::solve_lu (Matrix A, Vector b) |
| Solve the linear system \(A \vec x = \vec b\), finding \(\vec x\). More... | |
| template<typename Matrix1 , typename Matrix2 , typename Vector > | |
| Vector | theoretica::algebra::solve_lu (const Matrix1 &L, const Matrix2 &U, const Vector &b) |
| Use the LU decomposition of a matrix to solve its associated linear system, solving \(A \vec x = \vec b\) for \(\vec b\). More... | |
| template<typename Matrix , typename Vector > | |
| Vector | theoretica::algebra::solve_cholesky (const Matrix &L, const Vector &b) |
| Solve a linear system \(A \vec x = \vec b\) defined by a symmetric positive definite matrix, using the Cholesky decomposition \(L\) constructed so that \(A = LL^T\). More... | |
| template<typename Matrix , typename Vector > | |
| Vector | theoretica::algebra::solve (const Matrix &A, const Vector &b) |
| Solve the linear system \(A \vec x = \vec b\), finding \(\vec x\) using the best available algorithm. More... | |
| template<typename Matrix1 , typename Matrix2 > | |
| Matrix1 & | theoretica::algebra::inverse (Matrix1 &dest, const Matrix2 &src) |
| Invert the given matrix. More... | |
| template<typename Matrix , typename MatrixInv = Matrix> | |
| MatrixInv | theoretica::algebra::inverse (const Matrix &m) |
| Returns the inverse of the given matrix. More... | |
| template<typename Matrix > | |
| Matrix & | theoretica::algebra::invert (Matrix &m) |
| Invert the given matrix and overwrite it. More... | |
| template<typename Matrix > | |
| auto | theoretica::algebra::det (const Matrix &A) |
| Compute the determinant of a square matrix. More... | |
| template<typename Matrix , typename Vector > | |
| auto | theoretica::algebra::rayleigh_quotient (const Matrix &A, const Vector &x) |
| Compute the Rayleigh quotient \(\frac{x^T A x}{x^T x}\) of a vector with respect to a matrix. More... | |
| template<typename Matrix , typename Vector > | |
| auto | theoretica::algebra::eigenvalue_power (const Matrix &A, const Vector &x, real tolerance=ALGEBRA_EIGEN_TOL, unsigned int max_iter=ALGEBRA_EIGEN_ITER) |
| Find the biggest eigenvalue in module \(|\lambda_i|\) of a square matrix using the power method (Von Mises iteration). More... | |
| template<typename Matrix , typename Vector1 , typename Vector2 = Vector1> | |
| auto | theoretica::algebra::eigenpair_power (const Matrix &A, const Vector1 &x, Vector2 &v, real tolerance=ALGEBRA_EIGEN_TOL, unsigned int max_iter=ALGEBRA_EIGEN_ITER) |
| Find the biggest eigenvalue in module \(|\lambda_i|\) of a square matrix and its corresponding eigenvector (eigenpair), using the power method (Von Mises iteration). More... | |
| template<typename Matrix , typename Vector > | |
| auto | theoretica::algebra::eigenvalue_inverse (const Matrix &A, const Vector &x, real tolerance=ALGEBRA_EIGEN_TOL, unsigned int max_iter=ALGEBRA_EIGEN_ITER) |
| Find the eigenvalue with the smallest inverse \(\frac{1}{|\lambda_i|}\) of a square matrix, using the inverse power method with parameter equal to 0. More... | |
| template<typename Matrix , typename Vector1 , typename Vector2 > | |
| auto | theoretica::algebra::eigenpair_inverse (const Matrix &A, const Vector1 &x, Vector2 &v, real tolerance=ALGEBRA_EIGEN_TOL, unsigned int max_iter=ALGEBRA_EIGEN_ITER) |
| Find the eigenvalue with the smallest inverse \(\frac{1}{|\lambda_i|}\) of a square matrix and its corresponding eigenvector, using the inverse power method with parameter equal to 0. More... | |
| template<typename Matrix , typename Vector , typename T = matrix_element_t<Matrix>> | |
| Vector | theoretica::algebra::eigenvector_inverse (const Matrix &A, const T &lambda, const Vector &x, real tolerance=ALGEBRA_EIGEN_TOL, unsigned int max_iter=ALGEBRA_EIGEN_ITER) |
| Find the eigenvector associated with a given approximated eigenvalue using the inverse power method. More... | |
| template<typename Matrix , typename Vector , typename T = matrix_element_t<Matrix>> | |
| auto | theoretica::algebra::eigenvalue_rayleigh (const Matrix &A, const T &lambda, const Vector &x, real tolerance=ALGEBRA_EIGEN_TOL, unsigned int max_iter=ALGEBRA_EIGEN_ITER) |
| Compute an eigenvalue of a square matrix using the Rayleigh quotient iteration method. More... | |
| template<typename Matrix , typename Vector1 , typename Vector2 , typename T = matrix_element_t<Matrix>> | |
| auto | theoretica::algebra::eigenpair_rayleigh (const Matrix &A, const T &lambda, const Vector1 &x, Vector2 &v, real tolerance=ALGEBRA_EIGEN_TOL, unsigned int max_iter=ALGEBRA_EIGEN_ITER) |
| Compute an eigenvalue of a square matrix and its corresponding eigenvector (eigenpair) using the Rayleigh quotient iteration method. More... | |
Linear algebra routines.
This file implements all linear algebra routines of the library, using templates and type traits. The Matrix type must be a class with these methods:
1.9.1