#include <Monomials_Utilities.hpp>
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| Eigen::MatrixXi | Exponents (const Polydim::Utilities::Monomials_Data &data) const |
| | Collect the monomial exponents into a single matrix.
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| Eigen::MatrixXd | Vander (const Polydim::Utilities::Monomials_Data &data, const Eigen::MatrixXd &points, const Eigen::Vector3d ¢roid, const double &diam) const |
| | Evaluate the Vandermonde matrix of the scaled monomial basis.
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| template<typename MonomialType > |
| std::vector< Eigen::MatrixXd > | VanderDerivatives (const Polydim::Utilities::Monomials_Data &data, const MonomialType &monomials, const Eigen::MatrixXd &Vander, const double &diam) const |
| | Evaluate the Vandermonde matrices of the first-order partial derivatives.
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| template<typename MonomialType > |
| Eigen::MatrixXd | VanderLaplacian (const Polydim::Utilities::Monomials_Data &data, const MonomialType &monomials, const Eigen::MatrixXd &Vander, const double &diam) const |
| | Evaluate the Vandermonde matrix of the monomial Laplacian.
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| void | MGSOrthonormalize (const Eigen::VectorXd &weights, const Eigen::MatrixXd &Vander, Eigen::MatrixXd &Hmatrix, Eigen::MatrixXd &QmatrixInv, Eigen::MatrixXd &Qmatrix) const |
| | \(L^2(E)\)-orthonormalize the monomial basis via modified Gram–Schmidt.
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◆ Exponents()
template<unsigned short dimension>
Collect the monomial exponents into a single matrix.
- Parameters
-
- Returns
- A \(\text{dimension} \times N_{\text{mon}}\) integer matrix whose \(m\)-th column is the exponent multi-index \(\alpha\) of the \(m\)-th monomial.
◆ MGSOrthonormalize()
template<unsigned short dimension>
| void Polydim::Utilities::Monomials_Utilities< dimension >::MGSOrthonormalize |
( |
const Eigen::VectorXd & |
weights, |
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const Eigen::MatrixXd & |
Vander, |
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Eigen::MatrixXd & |
Hmatrix, |
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Eigen::MatrixXd & |
QmatrixInv, |
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Eigen::MatrixXd & |
Qmatrix |
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) |
| const |
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inline |
\(L^2(E)\)-orthonormalize the monomial basis via modified Gram–Schmidt.
Orthonormalizes the scaled monomials with respect to the weighted \(L^2(E)\) inner product defined by the quadrature weights, using a modified Gram–Schmidt factorization followed by a re-orthogonalization step for numerical stability. The routine returns the change-of-basis matrices mapping between the monomial and the orthonormal basis.
- Parameters
-
| weights | Quadrature weights defining the \(L^2(E)\) inner product. |
| Vander | Vandermonde matrix of the monomials at the quadrature points. |
| [out] | Hmatrix | Mass matrix in the orthonormal basis, \(Q_2^\top Q_2\) (close to the identity). |
| [out] | QmatrixInv | Lower-triangular inverse change-of-basis matrix, \((R_2 R_1)^\top\). |
| [out] | Qmatrix | Change-of-basis matrix to the orthonormal basis (inverse of QmatrixInv). |
◆ Vander()
template<unsigned short dimension>
Evaluate the Vandermonde matrix of the scaled monomial basis.
Builds the matrix whose \((p, m)\) entry is the \(m\)-th scaled monomial evaluated at the \(p\)-th point, \(\prod_d \left(\frac{x_d - x_{E,d}}{h_E}\right)^{\alpha_d}\). Per-coordinate integer powers are precomputed once (VanderPartial) and combined according to each monomial's exponent multi-index.
- Parameters
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| data | Monomial basis data (degree, exponents, number of monomials). |
| points | Evaluation points, one per column. |
| centroid | Element centroid \(\boldsymbol{x}_E\). |
| diam | Element diameter \(h_E\) used to scale the coordinates. |
- Returns
- A \(N_{\text{points}} \times N_{\text{mon}}\) Vandermonde matrix; a column of ones is returned when the basis reduces to the constant.
◆ VanderDerivatives()
template<unsigned short dimension>
template<typename MonomialType >
Evaluate the Vandermonde matrices of the first-order partial derivatives.
For each spatial direction \(i\), returns the Vandermonde matrix of \(\partial_{x_i} m_\alpha\), reusing the already-evaluated monomial values in Vander. Each derivative maps to a lower-degree monomial (via DerivativeIndices) scaled by the derivative coefficient and by the chain-rule factor \(1/h_E\); directions with no contribution yield a zero column.
- Template Parameters
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| MonomialType | Monomial type exposing DerivativeIndices and DerivativeMatrix. |
- Parameters
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| data | Monomial basis data. |
| monomials | Monomial object providing the derivative maps. |
| Vander | Vandermonde matrix of the monomials at the same points (see Vander()). |
| diam | Element diameter \(h_E\). |
- Returns
- A vector of
dimension matrices, the \(i\)-th being the Vandermonde matrix of \(\partial_{x_i}\) of the basis (same shape as Vander).
◆ VanderLaplacian()
template<unsigned short dimension>
template<typename MonomialType >
Evaluate the Vandermonde matrix of the monomial Laplacian.
Returns the matrix whose \(k\)-th column holds \(\Delta m_k\) evaluated at the points, assembled from the precomputed data.Laplacian coefficients and the second-derivative maps (SecondDerivativeIndices), scaled by \(1/h_E^2\). The constant and linear monomials, whose Laplacian vanishes, give zero columns.
- Template Parameters
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| MonomialType | Monomial type exposing SecondDerivativeIndices. |
- Parameters
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| data | Monomial basis data (must provide the Laplacian matrix). |
| monomials | Monomial object providing the second-derivative maps. |
| Vander | Vandermonde matrix of the monomials at the same points (see Vander()). |
| diam | Element diameter \(h_E\). |
- Returns
- A matrix, shaped like
Vander, holding \(\Delta\) of the basis.
The documentation for this struct was generated from the following file:
1.9.8