MatrixPotentialMS¶

About the `MatrixPotentialMS` class¶

The WaveBlocks Project

Inheritance diagram¶

Inheritance diagram of MatrixPotentialMS

Class documentation¶

class WaveBlocks.MatrixPotentialMS(expression, variables)¶

This class represents a matrix potential $V\left(x\right)$ . The potential is given as an analytical expression with a matrix of size bigger than $2 \times 2$ . Some calculations with the potential are supported. For example calculation of eigenvalues and exponentials and numerical evaluation. Further, there are methods for splitting the potential into a Taylor expansion and for basis transformations between canonical and eigenbasis. All methods use numerical techniques because symbolical calculations are unfeasible.

calculate_eigenvalues()¶: Calculate the eigenvalues $\lambda_i\left(x\right)$ of the potential $V\left(x\right)$ . We do the calculations with numerical tools. The multiplicities are taken into account.

Note

Note: the eigenvalues are memoized for later reuse.

calculate_eigenvectors()¶: Calculate the two eigenvectors $nu_i\left(x\right)$ of the potential $V\left(x\right)$ . We do the calculations with numerical tools.

Note

The eigenvectors are memoized for later reuse.

calculate_exponential(factor=1)¶

Calculate the matrix exponential $E = \exp\left(\alpha M\right)$ . In the case where the matrix is of size bigger than $2 \times 2$ symbolical calculations become unfeasible. We use numerical approximations to determine the matrix exponential.

Parameters:	factor – A prefactor $\alpha$ in the exponential.

calculate_hessian()¶: Calculate the hessian matrix for each component $V_{i,j}$ of the potential. For potentials which depend only one variable $x$ , this equals the second derivative.

calculate_jacobian()¶: Calculate the jacobian matrix for each component $V_{i,j}$ of the potential. For potentials which depend only one variable $x$ , this equals the first derivative.

calculate_local_quadratic(diagonal_component=None)¶

Calculate the local quadratic approximation matrix $U$ of the potential’s eigenvalues in $\Lambda$ . This function can be used for the homogeneous case and takes into account the leading component $\chi$ . If the parameter $\chi$ is not given, calculate the local quadratic approximation matrix $U$ of all the potential’s eigenvalues in $\Lambda$ . This function is used for the inhomogeneous case.

Parameters:	diagonal_component – Specifies the index $i$ of the eigenvalue $\lambda_i$ that gets expanded into a Taylor series $u_i$ .

calculate_local_remainder(diagonal_component=None)¶

Calculate the non-quadratic remainder matrix $W$ of the quadratic approximation matrix $U$ of the potential’s eigenvalue matrix $\Lambda$ . This function is used for the homogeneous case and takes into account the leading component $\chi$ .

Parameters:	diagonal_component – Specifies the index $\chi$ of the leading component $\lambda_\chi$ .

evaluate_at(nodes, component=None, as_matrix=True)¶

Evaluate the potential matrix elementwise at some given grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the potential at. component – The component $V_{i,j}$ that gets evaluated or ‘None’ to evaluate all. as_matrix – Returns the whole matrix $\Lambda$ instead of only a list with the eigenvalues $\lambda_i$ .
Returns:	A list with the $N^2$ entries evaluated at the nodes.

evaluate_eigenvalues_at(nodes, component=None, as_matrix=False)¶

Evaluate the eigenvalues $\lambda_i\left(x\right)$ at some grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the eigenvalues at. component – The index $i$ of the eigenvalue $\lambda_i$ that gets evaluated. as_matrix – Returns the whole matrix $\Lambda$ instead of only a list with the eigenvalues $\lambda_i$ .
Returns:	A sorted list with $N$ entries for all the eigenvalues evaluated at the nodes. Or a single value if a component was specified.

evaluate_eigenvectors_at(nodes)¶

Evaluate the eigenvectors $nu_i\left(x\right)$ at some grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the eigenvectors at.
Returns:	A list with the $N$ eigenvectors evaluated at the given nodes.

evaluate_exponential_at(nodes)¶

Evaluate the exponential of the potential matrix $V$ at some grid nodes $\gamma$ . For matrices of size $> 2$ we do completely numerical exponentation.

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the exponential at.
Returns:	The numerical approximation of the matrix exponential at the given grid nodes.

evaluate_hessian_at(nodes, component=None)¶

Evaluate the hessian at some grid nodes $\gamma$ for each component $V_{i,j}$ of the potential.

Parameters:	nodes – The grid nodes $\gamma$ the hessian gets evaluated at. component – The index tuple $\left(i,j\right)$ that specifies the potential’s entry of which the hessian is evaluated. (Or ‘None’ to evaluate all)
Returns:	Either a list or a single value depending on the optional parameters.

evaluate_jacobian_at(nodes, component=None)¶

Evaluate the jacobian at some grid nodes $\gamma$ for each component $V_{i,j}$ of the potential.

Parameters:	nodes – The grid nodes $\gamma$ the jacobian gets evaluated at. component – The index tuple $\left(i,j\right)$ that specifies the potential’s entry of which the jacobian is evaluated. (Defaults to ‘None’ to evaluate all)
Returns:	Either a list or a single value depending on the optional parameters.

evaluate_local_quadratic_at(nodes, diagonal_component)¶

Numerically evaluate the local quadratic approximation matrix $U$ of the potential’s eigenvalues in $\Lambda$ at the given grid nodes $\gamma$ .

Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the quadratic approximation at. diagonal_component – Specifies the index $i$ of the eigenvalue $\lambda_i$ that gets expanded into a Taylor series $u_i$ .
Returns:	A list of arrays or a single array containing the values of $U_{i,j}$ at the nodes $\gamma$ .

evaluate_local_remainder_at(position, nodes, diagonal_component=None, component=None)¶

Numerically evaluate the non-quadratic remainder matrix $W$ of the quadratic approximation matrix $U$ of the potential’s eigenvalues in $\Lambda$ at the given nodes $\gamma$ . This function is used for the homogeneous and the inhomogeneous case and just evaluates the remainder matrix $W$ .

Parameters:	position – The point $q$ where the Taylor series is computed. nodes – The grid nodes $\gamma$ we want to evaluate the potential at. component – The component $\left(i,j\right)$ of the remainder matrix $W$ that is evaluated.
Returns:	A list with a single entry consisting of an array containing the values of $W$ at the nodes $\gamma$ .

get_number_components()¶

Returns:	The number $N$ of components the potential supports. This is also the size of the matrix.

potential = None¶: The matrix of the potential $V\left(x\right)$ .

project_to_canonical(nodes, values, basis=None)¶

Project a given vector from the potential’s eigenbasis to the canonical basis.

Parameters:	nodes – The grid nodes $\gamma$ for the pointwise transformation. values – The list of vectors $\varphi_i$ containing the values we want to transform. basis – A list of basis vectors $nu_i$ . Allows to use this function for external data, similar to a static function.
Returns:	Returned is another list containing the projection of the values into the eigenbasis.

project_to_eigen(nodes, values, basis=None)¶

Project a given vector from the canonical basis to the eigenbasis of the potential.

Parameters:	nodes – The grid nodes $\gamma$ for the pointwise transformation. values – The list of vectors $\phi_i$ containing the values we want to transform. basis – A list of basisvectors $nu_i$ . Allows to use this function for external data, similar to a static function.
Returns:	Returned is another list containing the projection of the values into the eigenbasis.

x = None¶: The variable $x$ that represents position space.

MatrixPotentialMS¶

About the `MatrixPotentialMS` class¶

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MatrixPotentialMS¶

About the MatrixPotentialMS class¶

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About the `MatrixPotentialMS` class¶