MatrixPotential¶

About the `MatrixPotential` class¶

The WaveBlocks Project

Inheritance diagram¶

Inheritance diagram of MatrixPotential

Class documentation¶

class WaveBlocks.MatrixPotential¶

This class represents a potential $V\left(x\right)$ . The potential is given as an analytic expression. 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.

calculate_eigenvalues()¶

Calculate the eigenvalues $\lambda_i\left(x\right)$ of the potential $V\left(x\right)$ .

Raises NotImplementedError:
	This is an abstract base class.

calculate_eigenvectors()¶

Calculate the eigenvectors $\nu_i\left(x\right)$ of the potential $V\left(x\right)$ .

Raises NotImplementedError:
	This is an abstract base class.

calculate_exponential(factor=1)¶

Calculate the matrix exponential $E = \exp\left(\alpha M\right)$ .

Raises NotImplementedError:
Parameters:	factor – A prefactor $\alpha$ in the exponential.
	This is an abstract base class.

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.

Raises NotImplementedError:
	This is an abstract base class.

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.

Raises NotImplementedError:
	This is an abstract base class.

calculate_local_quadratic(diagonal_component=None)¶

Calculate the local quadratic approximation matrix $U$ of the potential’s eigenvalues in $\Lambda$ . This function is used for the homogeneous case and takes into account the leading component $\chi$ .

Raises NotImplementedError:
Parameters:	diagonal_component – Specifies the index $i$ of the eigenvalue $\lambda_i$ that gets expanded into a Taylor series $u_i$ .
	This is an abstract base class.

calculate_local_remainder(diagonal_component=0)¶

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 $\chi$ .

Raises NotImplementedError:
Parameters:	diagonal_component – Specifies the index $\chi$ of the leading component $\lambda_\chi$ .
	This is an abstract base class.

evaluate_at(nodes, component=None)¶

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

Raises NotImplementedError:
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.
	This is an abstract base class.

evaluate_eigenvalues_at(nodes, diagonal_component=None)¶

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

Raises NotImplementedError:
Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the eigenvalues at. diagonal_component – The index $i$ of the eigenvalue $\lambda_i$ that gets evaluated or ‘None’ to evaluate all.
	This is an abstract base class.

evaluate_eigenvectors_at(nodes)¶

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

Raises NotImplementedError:
Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the eigenvectors at.
	This is an abstract base class.

evaluate_exponential_at(nodes)¶

Evaluate the exponential of the potential matrix $V$ at some grid nodes $\gamma$ .

Raises NotImplementedError:
Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the exponential at.
	This is an abstract base class.

evaluate_hessian_at(nodes, component=None)¶

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

Raises NotImplementedError:
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)
	This is an abstract base class.

evaluate_jacobian_at(nodes, component=None)¶

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

Raises NotImplementedError:
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)
	This is an abstract base class.

evaluate_local_quadratic_at(nodes)¶

Numerically evaluate the local quadratic approximation matrix $U$ of the potential’s eigenvalues in $\Lambda$ at the given grid nodes $\gamma$ . This function is used for the homogeneous case and takes into account the leading component $\chi$ .

Raises NotImplementedError:
Parameters:	nodes – The grid nodes $\gamma$ we want to evaluate the quadratic approximation at.
	This is an abstract base class.

evaluate_local_remainder_at(position, nodes, 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 $W$ .

Raises NotImplementedError:
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.
	This is an abstract base class.

get_number_components()¶

Raises NotImplementedError:
Returns:	The number $N$ of components the potential supports.
	This is an abstract base class.

project_to_canonical(nodes, values, basis=None)¶

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

Raises NotImplementedError:
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.
	This is an abstract base class.

project_to_eigen(nodes, values, basis=None)¶

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

Raises NotImplementedError:
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 basisvectors $nu_i$ . Allows to use this function for external data, similar to a static function.
	This is an abstract base class.

MatrixPotential¶

About the `MatrixPotential` class¶

Inheritance diagram¶

Class documentation¶

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MatrixPotential¶

About the MatrixPotential class¶

Inheritance diagram¶

Class documentation¶

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About the `MatrixPotential` class¶