Eigenfunctions¶

This modules provides eigenfunctions for a certain set of second order spatial operators. Therefore functions for the computation of the corresponding eigenvalues are included. The functions which compute the eigenvalues are deliberately separated from the predefined eigenfunctions in order to handle transformations and reduce effort within the controller implementation.

class AddMulFunction(function)¶

Bases: object

(Temporary) Function class which can multiplied with scalars and added with functions. Only needed to compute the matrix (of scalars) vector (of functions) product in FiniteTransformFunction. Will be no longer needed when Function is overloaded with __add__ and __mul__ operator.

Parameters: function (callable) –

class Base(fractions, matching_base_lbls=None, intermediate_base_lbls=None)¶

Bases: pyinduct.core.ApproximationBasis

Base class for approximation bases.

In general, a Base is formed by a certain amount of BaseFractions and therefore forms finite-dimensional subspace of the distributed problem’s domain. Most of the time, the user does not need to interact with this class.

Parameters

fractions (iterable of BaseFraction) – List, array or dict of BaseFraction’s
matching_base_lbls (list of str) – List of labels from exactly matching bases, for which no transformation is necessary. Useful for transformations from bases that ‘live’ in different function spaces but evolve with the same time dynamic/coefficients (e.g. modal bases).
intermediate_base_lbls (list of str) – If it is certain that this base instance will be asked (as destination base) to return a transformation to a source base, whose implementation is cumbersome, its label can be provided here. This will trigger the generation of the transformation using build-in features. The algorithm, implemented in get_weights_transformation is then called again with the intermediate base as destination base and the ‘old’ source base. With this technique arbitrary long transformation chains are possible, if the provided intermediate bases again define intermediate bases.

derive(self, order)¶

Basic implementation of derive function. Empty implementation, overwrite to use this functionality.

Parameters: order (numbers.Number) – derivative order
Returns: derived object
Return type: Base

function_space_hint(self)¶: Hint that returns properties that characterize the functional space of the fractions. It can be used to determine if function spaces match.

Note

Overwrite to implement custom functionality.

get_attribute(self, attr)¶

Retrieve an attribute from the fractions of the base.

Parameters: attr (str) – Attribute to query the fractions for.
Returns: Array of len(fractions) holding the attributes. With None entries if the attribute is missing.
Return type: np.ndarray

raise_to(self, power)¶

Factory method to obtain instances of this base, raised by the given power.

Parameters: power – power to raise the basis onto.

scalar_product_hint(self)¶: Hint that returns steps for scalar product calculation with elements of this base.

Note

Overwrite to implement custom functionality.

scale(self, factor)¶

Factory method to obtain instances of this base, scaled by the given factor.

Parameters: factor – factor or function to scale this base with.

transformation_hint(self, info)¶

Method that provides a information about how to transform weights from one BaseFraction into another.

In Detail this function has to return a callable, which will take the weights of the source- and return the weights of the target system. It may have keyword arguments for other data which is required to perform the transformation. Information about these extra keyword arguments should be provided in form of a dictionary whose keys are keyword arguments of the returned transformation handle.

Note

This implementation covers the most basic case, where the two BaseFraction’s are of same type. For any other case it will raise an exception. Overwrite this Method in your implementation to support conversion between bases that differ from yours.

Parameters: info – TransformationInfo
Raises: NotImplementedError –
Returns: Transformation handle

class Domain(bounds=None, num=None, step=None, points=None)¶

Bases: object

Helper class that manages ranges for data evaluation, containing parameters.

Parameters

bounds (tuple) – Interval bounds.
num (int) – Number of points in interval.
step (numbers.Number) – Distance between points (if homogeneous).
points (array_like) – Points themselves.

Note

If num and step are given, num will take precedence.

bounds(self)¶

ndim(self)¶

points(self)¶

step(self)¶

class FiniteTransformFunction(function, M, l, scale_func=None, nested_lambda=False)¶

Bases: pyinduct.core.Function

This class provides a transformed Function $\bar x(z)$ through the transformation $\bar{\boldsymbol{\xi}} = T * \boldsymbol \xi$ , with the function vector $\boldsymbol \xi\in\mathbb R^{2n}$ and with a given matrix $T\in\mathbb R^{2n\times 2n}$ . The operator $*$ denotes the matrix (of scalars) vector (of functions) product. The interim result $\bar{\boldsymbol{\xi}}$ is a vector $\bar{\boldsymbol{\xi}} = (\bar\xi_{1,0},...,\bar\xi_{1,n-1}, \bar\xi_{2,0},...,\bar\xi_{2,n-1})^T$ of functions

$&\bar\xi_{1,j} = \bar x(jl_0 + z), \qquad j=0,...,n-1, \quad l_0=l/n, \quad z\in[0,l_0] &\bar\xi_{2,j} = \bar x(l - jl_0 + z).$

Finally, the provided function $\bar x(z)$ is given through $\bar\xi_{1,0},...,\bar\xi_{1,n-1}$ .

Note

For a more extensive documentation see section 4.2 in:

Wang, S. und F. Woittennek: Backstepping-Methode für parabolische Systeme mit punktförmigem inneren Eingriff. Automatisierungstechnik, 2015. http://dx.doi.org/10.1515/auto-2015-0023

Parameters

function (callable) – Function $x(z)$ that will act as start for the generation of $2n$ Functions $\xi_{i,j}$ in $\boldsymbol\xi = (\xi_{1,0},...,\xi_{1,n-1}, \xi_{2,0},...,\xi_{2,n-1})^T$ .
M (numpy.ndarray) – Matrix $T\in\mathbb R^{2n\times 2n}$ of scalars.
l (numbers.Number) – Length of the domain ( $z\in [0,l]$ ).

class Function(eval_handle, domain=- np.inf, np.inf, nonzero=- np.inf, np.inf, derivative_handles=None)¶

Bases: pyinduct.core.BaseFraction

Most common instance of a BaseFraction. This class handles all tasks concerning derivation and evaluation of functions. It is used broad across the toolbox and therefore incorporates some very specific attributes. For example, to ensure the accurateness of numerical handling functions may only evaluated in areas where they provide nonzero return values. Also their domain has to be taken into account. Therefore the attributes domain and nonzero are provided.

To save implementation time, ready to go version like LagrangeFirstOrder are provided in the pyinduct.simulation module.

For the implementation of new shape functions subclass this implementation or directly provide a callable eval_handle and callable derivative_handles if spatial derivatives are required for the application.

Parameters

eval_handle (callable) – Callable object that can be evaluated.
domain ((list of) tuples) – Domain on which the eval_handle is defined.
nonzero (tuple) – Region in which the eval_handle will return
output. Must be a subset of domain (nonzero) –
derivative_handles (list) – List of callable(s) that contain
of eval_handle (derivatives) –

add_neutral_element(self)¶

Return the neutral element of addition for this object.

In other words: self + ret_val == self.

derivative_handles(self)¶

derive(self, order=1)¶

Spatially derive this Function.

This is done by neglecting order derivative handles and to select handle $\text{order} - 1$ as the new evaluation_handle.

Parameters

order (int) – the amount of derivations to perform

Raises

TypeError – If order is not of type int.
ValueError – If the requested derivative order is higher than the provided one.

Returns

Function the derived function.

static from_data(x, y, **kwargs)¶

Create a Function based on discrete data by interpolating.

The interpolation is done by using interp1d from scipy, the kwargs will be passed.

Parameters

x (array-like) – Places where the function has been evaluated .
y (array-like) – Function values at x.
**kwargs – all kwargs get passed to Function .

Returns

An interpolating function.

Return type

Function

function_handle(self)¶

function_space_hint(self)¶: Return the hint that this function is an element of the an scalar product space which is uniquely defined by the scalar product scalar_product_hint().

Note

If you are working on different function spaces, you have to overwrite this hint in order to provide more properties which characterize your specific function space. For example the domain of the functions.

get_member(self, idx)¶

Implementation of the abstract parent method.

Since the Function has only one member (itself) the parameter idx is ignored and self is returned.

Parameters: idx – ignored.
Returns: self

mul_neutral_element(self)¶

Return the neutral element of multiplication for this object.

In other words: self * ret_val == self.

raise_to(self, power)¶

Raises the function to the given power.

Warning

Derivatives are lost after this action is performed.

Parameters: power (numbers.Number) – power to raise the function to
Returns: raised function

scalar_product_hint(self)¶: Return the hint that the _dot_product_l2() has to calculated to gain the scalar product.

scale(self, factor)¶

Factory method to scale a Function.

Parameters: factor – numbers.Number or a callable.

class LambdifiedSympyExpression(sympy_funcs, spat_symbol, spatial_domain, complex_=False)¶

Bases: pyinduct.core.Function

This class provides a Function $\varphi(z)$ based on a lambdified sympy expression. The sympy expressions for the function and it’s spatial derivatives must be provided as the list sympy_funcs. The expressions must be provided with increasing derivative order, starting with order 0.

Parameters

sympy_funcs (array_like) – Sympy expressions for the function and the derivatives: $\varphi(z), \varphi'(z), ...$ .
spat_symbol – Sympy symbol for the spatial variable $z$ .
spatial_domain (tuple) – Domain on which $\varphi(z)$ is defined (e.g.: spatial_domain=(0, 1)).
complex (bool) – If False the Function raises an Error if it returns complex values. Default: False.

class SecondOrderDirichletEigenfunction(om, param, l, scale=1, max_der_order=2)¶

Bases: pyinduct.eigenfunctions.SecondOrderEigenfunction

This class provides an eigenfunction $\varphi(z)$ to eigenvalue problems of the form

$a_2\varphi''(z) + a_1&\varphi'(z) + a_0\varphi(z) = \lambda\varphi(z) \\ \varphi(0) &= 0 \\ \varphi(l) &= 0.$

The eigenfrequency

$\omega = \sqrt{-\frac{a_1^2}{4a_2^2}+\frac{a_0-\lambda}{a_2}}$

must be provided (for example with the eigfreq_eigval_hint() of this class).

Parameters

om (numbers.Number) – eigenfrequency $\omega$
param (array_like) – $\Big( a_2, a_1, a_0, None, None \Big)^T$
l (numbers.Number) – End of the domain $z\in [0,l]$ .
scale (numbers.Number) – Factor to scale the eigenfunctions.
max_der_order (int) – Number of derivative handles that are needed.

static eigfreq_eigval_hint(param, l, n_roots)¶

Return the first n_roots eigenfrequencies $\omega$ and eigenvalues $\lambda$ .

$\omega_i = \sqrt{-\frac{a_1^2}{4a_2^2}+ \frac{a_0-\lambda_i}{a_2}} \quad i = 1,...,\text{n\_roots}$

to the considered eigenvalue problem.

Parameters

param (array_like) – $\Big( a_2, a_1, a_0, None, None \Big)^T$
l (numbers.Number) – Right boundary value of the domain $[0,l]\ni z$ .
n_roots (int) – Amount of eigenfrequencies to be compute.

Returns

$\Big(\big[\omega_1,...,\omega_\text{n\_roots}\Big], \Big[\lambda_1,...,\lambda_\text{n\_roots}\big]\Big)$

Return type

tuple –> two numpy.ndarrays of length n_roots

class SecondOrderEigenVector(char_pair, coefficients, domain, derivative_order)¶

Bases: pyinduct.shapefunctions.ShapeFunction

This class provides eigenvectors of the form

$\varphi(z) = e^{\eta z} \big(\kappa_1 \cos(\nu z) + \sin(\nu z) \big) \:,$

of a linear second order spatial operator $\textup{A}$ denoted by

$(\textup{A}\varphi)(z) = a_2 \partial_z^2 \varphi(z) + a_1 \partial_z \varphi(z) + a_0 \varphi(z)$

where the $a_i$ are constant and whose boundary conditions are given by

$\alpha_1 \partial_z x(z_1) + \alpha_0 x(z_1) &= 0 \\ \beta_1 \partial_z x(z_2) + \beta_0 x(z_2) &= 0 .$

To calculate the corresponding eigenvectors, the problem

$(\textup{A}\varphi)(z) = \lambda \varphi(z)$

is solved for the eigenvalues $\lambda$ , making use of the characteristic roots $p$ given by

$p = \underbrace{-\frac{a_1}{a_2}}_{=:\eta} + j\underbrace{\sqrt{ \frac{a_0 - \lambda}{a_2} - \left(\frac{a_1}{2a_2}\right)^2 }}_{=:\nu}$

Note

To easily instantiate a set of eigenvectors for a certain system, use the cure_hint() of this class or even better the helper-function cure_interval() .

Warns

Since an eigenvalue corresponds to a pair of conjugate complex
characteristic roots, latter are only calculated for the positive
half-plane since the can be mirrored.
To obtain the orthonormal properties of the generated
eigenvectors, the eigenvalue corresponding to the characteristic
root 0+0j is ignored, since it leads to the zero function.

Parameters

char_pair (tuple of complex) – Characteristic root, corresponding to the eigenvalue $\lambda$ for which the eigenvector is to be determined. (Can be obtained by convert_to_characteristic_root())
coefficients (tuple) – Constants of the exponential ansatz solution.

Returns

The eigenvector.

Return type

SecondOrderEigenVector

static calculate_eigenvalues(domain, params, count, extended_output=False, **kwargs)¶

Determine the eigenvalues of the problem given by parameters defined on domain .

Parameters

domain (Domain) – Domain of the spatial problem.
params (bunch-like) – Parameters of the system, see __init__() for details on their definition. Long story short, it must contain $a_2, a_1, a_0, \alpha_0, \alpha_1, \beta_0 \text{ and } \beta_1$ .
count (int) – Amount of eigenvalues to generate.
extended_output (bool) – If true, not only eigenvalues but also the corresponding characteristic roots and coefficients of the eigenvectors are returned. Defaults to False.

Keyword Arguments

debug (bool) – If provided, this parameter will cause several debug windows to open.

Returns

$\lambda$ , ordered in increasing order or tuple of ( $\lambda, p, \boldsymbol{\kappa}$ ) if extended_output is True.

Return type

array or tuple of arrays

static convert_to_characteristic_root(params, eigenvalue)¶

Converts a given eigenvalue $\lambda$ into a characteristic root $p$ by using the provided parameters. The relation is given by

$p = -\frac{a_1}{a_2} + j\sqrt{ \frac{a_0 - \lambda}{a_2} - \left(\frac{a_1}{2a_2}\right)^2 }$

Parameters

params (bunch) – system parameters, see cure_hint() .
eigenvalue (real) – eigenvalue $\lambda$

Returns

characteristic root $p$

Return type

complex number

static convert_to_eigenvalue(params, char_roots)¶

Converts a pair of characteristic roots $p_{1,2}$ into an eigenvalue $\lambda$ by using the provided parameters. The relation is given by

$\lambda = a_2 p^2 + a_1 p + a_0$

Parameters

params (SecondOrderOperator) – System parameters.
char_roots (tuple or array of tuples) – Characteristic roots

static cure_interval(interval, params, count, derivative_order, **kwargs)¶

Helper to cure an interval with eigenvectors.

Parameters

interval (Domain) – Domain of the spatial problem.
params (SecondOrderOperator) – Parameters of the system, see __init__() for details on their definition. Long story short, it must contain $a_2, a_1, a_0, \alpha_0, \alpha_1, \beta_0 \text{ and } \beta_1$ .
count (int) – Amount of eigenvectors to generate.
derivative_order (int) – Amount of derivative handles to provide.
kwargs – will be passed to calculate_eigenvalues()

Keyword Arguments

debug (bool) – If provided, this parameter will cause several debug windows to open.

Returns

An array holding the eigenvalues paired with a basis spanned by the eigenvectors.

Return type

tuple of (array, Base)

class SecondOrderEigenfunction(*args, **kwargs)¶

Bases: pyinduct.shapefunctions.ShapeFunction

Wrapper for all eigenvalue problems of the form

$a_2\varphi''(z) + a_1&\varphi'(z) + a_0\varphi(z) = \lambda\varphi(z), \qquad a_2, a_1, a_0, \lambda \in \mathbb C$

with eigenfunctions $\varphi$ and eigenvalues $\lambda$ . The roots of the characteristic equation (belonging to the ode) are denoted by

$p = \eta \pm j\omega, \qquad \eta \in \mathbb R, \quad \omega \in \mathbb C$

$\eta = -\frac{a_1}{2a_2}, \quad \omega = \sqrt{-\frac{a_1^2}{4 a_2^2} + \frac{a_0 - \lambda}{a_2}}$

In the following the variable $\omega$ is called an eigenfrequency.

classmethod cure_interval(cls, interval, param=None, n=None, eig_val=None, eig_freq=None, max_order=2, scale=None)¶

Provide the first n eigenvalues and eigenfunctions (wraped inside a pyinduct base). For the exact formulation of the considered eigenvalue problem, have a look at the docstring from the eigenfunction class from which you will call this method.

You must call this classmethod with one and only one of the kwargs:

n (eig_val and eig_freq will be computed with the eigfreq_eigval_hint())

eig_val (eig_freq will be calculated with eigval_tf_eigfreq())

eig_freq (eig_val will be calculated with eigval_tf_eigfreq()),

or (and) pass the kwarg scale (then n is set to len(scale)). If you have the kwargs eig_val and eig_freq already calculated then these are preferable, in the sense of performance.

Parameters

interval (Domain) – Domain/Interval of the eigenvalue problem.

Keyword Arguments

param – Parameters $(a_2, a_1, a_0, ...)$ see evp_class.__doc__.
n – Number of eigenvalues/eigenfunctions to compute.
eig_freq (array_like) – Pass your own choice of eigenfrequencies here.
eig_val (array_like) – Pass your own choice of eigenvalues here.
max_order – Maximum derivative order which must provided by the eigenfunctions.
scale (array_like) – Here you can pass a list of values to scale the eigenfunctions.

Returns

eigenvalues (numpy.array)
eigenfunctions (Base)

Return type

tuple

static eigfreq_eigval_hint(param, l, n_roots)¶

Parameters

param (array_like) – Parameters $(a_2, a_1, a_0, None, None)$ .
l – End of the domain $z\in[0, 1]$ .
n_roots (int) – Number of eigenfrequencies/eigenvalues to be compute.

Returns

Booth tuple elements are numpy.ndarrays of the same length, one for eigenfrequencies and one for eigenvalues.

$\Big(\big[\omega_1,...,\omega_\text{n\_roots}\Big], \Big[\lambda_1,...,\lambda_\text{n\_roots}\big]\Big)$

Return type

tuple

static eigval_tf_eigfreq(param, eig_val=None, eig_freq=None)¶

Provide corresponding of eigenvalues/eigenfrequencies for given eigenfreqeuncies/eigenvalues, depending on which type is given.

$\omega = \sqrt{-\frac{a_1^2}{4a_2^2}+\frac{a_0-\lambda}{a_2}}$

respectively

$\lambda = -\frac{a_1^2}{4a_2}+a_0 - a_2 \omega.$

Parameters

param (array_like) – Parameters $(a_2, a_1, a_0, None, None)$ .
eig_val (array_like) – Eigenvalues $\lambda$ .
eig_freq (array_like) – Eigenfrequencies $\omega$ .

Returns

Eigenfrequencies $\omega$ or eigenvalues $\lambda$ .

Return type

numpy.array

static get_adjoint_problem(param)¶

Return the parameters of the adjoint eigenvalue problem for the given parameter set. Hereby, dirichlet or robin boundary condition at $z=0$

$\varphi(0) = 0 \quad &\text{or} \quad \varphi'(0) = \alpha\varphi(0)$

and dirichlet or robin boundary condition at $z=l$

$\varphi`(l) = 0 \quad &\text{or} \quad \varphi'(l) = -\beta\varphi(l)$

can be imposed.

Parameters

param (array_like) –

To define a homogeneous dirichlet boundary condition set alpha or beta to None at the corresponding side. Possibilities:

$\Big( a_2, a_1, a_0, \alpha, \beta \Big)^T$ ,
$\Big( a_2, a_1, a_0, None, \beta \Big)^T$ ,
$\Big( a_2, a_1, a_0, \alpha, None \Big)^T$ or
$\Big( a_2, a_1, a_0, None, None \Big)^T$ .

Returns

Parameters $\big(a_2, \tilde a_1, a_0, \tilde \alpha, \tilde \beta \big)$ for the adjoint problem

$a_2\psi''(z) + \tilde a_1&\psi'(z) + a_0\psi(z) = \lambda\psi(z) \\ \psi(0) = 0 \quad &\text{or} \quad \psi'(0) = \tilde\alpha\psi(0) \\ \psi`(l) = 0 \quad &\text{or} \quad \psi'(l) = -\tilde\beta\psi(l)$

with

$\tilde a_1 = -a_1, \quad \tilde\alpha = \frac{a_1}{a_2}\alpha, \quad \tilde\beta = -\frac{a_1}{a_2}\beta.$

Return type

tuple

class SecondOrderOperator(a2=0, a1=0, a0=0, alpha1=0, alpha0=0, beta1=0, beta0=0, domain=- np.inf, np.inf)¶

Interface class to collect all important parameters that describe a second order ordinary differential equation.

Parameters

a2 (Number or callable) – coefficient $a_2$ .
a1 (Number or callable) – coefficient $a_1$ .
a0 (Number or callable) – coefficient $a_0$ .
alpha1 (Number) – coefficient $\alpha_1$ .
alpha0 (Number) – coefficient $\alpha_0$ .
beta1 (Number) – coefficient $\beta_1$ .
beta0 (Number) – coefficient $\beta_0$ .

static from_dict(param_dict, domain=None)¶

static from_list(param_list, domain=None)¶

get_adjoint_problem(self)¶

Return the parameters of the operator $A^*$ describing the the problem

$(\textup{A}^*\psi)(z) = \bar a_2 \partial_z^2 \psi(z) + \bar a_1 \partial_z \psi(z) + \bar a_0 \psi(z) \:,$

where the $\bar a_i$ are constant and whose boundary conditions are given by

$\bar\alpha_1 \partial_z \psi(z_1) + \bar\alpha_0 \psi(z_1) &= 0 \\ \bar\beta_1 \partial_z \psi(z_2) + \bar\beta_0 \psi(z_2) &= 0 .$

The following mapping is used:

$\bar a_2 = a_2, \quad\bar a_1 = -a_1, \quad\bar a_0 = a_0, \\ \bar\alpha_1 = -1, \quad \bar\alpha_0 = \frac{a_1}{a_2} - \frac{\alpha_0}{\alpha_1} , \\ \bar\beta_1 = -1, \quad \bar\beta_0 = \frac{a_1}{a_2} - \frac{\beta_0}{\beta_1} \:.$

Returns: Parameter set describing $A^*$ .
Return type: SecondOrderOperator

class SecondOrderRobinEigenfunction(om, param, l, scale=1, max_der_order=2)¶

Bases: pyinduct.eigenfunctions.SecondOrderEigenfunction

This class provides an eigenfunction $\varphi(z)$ to the eigenvalue problem given by

$a_2\varphi''(z) + a_1&\varphi'(z) + a_0\varphi(z) = \lambda\varphi(z) \varphi'(0) &= \alpha \varphi(0) \varphi'(l) &= -\beta \varphi(l) .$

The eigenfrequency $\omega = \sqrt{-\frac{a_1^2}{4a_2^2}+\frac{a_0-\lambda}{a_2}}$ must be provided (for example with the eigfreq_eigval_hint() of this class).

Parameters

om (numbers.Number) – eigenfrequency $\omega$
param (array_like) – $\Big( a_2, a_1, a_0, \alpha, \beta \Big)^T$
l (numbers.Number) – End of the domain $z\in [0,l]$ .
scale (numbers.Number) – Factor to scale the eigenfunctions (corresponds to $\varphi(0)=\text{phi\_0}$ ).
max_der_order (int) – Number of derivative handles that are needed.

static eigfreq_eigval_hint(param, l, n_roots, show_plot=False)¶

Return the first n_roots eigenfrequencies $\omega$ and eigenvalues $\lambda$ .

$\omega_i = \sqrt{ - \frac{a_1^2}{4a_2^2} + \frac{a_0 - \lambda_i}{a_2}} \quad i = 1, \dotsc, \text{n\_roots}$

to the considered eigenvalue problem.

Parameters

param (array_like) – Parameters $\big( a_2, a_1, a_0, \alpha, \beta \big)^T$
l (numbers.Number) – Right boundary value of the domain $[0,l]\ni z$ .
n_roots (int) – Amount of eigenfrequencies to compute.
show_plot (bool) – Show a plot window of the characteristic equation.

Returns

$\Big(\big[\omega_1, \dotsc, \omega_{\text{n\_roots}}\Big], \Big[\lambda_1, \dotsc, \lambda_{\text{n\_roots}}\big]\Big)$

Return type

tuple –> booth tuple elements are numpy.ndarrays of length nroots

class ShapeFunction(*args, **kwargs)¶

Bases: pyinduct.core.Function

Base class for approximation functions with compact support.

When a continuous variable of e.g. space and time $x(z,t)$ is decomposed in a series $\tilde x = \sum\limits_{i=1}^{\infty} \varphi_i(z) c_i(t)$ the $\varphi_i(z)$ denote the shape functions.

classmethod cure_interval(cls, interval, **kwargs)¶

Create a network or set of functions from this class and return an approximation base (Base) on the given interval.

The kwargs may hold the order of approximation or the amount of functions to use. Use them in your child class as needed.

If you don’t need to now from which class this method is called, overwrite the @classmethod decorator in the child class with the @staticmethod decorator.

Short reference: Inside a @staticmethod you know nothing about the class from which it is called and you can just play with the given parameters. Inside a @classmethod you can additionally operate on the class, since the first parameter is always the class itself.

Parameters

interval (Domain) – Interval to cure.
**kwargs – Various arguments, depending on the implementation.

Returns

Approximation base, generated by the created shape functions.

Return type

Base

class TransformedSecondOrderEigenfunction(target_eigenvalue, init_state_vector, dgl_coefficients, domain)¶

Bases: pyinduct.core.Function

This class provides an eigenfunction $\varphi(z)$ to the eigenvalue problem given by

$a_2(z)\varphi''(z) + a_1(z)\varphi'(z) + a_0(z)\varphi(z) = \lambda\varphi(z) \quad,$

where $\lambda \in \mathbb{C}$ denotes an eigenvalue and $z \in [z_0, \dotsc, z_n]$ the domain.

Parameters

target_eigenvalue (numbers.Number) – $\lambda$
init_state_vector (array_like) –

$\Big(\text{Re}\{\varphi(0)\}, \text{Re}\{\varphi'(0)\}, \text{Im}\{\varphi(0)\}, \text{Im}\{\varphi'(0)\}\Big)^T$
dgl_coefficients (array_like) – Function handles $\Big( a2(z), a1(z), a0(z) \Big)^T$ .
domain (Domain) – Spatial domain of the problem.

find_roots(function, grid, n_roots=None, rtol=1e-05, atol=1e-08, cmplx=False, sort_mode='norm')¶

Searches n_roots roots of the function $f(\boldsymbol{x})$ on the given grid and checks them for uniqueness with aid of rtol.

In Detail scipy.optimize.root() is used to find initial candidates for roots of $f(\boldsymbol{x})$ . If a root satisfies the criteria given by atol and rtol it is added. If it is already in the list, a comprehension between the already present entries’ error and the current error is performed. If the newly calculated root comes with a smaller error it supersedes the present entry.

Raises

ValueError – If the demanded amount of roots can’t be found.

Parameters

function (callable) – Function handle for math:f(boldsymbol{x}) whose roots shall be found.
grid (list) – Grid to use as starting point for root detection. The $i$ th element of this list provides sample points for the $i$ th parameter of $\boldsymbol{x}$ .
n_roots (int) – Number of roots to find. If none is given, return all roots that could be found in the given area.
rtol – Tolerance to be exceeded for the difference of two roots to be unique: $f(r1) - f(r2) > \textrm{rtol}$ .
atol – Absolute tolerance to zero: $f(x^0) < \textrm{atol}$ .
cmplx (bool) – Set to True if the given function is complex valued.
sort_mode (str) – Specify tho order in which the extracted roots shall be sorted. Default “norm” sorts entries by their $l_2$ norm, while “component” will sort them in increasing order by every component.

Returns

numpy.ndarray of roots; sorted in the order they are returned by $f(\boldsymbol{x})$ .

generic_scalar_product(b1, b2=None, scalar_product=None)¶

Calculates the pairwise scalar product between the elements of the ApproximationBase b1 and b2.

Parameters

b1 (ApproximationBase) – first basis
b2 (ApproximationBase) – second basis, if omitted defaults to b1
scalar_product (list of callable) – Callbacks for product calculation. Defaults to scalar_product_hint from b1.

Note

If b2 is omitted, the result can be used to normalize b1 in terms of its scalar product.

normalize_base(b1, b2=None)¶

Takes two ApproximationBase’s $\boldsymbol{b}_1$ , $\boldsymbol{b}_1$ and normalizes them so that $\langle\boldsymbol{b}_{1i}\, ,\:\boldsymbol{b}_{2i}\rangle = 1$ . If only one base is given, $\boldsymbol{b}_2$ defaults to $\boldsymbol{b}_1$ .

Parameters

b1 (ApproximationBase) – $\boldsymbol{b}_1$
b2 (ApproximationBase) – $\boldsymbol{b}_2$

Raises

ValueError – If $\boldsymbol{b}_1$ and $\boldsymbol{b}_2$ are orthogonal.

Returns

if b2 is None, otherwise: Tuple of 2 ApproximationBase’s.

Return type

ApproximationBase

real(data)¶

Check if the imaginary part of data vanishes and return its real part if it does.

Parameters: data (numbers.Number or array_like) – Possibly complex data to check.
Raises: ValueError – If provided data can’t be converted within the given tolerance limit.
Returns: Real part of data.
Return type: numbers.Number or array_like

visualize_roots(roots, grid, func, cmplx=False, return_window=False)¶

Visualize a given set of roots by examining the output of the generating function.

Parameters

roots (array like) – Roots to display, if None is given, no roots will be displayed, this is useful to get a view of func and choosing an appropriate grid.
grid (list) – List of arrays that form the grid, used for the evaluation of the given func.
func (callable) – Possibly vectorial function handle that will take input of of the shape (‘len(grid)’, ).
cmplx (bool) – If True, the complex valued func is handled as a vectorial function returning [Re(func), Im(func)].
return_window (bool) – If True the graphics window is not shown directly. In this case, a reference to the plot window is returned.

Returns: A PgPlotWindow if delay_exec is True.