AdaptiveStieltjesAlgorithm

class AdaptiveStieltjesAlgorithm(*args)

AdaptiveStieltjes algorithm used to build the orthonormal basis.

The algorithm builds a polynomial basis orthonormal with respect to a specific distribution.

Parameters:

measure : Distribution

A measure for which the orthonormal polynomial basis is built.

See also

OrthonormalizationAlgorithm, ChebychevAlgorithm, GramSchmidtAlgorithm

Notes

It implements an adaptive Stieltjes algorithm that builds the polynomial family orthonormal with respect to the distribution measure, using the GaussKronrod adaptive integration method to compute the following dot-products: R_n=\langle Q_n, Q_n\rangle and S_n=\langle xQ_n, Q_n\rangle where Q_n is the monic polynomial associated to the orthonormal polynomial P_n, needed to compute the coefficients of the three-terms recurrence relation that defines P_n (see OrthogonalUnivariatePolynomialFamily):

a_n = 1/\sqrt{\beta_{n+1}}\quad b_n = -\alpha_n/sqrt{\beta_{n+1}}\quad c_n = -\sqrt{\beta_n/\beta_{n+1}}

where \alpha_n = S_n / R_n and \beta_n = R_n / R_{n-1}, n>0, \beta_0 = 0.

Methods

getClassName() Accessor to the object’s name.
getId() Accessor to the object’s id.
getMeasure() Accessor to the measure.
getName() Accessor to the object’s name.
getRecurrenceCoefficients(n) Accessor to the recurrence coefficients.
getShadowedId() Accessor to the object’s shadowed id.
getVisibility() Accessor to the object’s visibility state.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
setMeasure(measure) Accessor to the measure.
setName(name) Accessor to the object’s name.
setShadowedId(id) Accessor to the object’s shadowed id.
setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)
getClassName()

Accessor to the object’s name.

Returns:

class_name : str

The object class name (object.__class__.__name__).
getId()

Accessor to the object’s id.

Returns:

id : int

Internal unique identifier.
getMeasure()

Accessor to the measure.

Returns:

m : Distribution

The measure for which the orthonormal polynomial basis is built.
getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.
getRecurrenceCoefficients(n)

Accessor to the recurrence coefficients.

Parameters:

n : integer

Index ot the recurrence step.
Returns:

coef : sequence of float

Calculate the coefficients of recurrence a_0, a_1, a_2 such that P_{n+1}(x) = (a_0 \times x + a_1) \times P_n(x) + a_2 \times P_{n-1}(x).
getShadowedId()

Accessor to the object’s shadowed id.

Returns:

id : int

Internal unique identifier.
getVisibility()

Accessor to the object’s visibility state.

Returns:

visible : bool

Visibility flag.
hasName()

Test if the object is named.

Returns:

hasName : bool

True if the name is not empty.
hasVisibleName()

Test if the object has a distinguishable name.

Returns:

hasVisibleName : bool

True if the name is not empty and not the default one.
setMeasure(measure)

Accessor to the measure.

Parameters:

m : Distribution

The measure for which the orthonormal polynomial basis is built.
setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.
setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters:

id : int

Internal unique identifier.
setVisibility(visible)

Accessor to the object’s visibility state.

Parameters:

visible : bool

Visibility flag.