OrthogonalUniVariatePolynomial¶

class OrthogonalUniVariatePolynomial(*args)¶

Base class for orthogonal univariate polynomials.

Warning

Orthogonal univariate polynomials are not intended to be created manually. They should be constructed with the subclasses of OrthogonalUniVariatePolynomialFamily (like e.g. HermiteFactory). Constructor parameters are therefore intentionally not documented.

See also

OrthogonalUniVariatePolynomialFamily

Examples

>>> import openturns as ot

Create a univariate polynomial from a list of coefficients:

>>> for i in range(3):
...     print(ot.OrthogonalUniVariatePolynomialFamily().build(i))
1
X
-0.707107 + 0.707107 * X^2

Methods

`derivate`()	Build the first-order derivative polynomial.
`draw`(xMin, xMax, pointNumber)	Draw the function.
`getClassName`()	Accessor to the object’s name.
`getCoefficients`()	Accessor to the polynomials’s coefficients.
`getDegree`()	Accessor to the polynomials’s degree.
`getId`()	Accessor to the object’s id.
`getName`()	Accessor to the object’s name.
`getRecurrenceCoefficients`()	Accessor to the recurrence coefficients.
`getRoots`()	Compute the roots of the polynomial.
`getShadowedId`()	Accessor to the object’s shadowed id.
`getVisibility`()	Accessor to the object’s visibility state.
`gradient`(x)	Compute the gradient at point $x$ .
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`hessian`(x)	Compute the hessian at point $x$ .
`incrementDegree`([degree])	Multiply the polynomial by $x^k$ .
`setCoefficients`(coefficients)	Accessor to the polynomials’s coefficients.
`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.

__call__

__init__(*args)¶: Initialize self. See help(type(self)) for accurate signature.

derivate()¶

Build the first-order derivative polynomial.

Returns:	derivated_polynomial : `Univariate` The first-order derivated polynomial.

Examples

>>> import openturns as ot
>>> P = ot.UniVariatePolynomial([1.0, 2.0, 3.0])
>>> print(P.derivate())
2 + 6 * X

draw(xMin, xMax, pointNumber)¶

Draw the function.

Parameters:	x_min : float, optional The starting value that is used for meshing the x-axis. x_max : float, optional, $x_{\max} > x_{\min}$ The ending value that is used for meshing the x-axis. n_points : int, optional The number of points that is used for meshing the x-axis.

Examples

>>> import openturns as ot
>>> from openturns.viewer import View
>>> f = ot.UniVariatePolynomial([1.0, 2.0, -3.0, 5.0])
>>> View(f.draw(-10.0, 10.0, 100)).show()

getClassName()¶

Accessor to the object’s name.

Returns:	class_name : str The object class name (object.__class__.__name__).

getCoefficients()¶

Accessor to the polynomials’s coefficients.

Returns:	coefficients : `Point` Polynomial coefficients in increasing polynomial order.

See also

setCoefficients

Examples

>>> import openturns as ot
>>> P = ot.UniVariatePolynomial([1.0, 2.0, 3.0])
>>> print(P.getCoefficients())
[1,2,3]

getDegree()¶

Accessor to the polynomials’s degree.

Returns:	degree : int Polynomial’s degree.

Examples

>>> import openturns as ot
>>> P = ot.UniVariatePolynomial([1.0, 2.0, 3.0])
>>> print(P.getDegree())
2

getId()¶

Accessor to the object’s id.

Returns:	id : int Internal unique identifier.

getName()¶

Accessor to the object’s name.

Returns:	name : str The name of the object.

getRecurrenceCoefficients()¶

Accessor to the recurrence coefficients.

Returns:	recurrence_coefficients : list of `Point` The list of recurrence coefficients that defined the orthogonal univariate polynomial from the very first univariate orthogonal polynomial $P_0$ .

Notes

Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:

$P_{i + 1} = (a_i x + b_i) P_i + c_i P_{i - 1}, \quad i > 1$

Examples

>>> import openturns as ot
>>> polynomial = ot.OrthogonalUniVariatePolynomialFamily().build(2)
>>> print(polynomial.getRecurrenceCoefficients())
[[1,0,0],[0.707107,0,-0.707107]]

getRoots()¶

Compute the roots of the polynomial.

Returns:	roots : list of complex values Polynomial’s roots.

Examples

>>> import openturns as ot
>>> P = ot.UniVariatePolynomial([1.0, 2.0, 3.0])
>>> print(P.getRoots())
[(-0.333333,0.471405),(-0.333333,-0.471405)]

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.

gradient(x)¶

Compute the gradient at point $x$ .

Returns:	gradient : float The value of the function’s first-order derivative at point $x$ .

Examples

>>> import openturns as ot
>>> P = ot.UniVariatePolynomial([1.0, 2.0, 3.0])
>>> print(P.gradient(1.0))
8.0

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.

hessian(x)¶

Compute the hessian at point $x$ .

Parameters:	x : float Input value.
Returns:	hessian : float The value of the function’s second-order derivative at point $x$ .

incrementDegree(degree=1)¶

Multiply the polynomial by $x^k$ .

Parameters:	degree : int, optional The incremented degree $k$ . Default uses $k = 1$ .
Returns:	incremented_degree_polynomial : `UniVariatePolynomial` Polynomial with incremented degree.

Examples

>>> import openturns as ot
>>> P = ot.UniVariatePolynomial([1.0, 2.0, 3.0])
>>> print(P.incrementDegree())
X + 2 * X^2 + 3 * X^3
>>> print(P.incrementDegree(2))
X^2 + 2 * X^3 + 3 * X^4

setCoefficients(coefficients)¶

Accessor to the polynomials’s coefficients.

Parameters:	coefficients : sequence of float Polynomial coefficients in increasing polynomial order.

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