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
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 .
hasName
()Test if the object is named. hasVisibleName
()Test if the object has a distinguishable name. hessian
(x)Compute the hessian at point .
incrementDegree
([degree])Multiply the polynomial by .
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
- derivated_polynomial :
-
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,
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()
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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
Examples
>>> import openturns as ot >>> P = ot.UniVariatePolynomial([1.0, 2.0, 3.0]) >>> print(P.getCoefficients()) [1,2,3]
- coefficients :
-
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
.
Notes
Any sequence of orthogonal polynomials has a recurrence formula relating any three consecutive polynomials as follows:
Examples
>>> import openturns as ot >>> polynomial = ot.OrthogonalUniVariatePolynomialFamily().build(2) >>> print(polynomial.getRecurrenceCoefficients()) [[1,0,0],[0.707107,0,-0.707107]]
- recurrence_coefficients : list of
-
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)]
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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
.
Returns: - gradient : float
The value of the function’s first-order derivative at point
.
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
.
Parameters: - x : float
Input value.
Returns: - hessian : float
The value of the function’s second-order derivative at point
.
-
incrementDegree
(degree=1)¶ Multiply the polynomial by
.
Parameters: - degree : int, optional
The incremented degree
. Default uses
.
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
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setCoefficients
(coefficients)¶ Accessor to the polynomials’s coefficients.
Parameters: - coefficients : sequence of float
Polynomial coefficients in increasing polynomial order.
See also
Examples
>>> import openturns as ot >>> P = ot.UniVariatePolynomial([1.0, 2.0, 3.0]) >>> P.setCoefficients([4.0, 2.0, 1.0]) >>> print(P) 4 + 2 * X + X^2
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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.
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setVisibility
(visible)¶ Accessor to the object’s visibility state.
Parameters: - visible : bool
Visibility flag.
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