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

`__call__`(x)	Call self as a function.
`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.

__init__(*args)¶

derivate()¶

Build the first-order derivative polynomial.

Returns

derivated_polynomialUnivariate: 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_minfloat, optional: The starting value that is used for meshing the x-axis.
x_maxfloat, optional, $x_{\max} > x_{\min}$: The ending value that is used for meshing the x-axis.
n_pointsint, 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_namestr: The object class name (object.__class__.__name__).

getCoefficients()¶

Accessor to the polynomials’s coefficients.

Returns

coefficientsPoint: 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

degreeint: 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

idint: Internal unique identifier.

getName()¶

Accessor to the object’s name.

Returns

namestr: The name of the object.

getRecurrenceCoefficients()¶

Accessor to the recurrence coefficients.

Returns

recurrence_coefficientslist 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())
0 : [  1         0         0        ]
1 : [  0.707107  0        -0.707107 ]

getRoots()¶

Compute the roots of the polynomial.

Returns

rootslist 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

idint: Internal unique identifier.

getVisibility()¶

Accessor to the object’s visibility state.

Returns

visiblebool: Visibility flag.

gradient(x)¶

Compute the gradient at point $x$ .

Returns

gradientfloat: 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

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns

hasVisibleNamebool: True if the name is not empty and not the default one.

hessian(x)¶

Compute the hessian at point $x$ .

Parameters

xfloat: Input value.

Returns

hessianfloat: The value of the function’s second-order derivative at point $x$ .

incrementDegree(degree=1)¶

Multiply the polynomial by $x^k$ .

Parameters

degreeint, optional: The incremented degree $k$ . Default uses $k = 1$ .

Returns

incremented_degree_polynomialUniVariatePolynomial: 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

coefficientssequence of float: Polynomial coefficients in increasing polynomial order.

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