GaussKronrod

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../../_images/GaussKronrod.png
class GaussKronrod(*args)

Adaptive integration algorithm of Gauss-Kronrod.

Parameters:

maximumSubIntervals : int

The maximal number of subdivisions of the interval [a,b]

maximumError : float

The maximal error between Gauss and Kronrod approximations.

GKRule : GaussKronrodRule

The rule that fixes the number of points used in the Gauss and Kronrod approximations.

Notes

The Gauss-Kronrod algorithm enables to approximate the definite integral:

\int_{a}^b f(t)\di{t}

with f: \Rset \mapsto \Rset^p, using both approximations : Gauss and Kronrod ones defined by:

\int_{-1}^1 f(t)\d{t} \simeq  \omega_0f(0) + \sum_{k=1}^n \omega_k (f(\xi_k)+f(-\xi_k))

and:

\int_{-1}^1 f(t)\di{t}\simeq  \alpha_0f(0) + \sum_{k=1}^{m} \alpha_k (f(\zeta_k)+f(-\zeta_k))

where \xi_k>0, \zeta_k>0, \zeta_{2j}=\xi_j, \omega_k>0 and \alpha_k>0.

The Gauss-Kronrod algorithm evaluates the integral using the Gauss and the Konrod approximations. If the difference between both approximations is greater that maximumError, then the interval [a,b] is subdivided into 2 subintervals with the same length. The Gauss-Kronrod algorithm is then applied on both subintervals with the sames rules. The algorithm is iterative until the difference between both approximations is less that maximumError. In that case, the integral on the subinterval is approximated by the Kronrod sum. The subdivision process is limited by maximumSubIntervals that imposes the maximum number of subintervals.

The final integral is the sum of the integrals evaluated on the subintervals.

Examples

Create a Gauss-Kronrod algorithm:

>>> import openturns as ot
>>> algo = ot.GaussKronrod(100, 1e-8, ot.GaussKronrodRule(ot.GaussKronrodRule.G11K23))

Methods

getClassName() Accessor to the object’s name.
getId() Accessor to the object’s id.
getMaximumError() Accessor to the maximal error between Gauss and Kronrod approximations.
getMaximumSubIntervals() Accessor to the maximal number of subdivisions of [a,b].
getName() Accessor to the object’s name.
getRule() Accessor to the Gauss-Kronrod rule used in the integration algorithm.
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.
integrate(*args) Evaluation of the integral of f on an interval.
setMaximumError(maximumError) Set the maximal error between Gauss and Kronrod approximations.
setMaximumSubIntervals(maximumSubIntervals) Set the maximal number of subdivisions of [a,b].
setName(name) Accessor to the object’s name.
setRule(rule) Set the Gauss-Kronrod rule used in the integration algorithm.
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.

getMaximumError()

Accessor to the maximal error between Gauss and Kronrod approximations.

Returns:

maximumErrorvalue : float, positive

The maximal error between Gauss and Kronrod approximations.

getMaximumSubIntervals()

Accessor to the maximal number of subdivisions of [a,b].

Returns:

maximumSubIntervals : float, positive

The maximal number of subdivisions of the interval [a,b].

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getRule()

Accessor to the Gauss-Kronrod rule used in the integration algorithm.

Returns:

rule : GaussKronrodRule

The Gauss-Kronrod rule used in the integration algorithm.

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.

integrate(*args)

Evaluation of the integral of f on an interval.

Available usages:

integrate(f, interval)

integrate(f, interval, error)

integrate(f, a, b, error, ai, bi, fi, ei)

Parameters:

f : NumericalMathFunction, f: \Rset \mapsto \Rset^p

The integrand function.

interval : Interval, interval \in \Rset

The integration domain.

error : NumericalPoint

The error estimation of the approximation.

a,b : float

Bounds of the integration interval.

ai, bi, ei : NumericalPoint;

ai is the set of lower bounds of the subintervals;

bi the corresponding upper bounds;

ei the associated error estimation.

fi : NumericalSample

fi is the set of \int_{ai}^{bi} f(t)\di{t}

Returns:

value : NumericalPoint

Approximation of the integral.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x'], ['abs(sin(x))'])
>>> a = -2.5
>>> b = 4.5
>>> algoGK = ot.GaussKronrod(100, 1e-8, ot.GaussKronrodRule(ot.GaussKronrodRule.G11K23))

Use the high-level usage:

>>> value = algoGK.integrate(f, ot.Interval(a, b))[0]
>>> print(value)
4.590...

Use the low-level usage:

>>> error = ot.NumericalPoint()
>>> ai = ot.NumericalPoint()
>>> bi = ot.NumericalPoint()
>>> ei = ot.NumericalPoint()
>>> fi = ot.NumericalSample()
>>> value2 = algoGK.integrate(f, a, b, error, ai, bi, fi, ei)[0]
>>> print(value2)
4.590...
setMaximumError(maximumError)

Set the maximal error between Gauss and Kronrod approximations.

Parameters:

maximumErrorvalue : float, positive

The maximal error between Gauss and Kronrod approximations.

setMaximumSubIntervals(maximumSubIntervals)

Set the maximal number of subdivisions of [a,b].

Parameters:

maximumSubIntervals : float, positive

The maximal number of subdivisions of the interval [a,b].

setName(name)

Accessor to the object’s name.

Parameters:

name : str

The name of the object.

setRule(rule)

Set the Gauss-Kronrod rule used in the integration algorithm.

Parameters:

rule : GaussKronrodRule

The Gauss-Kronrod rule used in the integration algorithm.

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.