GaussKronrodRule

class GaussKronrodRule(*args)

Gauss-Kronrod rule used in the integration algorithm.

Parameters:

myGaussKronrodPair : GaussKronrodPair

It encodes the selected rule.

Available rules:

  • GaussKronrodRule.G1K3,
  • GaussKronrodRule.G3K7,
  • GaussKronrodRule.G7K15,
  • GaussKronrodRule.G11K23,
  • GaussKronrodRule.G15K31,
  • GaussKronrodRule.G25K51.

Notes

The Gauss-Kronrod rules G_mK_{2m+1} with m=2n+1 enable to build two approximations of the definite integral \int_{-1}^1 f(t)\di{t} defined by:

\int_{-1}^1 f(t)\di{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))

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

The rule G_mK_{2m+1} combines a m-point Gauss rule and a (2m+1)-point Kronrod rule (re-using the m nodes of the Gauss method). The nodes are defined on [-1, 1] and always contain the node 0 when m is odd.

Examples

Create an Gauss-Kronrod rule:

>>> import openturns as ot
>>> myRule = ot.GaussKronrodRule(ot.GaussKronrodRule.G15K31)

Methods

getClassName() Accessor to the object’s name.
getId() Accessor to the object’s id.
getName() Accessor to the object’s name.
getOrder() Accessor to m parameter.
getOtherGaussWeights() Accessor to the weights used in the Gauss approximation.
getOtherKronrodNodes() Accessor to the positive nodes used in the Gauss-Kronrod approximation.
getOtherKronrodWeights() Accessor to the positive nodes used in the Gauss-Kronrod approximation.
getPair() Accessor to pair definig the rule.
getShadowedId() Accessor to the object’s shadowed id.
getVisibility() Accessor to the object’s visibility state.
getZeroGaussWeight() Accessor to the first Gauss weight.
getZeroKronrodWeight() Accessor to the first Kronrod weight.
hasName() Test if the object is named.
hasVisibleName() Test if the object has a distinguishable name.
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)

x.__init__(…) initializes x; see help(type(x)) for signature

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.

getName()

Accessor to the object’s name.

Returns:

name : str

The name of the object.

getOrder()

Accessor to m parameter.

Returns:

m : int

The number of points used for the Gauss approximation.

getOtherGaussWeights()

Accessor to the weights used in the Gauss approximation.

Returns:

otherGaussWeights : Point

The weights (\omega_k)_{1 \leq k \leq n}

getOtherKronrodNodes()

Accessor to the positive nodes used in the Gauss-Kronrod approximation.

Returns:

otherKronrodNodes : Point

The positive nodes (\zeta_k)_{1 \leq k \leq m} It contains the positive Gauss nodes as we have \zeta_{2j}=\xi_j.

getOtherKronrodWeights()

Accessor to the positive nodes used in the Gauss-Kronrod approximation.

Returns:

otherKronrodWeights : Point

The weights (\alpha_k)_{1 \leq k \leq m}.

getPair()

Accessor to pair definig the rule.

Returns:

gkPair : GaussKronrodPair

Id of the Gauss-Kronrod rule.

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.

getZeroGaussWeight()

Accessor to the first Gauss weight.

Returns:

zeroKronrodWeight : float

The first weight \omega_0.

getZeroKronrodWeight()

Accessor to the first Kronrod weight.

Returns:

zeroKronrodWeight : float

The first weight \alpha_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.

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.