/usr/include/openturns/swig/GaussKronrodRule_doc.i is in libopenturns-dev 1.7-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | %feature("docstring") OT::GaussKronrodRule
"Gauss-Kronrod rule used in the integration algorithm.
Parameters
----------
myGaussKronrodPair : :class:`~openturns.GaussKronrodPair`
It encodes the selected rule.
Available rules:
- GaussKronrodRule.G3K7,
- GaussKronrodRule.G7K15,
- GaussKronrodRule.G11K23,
- GaussKronrodRule.G15K31,
- GaussKronrodRule.G25K51.
Notes
-----
The Gauss-Kronrod rules :math:`G_mK_{2m+1}` with :math:`m=2n+1` enable to build two approximations of the definite integral :math:`\\\\int_{-1}^1 f(t)\\\\, dt` defined by:
.. math::
\\\\int_{-1}^1 f(t)\\\\, dt \\\\simeq \\\\omega_0f(0) + \\\\sum_{k=1}^n \\\\omega_k (f(\\\\xi_k)+f(-\\\\xi_k))
and:
.. math::
\\\\int_{-1}^1 f(t)\\\\, dt\\\\simeq \\\\alpha_0f(0) + \\\\sum_{k=1}^{m} \\\\alpha_k (f(\\\\zeta_k)+f(-\\\\zeta_k))
We have :math:`\\\\xi_k>0`, :math:`\\\\zeta_k>0`, :math:`\\\\zeta_{2j}=\\\\xi_j`, :math:`\\\\omega_k>0` and :math:`\\\\alpha_k>0`.
The rule :math:`G_mK_{2m+1}` combines a :math:`m`-point Gauss rule and a :math:`(2m+1)`-point Kronrod rule (re-using the :math:`m` nodes of the Gauss method). The nodes are defined on :math:`[-1, 1]` and always contain the node 0 when :math:`m` is odd.
Examples
--------
Create an Gauss-Kronrod rule:
>>> import openturns as ot
>>> myRule = ot.GaussKronrodRule(ot.GaussKronrodRule.G15K31)"
// ---------------------------------------------------------------------
%feature("docstring") OT::GaussKronrodRule::getOrder
"Accessor to :math:`m` parameter.
Returns
-------
m : int
The number of points used for the Gauss approximation."
// ---------------------------------------------------------------------
%feature("docstring") OT::GaussKronrodRule::getPair
"Accessor to pair definig the rule.
Returns
-------
gkPair : :class:`~openturns.GaussKronrodPair`
Id of the Gauss-Kronrod rule."
// ---------------------------------------------------------------------
%feature("docstring") OT::GaussKronrodRule::getOtherGaussWeights
"Accessor to the weights used in the Gauss approximation.
Returns
-------
otherGaussWeights : :class:`~openturns.NumericalPoint`
The weights :math:`(\\\\omega_k)_{1 \\\\leq k \\\\leq n}`"
// ---------------------------------------------------------------------
%feature("docstring") OT::GaussKronrodRule::getOtherGaussNodes
"Accessor to the positive nodes used in the Gauss-Kronrod approximation.
Returns
-------
otherKronrodNodes : :class:`~openturns.NumericalPoint`
The positive nodes :math:`(\\\\xi_k)_{1 \\\\leq k \\\\leq n}`
"
// ---------------------------------------------------------------------
%feature("docstring") OT::GaussKronrodRule::getOtherKronrodWeights
"Accessor to the positive nodes used in the Gauss-Kronrod approximation.
Returns
-------
otherKronrodWeights : :class:`~openturns.NumericalPoint`
The weights :math:`(\\\\alpha_k)_{1 \\\\leq k \\\\leq m}`.
"
// ---------------------------------------------------------------------
%feature("docstring") OT::GaussKronrodRule::getOtherKronrodNodes
"Accessor to the positive nodes used in the Gauss-Kronrod approximation.
Returns
-------
otherKronrodNodes : :class:`~openturns.NumericalPoint`
The positive nodes :math:`(\\\\zeta_k)_{1 \\\\leq k \\\\leq m}`
It contains the positive Gauss nodes as we have :math:`\\\\zeta_{2j}=\\\\xi_j`.
"
// ---------------------------------------------------------------------
%feature("docstring") OT::GaussKronrodRule::getZeroGaussWeight
"Accessor to the first Gauss weight.
Returns
-------
zeroKronrodWeight : float
The first weight :math:`\\\\omega_0`.
"
// ---------------------------------------------------------------------
%feature("docstring") OT::GaussKronrodRule::getZeroKronrodWeight
"Accessor to the first Kronrod weight.
Returns
-------
zeroKronrodWeight : float
The first weight :math:`\\\\alpha_0`.
"
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