FilonQuadrature

class FilonQuadrature(*args)

Integration algorithm for oscillating integrand.

Parameters:
nint, n>0

The discretization used by the algorithm. The integration algorithm will be regularly discretized by 2n+1 points.

omegafloat

The default pulsation in the oscillating kernel. Default value is 1.0.

kindint, kind\geq 0

The type of oscillating kernel defining the integral, see notes. Default value is 0.

Methods

getClassName()

Accessor to the object's name.

getKind()

Accessor to the kind of oscillating weight defining the integral.

getN()

Accessor to the discretization of the algorithm.

getName()

Accessor to the object's name.

getOmega()

Accessor to the default pulsation.

hasName()

Test if the object is named.

integrate(*args)

Evaluation of the integral of f w on an interval.

setKind(kind)

Accessor to the kind of oscillating weight defining the integral.

setN(n)

Accessor to the discretization of the algorithm.

setName(name)

Accessor to the object's name.

setOmega(omega)

Accessor to the default pulsation.

Notes

The Filon algorithm enables to approximate the definite integral:

\int_a^b f(t)w(\omega t)\di{t}

with f: \Rset \mapsto \Rset^p, a, b\in\Rset, \omega\in\Rset and:

w(\omega t)=\left\{ \begin{array}{ll} \cos(\omega t) & \mathrm{if}\: kind=0 \\ \sin(\omega t) & \mathrm{if}\: kind=1 \\ \exp(i \omega t) & \mathrm{if}\: kind\geq 2 \end{array} \right.

This algorithm is based on a regular partition of the interval [a,b], the function f being approximated by a quadratic function on three consecutive points. This algorithm provides an approximation of order \cO(1/\omega^2) when \omega\rightarrow\infty. When w(\omega t)=\exp(i \omega t), the result is returned as a Point of dimension 2, the first component being the real part of the result and the second component the imaginary part.

Examples

Create a Filon algorithm:

>>> import openturns as ot
>>> algo = ot.FilonQuadrature(100)
>>> algo = ot.FilonQuadrature(100, 10.0)
__init__(*args)
getClassName()

Accessor to the object’s name.

Returns:
class_namestr

The object class name (object.__class__.__name__).

getKind()

Accessor to the kind of oscillating weight defining the integral.

Returns:
kindint

The oscillating weight function defining the integral, see the notes.

getN()

Accessor to the discretization of the algorithm.

Returns:
nint

The discretization used by the algorithm.

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getOmega()

Accessor to the default pulsation.

Returns:
omegafloat

The pulsation used in the integrate method if not explicitly given.

hasName()

Test if the object is named.

Returns:
hasNamebool

True if the name is not empty.

integrate(*args)

Evaluation of the integral of f w on an interval.

Available usages:

integrate(f, interval)

integrate(f, interval, omega)

Parameters:
fFunction, f: \Rset \mapsto \Rset^p

The integrand function.

omegafloat

The pulsation in the weight function. This value superseeds the value given in the constructor.

intervalInterval, interval \in \Rset

The integration domain.

Returns:
valuePoint

Approximation of the integral. Its dimension is p if kind\in\{0,1\}, otherwise it is 2p with the p first components corresponding to the real part of the integral and the remaining ones to the imaginary part.

Examples

>>> import openturns as ot
>>> import math
>>> f = ot.SymbolicFunction(['t'], ['log(1+t)'])
>>> a = 0.5
>>> b = a + 8.0 * math.pi
>>> n = 100
>>> omega = 1000.0
>>> kind = 0
>>> algoF = ot.FilonQuadrature(n, omega, kind)
>>> value = algoF.integrate(f, ot.Interval(a, b))
>>> print(value[0])
-0.00134...
>>> kind = 1
>>> algoF = ot.FilonQuadrature(n, omega, kind)
>>> value = algoF.integrate(f, ot.Interval(a, b))
>>> print(value[0])
0.00254...
>>> kind = 2
>>> algoF = ot.FilonQuadrature(n, omega, kind)
>>> value = algoF.integrate(f, ot.Interval(a, b))
>>> print(value[0])
-0.00134...
>>> print(value[1])
0.00254...
setKind(kind)

Accessor to the kind of oscillating weight defining the integral.

Parameters:
kindint

The oscillating weight function defining the integral, see the notes.

setN(n)

Accessor to the discretization of the algorithm.

Parameters:
nint, n>0

The discretization used by the algorithm.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

setOmega(omega)

Accessor to the default pulsation.

Parameters:
omegafloat

The pulsation used in the integrate method if not explicitly given.