Function¶

class Function(*args)

Function base class.

Notes

A function acts on points to produce points: .

A function enables to evaluate its gradient and its hessian when mathematically defined.

Examples

Create a Function from a list of analytical formulas and descriptions of the input vector and the output vector :

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x0', 'x1'],
...                         ['x0 + x1', 'x0 - x1'])
>>> print(f([1, 2]))
[3,-1]


Create a Function from strings:

>>> import openturns as ot
>>> f = ot.SymbolicFunction('x', '2.0*sqrt(x)')
>>> print(f(([16],[4])))
[ y0 ]
0 : [ 8  ]
1 : [ 4  ]


Create a Function from a Python function:

>>> def a_function(X):
...     return [X[0] + X[1]]
>>> f = ot.PythonFunction(2, 1, a_function)
>>> print(f(((10, 5),(6, 7))))
[ y0 ]
0 : [ 15 ]
1 : [ 13 ]


See PythonFunction for further details.

Create a Function from a Python class:

>>> class FUNC(OpenTURNSPythonFunction):
...     def __init__(self):
...         super(FUNC, self).__init__(2, 1)
...         self.setInputDescription(['R', 'S'])
...         self.setOutputDescription(['T'])
...     def _exec(self, X):
...         Y = [X[0] + X[1]]
...         return Y
>>> F = FUNC()
>>> myFunc = Function(F)
>>> print(myFunc((1.0, 2.0)))
[3]


See PythonFunction for further details.

Create a Function from another Function:

>>> f = ot.SymbolicFunction(ot.Description.BuildDefault(4, 'x'),
...                         ['x0', 'x0 + x1', 'x0 + x2 + x3'])


Then create another function by setting x1=4 and x3=10:

>>> g = ot.ParametricFunction(f, [3, 1], [10.0, 4.0], True)
>>> print(g.getInputDescription())
[x0,x2]
>>> print(g((1, 2)))
[1,5,13]


Or by setting x0=6 and x2=5:

>>> g = ot.ParametricFunction(f, [3, 1], [6.0, 5.0], False)
>>> print(g.getInputDescription())
[x3,x1]
>>> print(g((1, 2)))
[6,8,12]


Create a Function from another Function and by using a comparison operator:

>>> analytical = ot.SymbolicFunction(['x0','x1'], ['x0 + x1'])
>>> indicator = ot.IndicatorFunction(analytical, ot.Less(), 0.0)
>>> print(indicator([2, 3]))
[0]
>>> print(indicator([2, -3]))
[1]


Create a Function from a collection of functions:

>>> functions = list()
>>> functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'],
...                                      ['x1^2 + x2', 'x1 + x2 + x3']))
>>> functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'],
...                                      ['x1 + 2 * x2 + x3', 'x1 + x2 - x3']))
>>> myFunction = ot.AggregatedFunction(functions)
>>> print(myFunction([1.0, 2.0, 3.0]))
[3,6,8,0]


Create a Function which is the linear combination linComb of the functions defined in functionCollection with scalar weights defined in scalarCoefficientColl:

where and then the linear combination is:

>>> myFunction2 = ot.LinearCombinationFunction(functions, [2.0, 4.0])
>>> print(myFunction2([1.0, 2.0, 3.0]))
[38,12]


Create a Function which is the linear combination vectLinComb of the scalar functions defined in scalarFunctionCollection with vectorial weights defined in vectorCoefficientColl:

If where and where

>>> functions=list()
>>> functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'],
...                                      ['x1 + 2 * x2 + x3']))
>>> functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'],
...                                      ['x1^2 + x2']))
>>> myFunction2 = ot.DualLinearCombinationFunction(functions, [[2., 4.], [3., 1.]])
>>> print(myFunction2([1, 2, 3]))
[25,35]


Create a Function from values of the inputs and the outputs:

>>> inputSample = [[1.0, 1.0], [2.0, 2.0]]
>>> outputSample = [[4.0], [5.0]]
>>> database = ot.DatabaseFunction(inputSample, outputSample)
>>> x = [1.8]*database.getInputDimension()
>>> print(database(x))
[5]


Create a Function which is the composition function :

>>> g = ot.SymbolicFunction(['x1', 'x2'],
...                         ['x1 + x2','3 * x1 * x2'])
>>> f = ot.SymbolicFunction(['x1', 'x2'], ['2 * x1 - x2'])
>>> composed = ot.ComposedFunction(f, g)
>>> print(composed([3, 4]))
[-22]


Methods

 __call__(self, \*args) Call self as a function. draw(self, \*args) Draw the output of function as a Graph. getCallsNumber(self) Accessor to the number of times the function has been called. getClassName(self) Accessor to the object’s name. getDescription(self) Accessor to the description of the inputs and outputs. getEvaluation(self) Accessor to the evaluation function. Accessor to the number of times the function has been called. getGradient(self) Accessor to the gradient function. Accessor to the number of times the gradient of the function has been called. getHessian(self) Accessor to the hessian function. Accessor to the number of times the hessian of the function has been called. getId(self) Accessor to the object’s id. Accessor to the underlying implementation. Accessor to the description of the input vector. Accessor to the dimension of the input vector. getMarginal(self, \*args) Accessor to marginal. getName(self) Accessor to the object’s name. Accessor to the description of the output vector. Accessor to the number of the outputs. getParameter(self) Accessor to the parameter values. Accessor to the parameter description. Accessor to the dimension of the parameter. gradient(self, inP) Return the Jacobian transposed matrix of the function at a point. hessian(self, inP) Return the hessian of the function at a point. isLinear(self) Accessor to the linearity of the function. isLinearlyDependent(self, index) Accessor to the linearity of the function with regard to a specific variable. parameterGradient(self, inP) Accessor to the gradient against the parameter. setDescription(self, description) Accessor to the description of the inputs and outputs. setEvaluation(self, evaluation) Accessor to the evaluation function. setGradient(self, gradient) Accessor to the gradient function. setHessian(self, hessian) Accessor to the hessian function. setInputDescription(self, inputDescription) Accessor to the description of the input vector. setName(self, name) Accessor to the object’s name. setOutputDescription(self, inputDescription) Accessor to the description of the output vector. setParameter(self, parameter) Accessor to the parameter values. setParameterDescription(self, description) Accessor to the parameter description.
__init__(self, \*args)

Initialize self. See help(type(self)) for accurate signature.

draw(self, \*args)

Draw the output of function as a Graph.

Available usages:

draw(inputMarg, outputMarg, CP, xiMin, xiMax, ptNb)

draw(firstInputMarg, secondInputMarg, outputMarg, CP, xiMin_xjMin, xiMax_xjMax, ptNbs)

draw(xiMin, xiMax, ptNb)

draw(xiMin_xjMin, xiMax_xjMax, ptNbs)

Parameters
outputMarg, inputMargint,

outputMarg is the index of the marginal to draw as a function of the marginal with index inputMarg.

firstInputMarg, secondInputMargint,

In the 2D case, the marginal outputMarg is drawn as a function of the two marginals with indexes firstInputMarg and secondInputMarg.

CPsequence of float

Central point.

xiMin, xiMaxfloat

Define the interval where the curve is plotted.

xiMin_xjMin, xiMax_xjMaxsequence of float of dimension 2.

In the 2D case, define the intervals where the curves are plotted.

ptNbint or list of ints of dimension 2

The number of points to draw the curves.

Notes

We note where and , with and .

• In the first usage:

Draws graph of the given 1D outputMarg marginal as a function of the given 1D inputMarg marginal with respect to the variation of in the interval , when all the other components of are fixed to the corresponding ones of the central point CP. Then OpenTURNS draws the graph: .

• In the second usage:

Draws the iso-curves of the given outputMarg marginal as a function of the given 2D firstInputMarg and secondInputMarg marginals with respect to the variation of in the interval , when all the other components of are fixed to the corresponding ones of the central point CP. Then OpenTURNS draws the graph: .

• In the third usage:

The same as the first usage but only for function .

• In the fourth usage:

The same as the second usage but only for function .

Examples

>>> import openturns as ot
>>> from openturns.viewer import View
>>> f = ot.SymbolicFunction('x', 'sin(2*pi_*x)*exp(-x^2/2)')
>>> graph = f.draw(-1.2, 1.2, 100)
>>> View(graph).show()

getCallsNumber(self)

Accessor to the number of times the function has been called.

Returns
calls_numberint

Integer that counts the number of times the function has been called since its creation.

getClassName(self)

Accessor to the object’s name.

Returns
class_namestr

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

getDescription(self)

Accessor to the description of the inputs and outputs.

Returns
descriptionDescription

Description of the inputs and the outputs.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                         ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getDescription())
[x1,x2,y0]

getEvaluation(self)

Accessor to the evaluation function.

Returns
functionEvaluationImplementation

The evaluation function.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                         ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getEvaluation())
[x1,x2]->[2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6]

getEvaluationCallsNumber(self)

Accessor to the number of times the function has been called.

Returns
evaluation_calls_numberint

Integer that counts the number of times the function has been called since its creation.

getGradient(self)

Returns
gradientGradientImplementation

getGradientCallsNumber(self)

Accessor to the number of times the gradient of the function has been called.

Returns

Integer that counts the number of times the gradient of the Function has been called since its creation. Note that if the gradient is implemented by a finite difference method, the gradient calls number is equal to 0 and the different calls are counted in the evaluation calls number.

getHessian(self)

Accessor to the hessian function.

Returns
hessianHessianImplementation

The hessian function.

getHessianCallsNumber(self)

Accessor to the number of times the hessian of the function has been called.

Returns
hessian_calls_numberint

Integer that counts the number of times the hessian of the Function has been called since its creation. Note that if the hessian is implemented by a finite difference method, the hessian calls number is equal to 0 and the different calls are counted in the evaluation calls number.

getId(self)

Accessor to the object’s id.

Returns
idint

Internal unique identifier.

getImplementation(self)

Accessor to the underlying implementation.

Returns
implImplementation

The implementation class.

getInputDescription(self)

Accessor to the description of the input vector.

Returns
descriptionDescription

Description of the input vector.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getInputDescription())
[x1,x2]

getInputDimension(self)

Accessor to the dimension of the input vector.

Returns
inputDimint

Dimension of the input vector .

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getInputDimension())
2

getMarginal(self, \*args)

Accessor to marginal.

Parameters
indicesint or list of ints

Set of indices for which the marginal is extracted.

Returns
marginalFunction

Function corresponding to either or , with and .

getName(self)

Accessor to the object’s name.

Returns
namestr

The name of the object.

getOutputDescription(self)

Accessor to the description of the output vector.

Returns
descriptionDescription

Description of the output vector.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getOutputDescription())
[y0]

getOutputDimension(self)

Accessor to the number of the outputs.

Returns
number_outputsint

Dimension of the output vector .

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getOutputDimension())
1

getParameter(self)

Accessor to the parameter values.

Returns
parameterPoint

The parameter values.

getParameterDescription(self)

Accessor to the parameter description.

Returns
parameterDescription

The parameter description.

getParameterDimension(self)

Accessor to the dimension of the parameter.

Returns
parameterDimensionint

Dimension of the parameter.

gradient(self, inP)

Return the Jacobian transposed matrix of the function at a point.

Parameters
pointsequence of float

Point where the Jacobian transposed matrix is calculated.

Returns
gradientMatrix

The Jacobian transposed matrix of the function at point.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6','x1 + x2'])
[[ 13.5345   1       ]
[  4.00001  1       ]]

hessian(self, inP)

Return the hessian of the function at a point.

Parameters
pointsequence of float

Point where the hessian of the function is calculated.

Returns
hessianSymmetricTensor

Hessian of the function at point.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6','x1 + x2'])
>>> print(f.hessian([3.14, 4]))
sheet #0
[[ 20          -0.00637061 ]
[ -0.00637061  0          ]]
sheet #1
[[  0           0          ]
[  0           0          ]]

isLinear(self)

Accessor to the linearity of the function.

Returns
linearbool

True if the function is linear, False otherwise.

isLinearlyDependent(self, index)

Accessor to the linearity of the function with regard to a specific variable.

Parameters
indexint

The index of the variable with regard to which linearity is evaluated.

Returns
linearbool

True if the function is linearly dependent on the specified variable, False otherwise.

parameterGradient(self, inP)

Accessor to the gradient against the parameter.

Returns
gradientMatrix

setDescription(self, description)

Accessor to the description of the inputs and outputs.

Parameters
descriptionsequence of str

Description of the inputs and the outputs.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getDescription())
[x1,x2,y0]
>>> f.setDescription(['a','b','y'])
>>> print(f.getDescription())
[a,b,y]

setEvaluation(self, evaluation)

Accessor to the evaluation function.

Parameters
functionEvaluationImplementation

The evaluation function.

setGradient(self, gradient)

Parameters
gradient_functionGradientImplementation

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
...  f.getEvaluation()))

setHessian(self, hessian)

Accessor to the hessian function.

Parameters
hessian_functionHessianImplementation

The hessian function.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                         ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> f.setHessian(ot.CenteredFiniteDifferenceHessian(
...  ot.ResourceMap.GetAsScalar('CenteredFiniteDifferenceHessian-DefaultEpsilon'),
...  f.getEvaluation()))

setInputDescription(self, inputDescription)

Accessor to the description of the input vector.

Parameters
descriptionDescription

Description of the input vector.

setName(self, name)

Accessor to the object’s name.

Parameters
namestr

The name of the object.

setOutputDescription(self, inputDescription)

Accessor to the description of the output vector.

Parameters
descriptionDescription

Description of the output vector.

setParameter(self, parameter)

Accessor to the parameter values.

Parameters
parametersequence of float

The parameter values.

setParameterDescription(self, description)

Accessor to the parameter description.

Parameters
parameterDescription

The parameter description.