# 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__(…) <==> x(…) addCacheContent(inSample, outSample) Add input numerical points and associated output to the cache. clearCache() Empty the content of the cache. clearHistory() Empty the content of the history. disableCache() Disable the cache mechanism. disableHistory() Disable the history mechanism. draw(*args) Draw the output of function as a Graph. enableCache() Enable the cache mechanism. enableHistory() Enable the history mechanism. getCacheHits() Accessor to the number of computations saved thanks to the cache mecanism. getCacheInput() Accessor to all the input numerical points stored in the cache mecanism. getCacheOutput() Accessor to all the output numerical points stored in the cache mecanism. getCallsNumber() Accessor to the number of times the function has been called. getClassName() Accessor to the object’s name. getDescription() Accessor to the description of the inputs and outputs. getEvaluation() Accessor to the evaluation function. getEvaluationCallsNumber() Accessor to the number of times the function has been called. getGradient() Accessor to the gradient function. getGradientCallsNumber() Accessor to the number of times the gradient of the function has been called. getHessian() Accessor to the hessian function. getHessianCallsNumber() Accessor to the number of times the hessian of the function has been called. getHistoryInput() Accessor to the history of the input values. getHistoryOutput() Accessor to the history of the output values. getId() Accessor to the object’s id. getImplementation(*args) Accessor to the underlying implementation. getInputDescription() Accessor to the description of the input vector. getInputDimension() Accessor to the dimension of the input vector. getInputParameterHistory() Accessor to the history of the input parameter values. getInputPointHistory() Accessor to the history of the input point values. getMarginal(*args) Accessor to marginal. getName() Accessor to the object’s name. getOutputDescription() Accessor to the description of the output vector. getOutputDimension() Accessor to the number of the outputs. getParameter() Accessor to the parameter values. getParameterDescription() Accessor to the parameter description. getParameterDimension() Accessor to the dimension of the parameter. gradient(*args) Return the Jacobian transposed matrix of the function at a point. hessian(*args) Return the hessian of the function at a point. isCacheEnabled() Test whether the cache mechanism is enabled or not. isHistoryEnabled() Test whether the history mechanism is enabled or not. parameterGradient(*args) Accessor to the gradient against the parameter. setDescription(description) Accessor to the description of the inputs and outputs. setEvaluation(evaluation) Accessor to the evaluation function. setGradient(gradient) Accessor to the gradient function. setHessian(hessian) Accessor to the hessian function. setName(name) Accessor to the object’s name. setParameter(parameter) Accessor to the parameter values. setParameterDescription(description) Accessor to the parameter description.
__init__(*args)

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

addCacheContent(inSample, outSample)

Add input numerical points and associated output to the cache.

Parameters: input_sample : 2-d sequence of float Input numerical points to be added to the cache. output_sample : 2-d sequence of float Output numerical points associated with the input_sample to be added to the cache.
clearCache()

Empty the content of the cache.

clearHistory()

Empty the content of the history.

disableCache()

Disable the cache mechanism.

disableHistory()

Disable the history mechanism.

draw(*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, inputMarg : int, outputMarg is the index of the marginal to draw as a function of the marginal with index inputMarg. firstInputMarg, secondInputMarg : int, In the 2D case, the marginal outputMarg is drawn as a function of the two marginals with indexes firstInputMarg and secondInputMarg. CP : sequence of float Central point. xiMin, xiMax : float Define the interval where the curve is plotted. xiMin_xjMin, xiMax_xjMax : sequence of float of dimension 2. In the 2D case, define the intervals where the curves are plotted. ptNb : int 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()

enableCache()

Enable the cache mechanism.

enableHistory()

Enable the history mechanism.

getCacheHits()

Accessor to the number of computations saved thanks to the cache mecanism.

Returns: cacheHits : int Integer that counts the number of computations saved thanks to the cache mecanism.
getCacheInput()

Accessor to all the input numerical points stored in the cache mecanism.

Returns: cacheInput : Sample All the input numerical points stored in the cache mecanism.
getCacheOutput()

Accessor to all the output numerical points stored in the cache mecanism.

Returns: cacheInput : Sample All the output numerical points stored in the cache mecanism.
getCallsNumber()

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

Returns: calls_number : int Integer that counts the number of times the function has been called since its creation.
getClassName()

Accessor to the object’s name.

Returns: class_name : str The object class name (object.__class__.__name__).
getDescription()

Accessor to the description of the inputs and outputs.

Returns: description : Description 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()

Accessor to the evaluation function.

Returns: function : EvaluationImplementation 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()

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

Returns: evaluation_calls_number : int Integer that counts the number of times the function has been called since its creation.
getGradient()

Returns: gradient : GradientImplementation The gradient function.
getGradientCallsNumber()

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

Returns: gradient_calls_number : int 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()

Accessor to the hessian function.

Returns: hessian : HessianImplementation The hessian function.
getHessianCallsNumber()

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

Returns: hessian_calls_number : int 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.
getHistoryInput()

Accessor to the history of the input values.

Returns: input_history : Sample All the input numerical points stored in the history mecanism.
getHistoryOutput()

Accessor to the history of the output values.

Returns: output_history : Sample All the output numerical points stored in the history mecanism.
getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getImplementation(*args)

Accessor to the underlying implementation.

Returns: impl : Implementation The implementation class.
getInputDescription()

Accessor to the description of the input vector.

Returns: description : Description 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()

Accessor to the dimension of the input vector.

Returns: inputDim : int 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

getInputParameterHistory()

Accessor to the history of the input parameter values.

Returns: history : Sample All the input parameters stored in the history mecanism.
getInputPointHistory()

Accessor to the history of the input point values.

Returns: history : Sample All the input points stored in the history mecanism.
getMarginal(*args)

Accessor to marginal.

Parameters: indices : int or list of ints Set of indices for which the marginal is extracted. marginal : Function Function corresponding to either or , with and .
getName()

Accessor to the object’s name.

Returns: name : str The name of the object.
getOutputDescription()

Accessor to the description of the output vector.

Returns: description : Description 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()

Accessor to the number of the outputs.

Returns: number_outputs : int 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()

Accessor to the parameter values.

Returns: parameter : Point The parameter values.
getParameterDescription()

Accessor to the parameter description.

Returns: parameter : Description The parameter description.
getParameterDimension()

Accessor to the dimension of the parameter.

Returns: parameterDimension : int Dimension of the parameter.
gradient(*args)

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

Parameters: point : sequence of float Point where the Jacobian transposed matrix is calculated. gradient : Matrix 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(*args)

Return the hessian of the function at a point.

Parameters: point : sequence of float Point where the hessian of the function is calculated. hessian : SymmetricTensor 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          ]]

isCacheEnabled()

Test whether the cache mechanism is enabled or not.

Returns: isCacheEnabled : bool Flag telling whether the cache mechanism is enabled. It is disabled by default.
isHistoryEnabled()

Test whether the history mechanism is enabled or not.

Returns: isHistoryEnabled : bool Flag telling whether the history mechanism is enabled. It is disabled by default.
parameterGradient(*args)

Accessor to the gradient against the parameter.

Returns: gradient : Matrix The gradient.
setDescription(description)

Accessor to the description of the inputs and outputs.

Parameters: description : sequence 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(evaluation)

Accessor to the evaluation function.

Parameters: function : EvaluationImplementation The evaluation function.
setGradient(gradient)

Parameters: gradient_function : GradientImplementation The gradient function.

Examples

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

setHessian(hessian)

Accessor to the hessian function.

Parameters: hessian_function : HessianImplementation 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()))

setName(name)

Accessor to the object’s name.

Parameters: name : str The name of the object.
setParameter(parameter)

Accessor to the parameter values.

Parameters: parameter : sequence of float The parameter values.
setParameterDescription(description)

Accessor to the parameter description.

Parameters: parameter : Description The parameter description.