NumericalMathFunction¶
-
class
NumericalMathFunction
(*args)¶ Function base class.
- Available constructors:
NumericalMathFunction(inputs, outputs, formulas)
NumericalMathFunction(inputs, formulas)
NumericalMathFunction(inputString, formulaString, outputString = ‘outputVariable’)
NumericalMathFunction(f, g)
NumericalMathFunction(functionCollection)
NumericalMathFunction(functionCollection, scalarCoefficientColl)
NumericalMathFunction(scalarFunctionCollection, vectorCoefficientColl)
NumericalMathFunction(function, comparisonOperator, threshold)
NumericalMathFunction(function, indices, referencePoint, parametersSet=True)
NumericalMathFunction(inputSample, outputSample)
Parameters: inputs : sequence of str
Ordered list of input variables names of the NumericalMathFunction.
outputs : sequence of str
Ordered list of output variables names of the NumericalMathFunction. If it is not specified, default names are created for the output variables.
formulas : sequence of str
Ordered list of analytical formulas between the inputs and the outputs. The NumericalMathFunction is defined by ouputs = formulas(inputs).
inputString : str
Description of the NumericalMathFunction‘s input.
outputString : str
Description of the NumericalMathFunction‘s output.
formulaString : str
Analytical formula of the NumericalMathFunction. The NumericalMathFunction is defined by ouputString = formulaString(inputString).
Available functions:
- sin
- cos
- tan
- asin
- acos
- atan
- sinh
- cosh
- tanh
- asinh
- acosh
- atanh
- log2
- log10
- log
- ln
- lngamma
- gamma
- exp
- erf
- erfc
- sqrt
- cbrt
- besselJ0
- besselJ1
- besselY0
- besselY1
- sign
- rint
- abs
- min
- max
- sum
- avg
- floor
- ceil
- trunc
- round
Available operators:
- = (assignment)
- && (logical and)
- || (logical or)
- <= (less or equal)
- >= (greater or equal)
- != (not equal)
- == (equal)
- > (greater than)
- < (less than)
- + (addition)
- - (subtraction)
- * (multiplication)
- / (division)
- ^ (raise x to the power of y)
Available constants:
- _e (Euler’s constant)
- _pi (Pi)
f,g : two
NumericalMathFunction
The NumericalMathFunction is the composition function
.
functionCollection : list of
NumericalMathFunction
Collection of several NumericalMathFunction.
scalarCoefficientColl : sequence of float
Collection of scalar weights.
scalarFunctionCollection : list of
NumericalMathFunction
Collection of several scalar NumericalMathFunction.
vectorCoefficientColl : 2-d sequence of float
Collection of vectorial weights.
function :
NumericalMathFunction
Function from which another function is created.
comparisonOperator :
ComparisonOperator
Comparison operator.
threshold : float
Threshold from which values are compared.
indices : sequence of int
Indices of the set variables which are set to referencePoint‘s values.
parametersSet : bool
If True (default), the set variables are the ones referenced in indices. Otherwise, the set variables are the ones referenced in the complementary vector of indices.
referencePoint : sequence of float
Values of the set variables. Must be of size of indices if parametersSet is True (default), else its size should be the complementary size of indices.
inputSample : 2-d sequence of float
Values of the inputs.
outputSample : 2-d sequence of float
Values of the outputs.
Examples
Create a NumericalMathFunction from a list of analytical formulas and descriptions of the inputs and the outputs :
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x0', 'x1'], ['y0', 'y1'], ... ['x0 + x1', 'x0 - x1']) >>> print(f([1, 2])) [3,-1]
Create a NumericalMathFunction from strings:
>>> import openturns as ot >>> f = ot.NumericalMathFunction('x', '2.0*sqrt(x)', 'y') >>> print(f(([16],[4]))) [ y ] 0 : [ 8 ] 1 : [ 4 ]
Create a NumericalMathFunction 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 NumericalMathFunction from another NumericalMathFunction:
>>> f = ot.NumericalMathFunction(ot.Description.BuildDefault(4, 'x'), ... ['x0', 'x0 + x1', 'x0 + x2 + x3'])
Then create another function by setting x1=4 and x3=10:
>>> g = ot.NumericalMathFunction(f, [3, 1], [10, 4], True) >>> print(g.getInputDescription()) [x0,x2] >>> print(g((1, 2))) [1,5,13]
Or by setting x0=6 and x2=5:
>>> g = ot.NumericalMathFunction(f, [3, 1], [6, 5], False) >>> print(g.getInputDescription()) [x3,x1] >>> print(g((1, 2))) [6,8,12]
Create a NumericalMathFunction from another NumericalMathFunction and by using a comparison operator:
>>> analytical = ot.NumericalMathFunction(['x0','x1'], ['y'], ['x0 + x1']) >>> indicator = ot.NumericalMathFunction(analytical, ot.Less(), 0.0) >>> print(indicator([2, 3])) [0] >>> print(indicator([2, -3])) [1]
Create a NumericalMathFunction from a collection of functions:
>>> functions = list() >>> functions.append(ot.NumericalMathFunction(['x1', 'x2', 'x3'], ['y1', 'y2'], ... ['x1^2 + x2', 'x1 + x2 + x3'])) >>> functions.append(ot.NumericalMathFunction(['x1', 'x2', 'x3'], ['y1', 'y2'], ... ['x1 + 2 * x2 + x3', 'x1 + x2 - x3'])) >>> myFunction = ot.NumericalMathFunction(functions) >>> print(myFunction([1.0, 2.0, 3.0])) [3,6,8,0]
Create a NumericalMathFunction 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.NumericalMathFunction(functions, [2.0, 4.0]) >>> print(myFunction2([1.0, 2.0, 3.0])) [38,12]
Create a NumericalMathFunction 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.NumericalMathFunction(['x1', 'x2', 'x3'], ['y1'], ... ['x1 + 2 * x2 + x3'])) >>> functions.append(ot.NumericalMathFunction(['x1', 'x2', 'x3'], ['y1'], ... ['x1^2 + x2'])) >>> myFunction2 = ot.NumericalMathFunction(functions, [[2., 4.], [3., 1.]]) >>> print(myFunction2([1, 2, 3])) [25,35]
Create a NumericalMathFunction from values of the inputs and the outputs:
>>> inputSample = [[1.0, 1.0], [2.0, 2.0]] >>> outputSample = [[4.0], [5.0]] >>> database = ot.NumericalMathFunction(inputSample, outputSample) >>> x = [1.8]*database.getInputDimension() >>> print(database(x)) [5]
Create a NumericalMathFunction which is the composition function
:
>>> g = ot.NumericalMathFunction(['x1', 'x2'], ['y1', 'y2'], ... ['x1 + x2','3 * x1 * x2']) >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y'], ['2 * x1 - x2']) >>> composed = ot.NumericalMathFunction(f, g) >>> print(composed([3, 4])) [-22]
Methods
GetValidConstants
GetValidFunctions
GetValidOperators
__call__
(*args)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 inputs. getInputDimension
()Accessor to the number of the inputs. 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 outputs. 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)¶
-
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.NumericalMathFunction('x', 'sin(2*_pi*x)*exp(-x^2/2)', 'y') >>> 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 :
NumericalSample
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 :
NumericalSample
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.NumericalMathFunction(['x1', 'x2'], ['y'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getDescription()) [x1,x2,y]
-
getEvaluation
()¶ Accessor to the evaluation function.
Returns: function :
NumericalMathEvaluationImplementation
The evaluation function.
Examples
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y'], ... ['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
()¶ Accessor to the gradient function.
Returns: gradient :
NumericalMathGradientImplementation
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 NumericalMathFunction 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 :
NumericalMathHessianImplementation
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 NumericalMathFunction 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 :
NumericalSample
All the input numerical points stored in the history mecanism.
-
getHistoryOutput
()¶ Accessor to the history of the output values.
Returns: output_history :
NumericalSample
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 inputs.
Returns: description :
Description
Description of the inputs.
Examples
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getInputDescription()) [x1,x2]
-
getInputDimension
()¶ Accessor to the number of the inputs.
Returns: number_inputs : int
Number of inputs.
Examples
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y'], ... ['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 :
NumericalSample
All the input parameters stored in the history mecanism.
-
getInputPointHistory
()¶ Accessor to the history of the input point values.
Returns: history :
NumericalSample
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.
Returns: marginal :
NumericalMathFunction
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 outputs.
Returns: description :
Description
Description of the outputs.
Examples
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getOutputDescription()) [y]
-
getOutputDimension
()¶ Accessor to the number of the outputs.
Returns: number_outputs : int
Number of outputs.
Examples
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getOutputDimension()) 1
-
getParameter
()¶ Accessor to the parameter values.
Returns: parameter :
NumericalPoint
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.
Returns: gradient :
Matrix
The Jacobian transposed matrix of the function at point.
Examples
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y','z'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6','x1 + x2']) >>> print(f.gradient([3.14, 4])) [[ 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.
Returns: hessian :
SymmetricTensor
Hessian of the function at point.
Examples
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y','z'], ... ['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.NumericalMathFunction(['x1', 'x2'], ['y'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getDescription()) [x1,x2,y] >>> f.setDescription(['a','b','y']) >>> print(f.getDescription()) [a,b,y]
-
setEvaluation
(evaluation)¶ Accessor to the evaluation function.
Parameters: function :
NumericalMathEvaluationImplementation
The evaluation function.
-
setGradient
(gradient)¶ Accessor to the gradient function.
Parameters: gradient_function :
NumericalMathGradientImplementation
The gradient function.
Examples
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> f.setGradient(ot.CenteredFiniteDifferenceGradient( ... ot.ResourceMap.GetAsNumericalScalar('CenteredFiniteDifferenceGradient-DefaultEpsilon'), ... f.getEvaluation()))
-
setHessian
(hessian)¶ Accessor to the hessian function.
Parameters: hessian_function :
NumericalMathHessianImplementation
The hessian function.
Examples
>>> import openturns as ot >>> f = ot.NumericalMathFunction(['x1', 'x2'], ['y'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> f.setHessian(ot.CenteredFiniteDifferenceHessian( ... ot.ResourceMap.GetAsNumericalScalar('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.