SymbolicFunction

class SymbolicFunction(*args)

Symbolic function.

Available constructor:
SymbolicFunction(inputs, formulas)
Parameters:
inputssequence of str, or str

List of input variables names of the function.

formulassequence of str, or str

List of analytical formulas between the inputs and the outputs. The function is defined by ouputs = formulas(inputs).

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:
  • <= (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)

Notes

Up to version 1.10, OpenTURNS relied on muParser to parse analytical formulas. Since version 1.11, ExprTk is used by default, but both parsers can be used, if their support haved been compiled in. This is controlled by the ‘SymbolicParser-Backend’ ResourceMap entry.

Examples

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

ExprTk allows to write multiple outputs; in this case, the constructor has a special syntax, it contains input variables names, but also output variables names, and formula is a string:

>>> import openturns as ot
>>> ot.ResourceMap.Set('SymbolicParser-Backend', 'ExprTk')
>>> f = ot.SymbolicFunction(['x0', 'x1'], ['y0', 'y1'], 'y0 := x0 + x1; y1 := x0 - x1')
>>> print(f([1, 2]))
[3,-1]

See ExprTk documentation for details.

Attributes:
thisown

The membership flag

Methods

GetValidConstants() Return the list of valid constants.
GetValidFunctions() Return the list of valid functions.
GetValidOperators() Return the list of valid operators.
GetValidParsers() Return the list of built-in parsers.
__call__(*args) Call self as a function.
draw(*args) Draw the output of function as a Graph.
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.
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.
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(inP) Return the Jacobian transposed matrix of the function at a point.
hessian(inP) Return the hessian of the function at a point.
parameterGradient(inP) 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.
setInputDescription(inputDescription) Accessor to the description of the input vector.
setName(name) Accessor to the object’s name.
setOutputDescription(inputDescription) Accessor to the description of the output vector.
setParameter(parameter) Accessor to the parameter values.
setParameterDescription(description) Accessor to the parameter description.
__init__(*args)

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

static GetValidConstants()

Return the list of valid constants.

Returns:
list_constantsDescription

List of the available constants.

Examples

>>> import openturns as ot
>>> print(ot.SymbolicFunction.GetValidConstants()[0])
e_ -> Euler's constant (2.71828...)
static GetValidFunctions()

Return the list of valid functions.

Returns:
list_functionsDescription

List of the available functions.

Examples

>>> import openturns as ot
>>> print(ot.SymbolicFunction.GetValidFunctions()[0])
sin(arg) -> sine function
static GetValidOperators()

Return the list of valid operators.

Returns:
list_operatorsDescription

List of the available operators.

Examples

>>> import openturns as ot
>>> print(ot.SymbolicFunction.GetValidOperators()[0])
= -> assignement, can only be applied to variable names (priority -1)
static GetValidParsers()

Return the list of built-in parsers.

Analytical formulas can be parsed by ‘MuParser’ or ‘ExprTk’ parsers, but this support may be disabled at build-time. This method returns the list of parsers available at run-time. Parser can be switched by changing ‘SymbolicParser-Backend’ ResourceMap entry.

Returns:
list_constantsDescription

List of the available parsers.

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, inputMargint, outputMarg, inputMarg \geq 0

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

firstInputMarg, secondInputMargint, firstInputMarg, secondInputMarg \geq 0

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 ptNb > 0 or list of ints of dimension 2 ptNb_k > 0, k=1,2

The number of points to draw the curves.

Notes

We note f: \Rset^n \rightarrow \Rset^p where \vect{x} = (x_1, \dots, x_n) and f(\vect{x}) = (f_1(\vect{x}), \dots,f_p(\vect{x})), with n\geq 1 and p\geq 1.

  • In the first usage:

Draws graph of the given 1D outputMarg marginal f_k: \Rset^n \rightarrow \Rset as a function of the given 1D inputMarg marginal with respect to the variation of x_i in the interval [x_i^{min}, x_i^{max}], when all the other components of \vect{x} are fixed to the corresponding ones of the central point CP. Then OpenTURNS draws the graph: t\in [x_i^{min}, x_i^{max}] \mapsto f_k(CP_1, \dots, CP_{i-1}, t,  CP_{i+1} \dots, CP_n).

  • In the second usage:

Draws the iso-curves of the given outputMarg marginal f_k as a function of the given 2D firstInputMarg and secondInputMarg marginals with respect to the variation of (x_i, x_j) in the interval [x_i^{min}, x_i^{max}] \times [x_j^{min}, x_j^{max}], when all the other components of \vect{x} are fixed to the corresponding ones of the central point CP. Then OpenTURNS draws the graph: (t,u) \in [x_i^{min}, x_i^{max}] \times [x_j^{min}, x_j^{max}] \mapsto f_k(CP_1, \dots, CP_{i-1}, t, CP_{i+1}, \dots, CP_{j-1}, u,  CP_{j+1} \dots, CP_n).

  • In the third usage:

The same as the first usage but only for function f: \Rset \rightarrow \Rset.

  • In the fourth usage:

The same as the second usage but only for function f: \Rset^2 \rightarrow \Rset.

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()

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()

Accessor to the object’s name.

Returns:
class_namestr

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

getDescription()

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()

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()

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()

Accessor to the gradient function.

Returns:
gradientGradientImplementation

The gradient function.

getGradientCallsNumber()

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

Returns:
gradient_calls_numberint

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:
hessianHessianImplementation

The hessian function.

getHessianCallsNumber()

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()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

getImplementation(*args)

Accessor to the underlying implementation.

Returns:
implImplementation

The implementation class.

getInputDescription()

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()

Accessor to the dimension of the input vector.

Returns:
inputDimint

Dimension of the input vector d.

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(*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 f_i or (f_i)_{i \in indices}, with f:\Rset^n \rightarrow \Rset^p and f=(f_0 , \dots, f_{p-1}).

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getOutputDescription()

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()

Accessor to the number of the outputs.

Returns:
number_outputsint

Dimension of the output vector d'.

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:
parameterPoint

The parameter values.

getParameterDescription()

Accessor to the parameter description.

Returns:
parameterDescription

The parameter description.

getParameterDimension()

Accessor to the dimension of the parameter.

Returns:
parameterDimensionint

Dimension of the parameter.

gradient(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'])
>>> print(f.gradient([3.14, 4]))
[[ 13.5345   1       ]
 [  4.00001  1       ]]
hessian(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          ]]
parameterGradient(inP)

Accessor to the gradient against the parameter.

Returns:
gradientMatrix

The gradient.

setDescription(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(evaluation)

Accessor to the evaluation function.

Parameters:
functionEvaluationImplementation

The evaluation function.

setGradient(gradient)

Accessor to the gradient function.

Parameters:
gradient_functionGradientImplementation

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.setGradient(ot.CenteredFiniteDifferenceGradient(
...  ot.ResourceMap.GetAsScalar('CenteredFiniteDifferenceGradient-DefaultEpsilon'),
...  f.getEvaluation()))
setHessian(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(inputDescription)

Accessor to the description of the input vector.

Parameters:
descriptionDescription

Description of the input vector.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

setOutputDescription(inputDescription)

Accessor to the description of the output vector.

Parameters:
descriptionDescription

Description of the output vector.

setParameter(parameter)

Accessor to the parameter values.

Parameters:
parametersequence of float

The parameter values.

setParameterDescription(description)

Accessor to the parameter description.

Parameters:
parameterDescription

The parameter description.

thisown

The membership flag