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 have been compiled. 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
>>> f = ot.SymbolicFunction(['x0', 'x1'], ['y0', 'y1'], 'y0 := x0 + x1; y1 := x0 - x1')
>>> print(f([1, 2]))
[3,-1]

The following example uses the min and sqrt functions:

>>> formula = 'min(-x1 - x2 - x3 + 3 * sqrt(3), -x3 + 3)'
>>> limitStateFunction = ot.SymbolicFunction(['x1', 'x2', 'x3'], [formula])
>>> print(limitStateFunction([1, 2, 3]))
[-0.803848]

The following example splits the formula into four parts to manage its length:

>>> formula = '15.59 * 1e4 - x1 *x2^3 / (2 * x3^3) *'
>>> formula += '((x4^2 - 4 * x5 * x6 * x7^2 + '
>>> formula += 'x4 * (x6 + 4 * x5 + 2 *x6 * x7)) / '
>>> formula += '(x4 * x5 * (x4 + x6 + 2 *x6 *x7)))'
>>> input_variables = ['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7']
>>> limitStateFunction = ot.SymbolicFunction(input_variables, [formula])
>>> print(limitStateFunction([1, 2, 3, 4, 5, 6, 7]))
[155900]

ExprTk allows to manage intermediate variables with the var keyword. This is convenient in the situation where several outputs require the same intermediate calculation or if the output is a complex function of the input. In the following example, we compute the alpha variable which is the slope of the river in the flooding example. This slope is then used in the computation of the height H.

>>> import openturns as ot
>>> inputs = ['Q', 'Ks', 'Zv', 'Zm', 'Hd', 'Zb', 'L', 'B']
>>> outputs = ['H', 'S']
>>> formula = 'var alpha := (Zm - Zv)/L; H := (Q/(Ks*B*sqrt(alpha)))^(3.0/5.0);'
>>> formula += 'var Zc := H + Zv; var Zd := Zb + Hd; S := Zc - Zd'
>>> myFunction = ot.SymbolicFunction(inputs, outputs, formula)
>>> X = [1013.0, 30.0, 50.0, 55.0, 8, 55.5, 5000.0, 300.0]
>>> print(myFunction(X))
[2.142,-11.358]

The following example illustrates a function for a system of two components.

>>> equations = ['var g1 := x1^2 -8 * x2 + 16']
>>> equations.append('var g2 := -16 * x1 + x2 + 32')
>>> equations.append('gsys := max(g1, g2)')
>>> formula = ';'.join(equations)
>>> limitStateFunction = ot.SymbolicFunction(['x1', 'x2'], ['gsys'], formula)
>>> print(limitStateFunction([1, 2]))
[18]

See the ExprTk documentation for details.

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`()	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.
`isLinear`()	Accessor to the linearity of the function.
`isLinearlyDependent`(index)	Accessor to the linearity of the function with regard to a specific variable.
`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()¶

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          ]]

isLinear()¶

Accessor to the linearity of the function.

Returns

linearbool: True if the function is linear, False otherwise.

isLinearlyDependent(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(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.

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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