Manipulate Expressions

We start with setting the Feel++ environment and loading the Feel++ library.

Set the Feel++ environment with local repository

import feelpp
import sys,math
import plotly.express as px
import plotly.graph_objects as go

app = feelpp.Environment(["myapp"],config=feelpp.localRepository(""))
print("feelpp version: ",feelpp.Info.version())

python

Results

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
File :1
----> 1 import feelpp
      2 import sys,math
      3 import plotly.express as px

ModuleNotFoundError: No module named 'feelpp'

1. Create and evaluate expressions

An expression is a mathematical expression that can be evaluated at a given point and depend on multiple variables. It can be a scalar, a vectorial or a matricial expression. The expression can be created from a string. The expression can be then evaluated at a given point.

1.1. Scalar expression

Create a scalar expression abd evaluate the expression at a given point (1,2)

expr = feelpp.expr("x+y:x:y") (1)
expr.evaluate({"x":1,"y":2}) (2)

python

Results

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
File :1
----> 1 expr = feelpp.expr("x+y:x:y")
      2 expr.evaluate({"x":1,"y":2})

NameError: name 'feelpp' is not defined

1	Create a scalar expression from a string, the symbols are listed at the end of the string using `:` as separator.
2	Evaluate the expression at the point (1,2), it returns a array with one element.

The expression can also be evaluated at a set of points.

Evaluate the expression at a set of points

expr.setParameterValues({"y":2}) (1)
import numpy as np
x=np.linspace(0,1,10) (2)
expr.evaluate("x",x) (3)

python

Results

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
File :1
----> 1 expr.setParameterValues({"y":2})
      2 import numpy as np
      3 x=np.linspace(0,1,10)

NameError: name 'expr' is not defined

1	Set the value of the parameter `y` to 2
2	Create a numpy array with 10 points between 0 and 1
3	Evaluate the expression at the 10 equidistributed points between 0 and 1, it returns a array with two elements.

1.2. Vectorial expression

We start with an expression that depends on two variables x and y and create a vectorial expression with the expression x and y.

Create a vectorial expression abd evaluate the expression at a given point (1,2)

expr21 = feelpp.expr("{x,y}:x:y",row=2,col=1) (1)
expr21.evaluate({"x":1,"y":2}) (2)

python

Results

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
File :1
----> 1 expr21 = feelpp.expr("{x,y}:x:y",row=2,col=1)
      2 expr21.evaluate({"x":1,"y":2})

NameError: name 'feelpp' is not defined

1	Create a vectorial expression from a string, the symbols are listed at the end of the string using `:` as separator.
2	Evaluate the expression at the point (1,2), it returns a array with two elements.

Now we turn to a vectorial expression with 3 components.

expr31 = feelpp.expr("{x,y,z+x}:x:y:z",row=3,col=1) (1)
expr31.evaluate({"x":1,"y":2,"z":10}) (2)

python

Results

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
File :1
----> 1 expr31 = feelpp.expr("{x,y,z+x}:x:y:z",row=3,col=1)
      2 expr31.evaluate({"x":1,"y":2,"z":10})

NameError: name 'feelpp' is not defined

1	Create a vectorial expression from a string, the symbols are listed at the end of the string using `:` as separator.
2	Evaluate the expression at the point (1,2,10), it returns a array with three elements.

1.3. Matricial expression

We start with a 2x2 matricial expression.

Create a matricial expression abd evaluate the expression at a given point (1,2)

expr22 = feelpp.expr("{x+y,x,y,y-x}:x:y", row=2, col=2)  (1)
expr22.evaluate({"x":1,"y":2}) (2)

python

Results

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
File :1
----> 1 expr22 = feelpp.expr("{x+y,x,y,y-x}:x:y", row=2, col=2)
      2 expr22.evaluate({"x":1,"y":2})

NameError: name 'feelpp' is not defined

1	Create a matricial expression from a string, the symbols are listed at the end of the string using `:` as separator.
2	Evaluate the expression at the point (1,2), it returns a array with four elements.

We now turn to a 3x3 matricial expression.

expr33 = feelpp.expr("{x,y,z,x-y,y-y,z+x+y,z,y,x}:x:y:z",row=3,col=3) (1)
expr33.evaluate({"x":1,"y":2,"z":3}) (2)

python

Results

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
File :1
----> 1 expr33 = feelpp.expr("{x,y,z,x-y,y-y,z+x+y,z,y,x}:x:y:z",row=3,col=3)
      2 expr33.evaluate({"x":1,"y":2,"z":3})

NameError: name 'feelpp' is not defined

1	Create a matricial expression from a string, the symbols are listed at the end of the string using `:` as separator.
2	Evaluate the expression at the point (1,2,3), it returns a array with nine elements.

2. Differentiation

The member functions diff and diff2 allows to compute the first and second symbolic derivatives of a function. The first argument is the symbol with respect to which the derivative is computed.

ex=feelpp.expr("a*sin(x):x:a")
ex.setParameterValues({"a":1})
exd = ex.diff("x") (1)
exd2 = ex.diff2("x") (2)
exda = ex.diff("a") (3)
exdax = ex.diff("a").diff("x") (4)
x=np.linspace(0,2*math.pi,200)
print(f"   ex: {ex}")
print(f"  exd: {exd}")
print(f" exd2: {exd2}")
print(f" exda: {exda}")
print(f"exdaa: {exdax}")

python

Results

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
File :1
----> 1 ex=feelpp.expr("a*sin(x):x:a")
      2 ex.setParameterValues({"a":1})
      3 exd = ex.diff("x")

NameError: name 'feelpp' is not defined

1	First derivative with respect to `x` $\frac{\partial \cdot }{\partial x}$
2	Second derivative with respect to `x` $\frac{\partial^2 \cdot }{\partial^2 x}$
3	First derivative with respect to `a` $\frac{\partial \cdot }{\partial a}$
4	First derivative with respect to `a` and then with respect to `x` $\frac{\partial^2 \cdot}{\partial x \partial a}$

Now let’s plot the expression and its derivative. The first and last are the same.

fig = go.Figure()
fig.add_trace(go.Scatter(x=x, y=ex.evaluate("x",x), name="sin(x)"))
fig.add_trace(go.Scatter(x=x, y=exd.evaluate("x",x), name="cos(x)"))
fig.add_trace(go.Scatter(x=x, y=exd2.evaluate("x",x), name="-sin(x)"))
fig.add_trace(go.Scatter(x=x, y=exda.evaluate("x",x), name="sin(x)"))
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 fig = go.Figure() 2 fig.add_trace(go.Scatter(x=x, y=ex.evaluate("x",x), name="sin(x)")) 3 fig.add_trace(go.Scatter(x=x, y=exd.evaluate("x",x), name="cos(x)")) NameError: name 'go' is not defined

2.1. Derivative of a vectorial expression

Here is an example of a vectorial expression and its derivative with respect to x and y.

expr21 = feelpp.expr("{x^3,x*y^2}:x:y", row=2, col=1)  (1)
print("d(x,y)/dx = ",expr21.diff("x")) (2)
print("d(x,y)/dy = ", expr21.diff("y")) (3)
print("d2(x,y)/dxdy = ", expr21.diff("x").diff("y")) (4)
print("d2(x,y)/dydx = ", expr21.diff("y").diff("x"))   (5)

python

Results

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
File :1
----> 1 expr21 = feelpp.expr("{x^3,x*y^2}:x:y", row=2, col=1)
      2 print("d(x,y)/dx = ",expr21.diff("x"))
      3 print("d(x,y)/dy = ", expr21.diff("y"))

NameError: name 'feelpp' is not defined

1	Create a vectorial expression from a string, the symbols are listed at the end of the string using `:` as separator.
2	First derivative with respect to `x` $\frac{\partial \cdot }{\partial x}$
3	First derivative with respect to `y` $\frac{\partial \cdot }{\partial y}$
4	First derivative with respect to `x` and then with respect to `y` $\frac{\partial^2 \cdot}{\partial x \partial y}$
5	First derivative with respect to `y` and then with respect to `x` $\frac{\partial^2 \cdot}{\partial y \partial x}$

similar results are obtained for matricia expressions.

3. Special functions

The Feel++ library provides a set of special functions that can be used in expressions.

3.1. function with multiple parameters

ex=feelpp.expr("sin(w*x+b):x:w:b")
ex.setParameterValues({"w":2*math.pi,"b":0.5})
x=np.linspace(0,2*math.pi,100)
y=ex.evaluate("x",x)

fig = px.line(x=x, y=y, title="sin",labels={"x":"x","y":r'y'})
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 ex=feelpp.expr("sin(w*x+b):x:w:b") 2 ex.setParameterValues({"w":2*math.pi,"b":0.5}) 3 x=np.linspace(0,2*math.pi,100) NameError: name 'feelpp' is not defined

3.2. clamp

The function clamp allows to clamp a value between two bounds.

ex=feelpp.expr("clamp(sin(w*x+b),-0.3,0.4):x:w:b")
ex.setParameterValues({"w":2*math.pi,"b":0.5})
x=np.linspace(0,2*math.pi,100)
y=ex.evaluate("x",x)

fig = px.line(x=x, y=y, title="f",labels={"x":"x","y":r'y'})
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 ex=feelpp.expr("clamp(sin(w*x+b),-0.3,0.4):x:w:b") 2 ex.setParameterValues({"w":2*math.pi,"b":0.5}) 3 x=np.linspace(0,2*math.pi,100) NameError: name 'feelpp' is not defined

3.3. mapabcd

The function mapabcd allows to map a value from a given interval to another interval. The first argument is the value to map, the second and third arguments are the lower and upper bounds of the interval to map from, the fourth and fifth arguments are the lower and upper bounds of the interval to map to.

ex=feelpp.expr("mapabcd(x,-1,1,0,1):x")
x=np.linspace(-1,1,100)
y=ex.evaluate("x",x)

fig = px.line(x=x, y=y, title="f",labels={"x":"x","y":r'y'})
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 ex=feelpp.expr("mapabcd(x,-1,1,0,1):x") 2 x=np.linspace(-1,1,100) 3 y=ex.evaluate("x",x) NameError: name 'feelpp' is not defined

3.4. step1

The function step1 is a step function that is equal to 1 if the argument is greater than 0 and 0 otherwise.

ex=feelpp.expr("step1(x,2):x")
x=np.linspace(0,2*math.pi,100)
y=ex.evaluate("x",x)

fig = px.line(x=x, y=y, title="step1",labels={"x":"x","y":r'y'})
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 ex=feelpp.expr("step1(x,2):x") 2 x=np.linspace(0,2*math.pi,100) 3 y=ex.evaluate("x",x) NameError: name 'feelpp' is not defined

3.5. smoothstep

The smoothstep function is a smooth approximation of the step function.

ex=feelpp.expr("smoothstep(sin(x),0,0.5):x")
x=np.linspace(0,2*math.pi,100)
y=ex.evaluate("x",x)

fig = px.line(x=x, y=y, title="smoothstep",labels={"x":"x","y":r'y'})
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 ex=feelpp.expr("smoothstep(sin(x),0,0.5):x") 2 x=np.linspace(0,2*math.pi,100) 3 y=ex.evaluate("x",x) NameError: name 'feelpp' is not defined

3.6. pulse

The pulse function is defined as $\text{pulse}(x)=\begin{cases} 1 & \text{if } x\in[0,1] \\ 0 & \text{otherwise} \end{cases}$

ex=feelpp.expr("pulse(x,0,1,2):x")
x=np.linspace(0,2*math.pi,100)
y=ex.evaluate("x",x)

fig = px.line(x=x, y=y, title="rectangle pulse(0,1,2)",labels={"x":"x","y":r'y'})
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 ex=feelpp.expr("pulse(x,0,1,2):x") 2 x=np.linspace(0,2*math.pi,100) 3 y=ex.evaluate("x",x) NameError: name 'feelpp' is not defined

3.7. rectangle

The rectangle function is defined as $\text{rectangle}(x)=\begin{cases} 1 & \text{if } x\in[0,1] \\ 0 & \text{otherwise} \end{cases}$

ex=feelpp.expr("rectangle(x,0,1):x")
x=np.linspace(0,2*math.pi,100)
y=ex.evaluate("x",x)

fig = px.line(x=x, y=y, title="rectangle rectangle(0,1)",labels={"x":"x","y":r'y'})
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 ex=feelpp.expr("rectangle(x,0,1):x") 2 x=np.linspace(0,2*math.pi,100) 3 y=ex.evaluate("x",x) NameError: name 'feelpp' is not defined

3.8. triangle

The triangle function is defined as $\text{triangle}(x)=1-\left|\chi_{[a,b]} \left(-1+2\frac{x-a}{b-a} \right) \right| ], \quad x\in[a,b]$

ex=feelpp.expr("triangle(x,0,1):x")
x=np.linspace(0,2*math.pi,100)
y=ex.evaluate("x",x)

fig = px.line(x=x, y=y, title="rectangle triangle(0,1)",labels={"x":"x","y":r'y'})
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 ex=feelpp.expr("triangle(x,0,1):x") 2 x=np.linspace(0,2*math.pi,100) 3 y=ex.evaluate("x",x) NameError: name 'feelpp' is not defined

3.9. sinwave

The sinwave function is a periodic function with a given period and amplitude.

ex=feelpp.expr("sinewave(x,2,0.5):x")
x=np.linspace(0,2*math.pi,200)
y=ex.evaluate("x",x)

fig = px.line(x=x, y=y, title="sinewave(4*pi*x+0.5)",labels={"x":"x","y":r'y'})
fig.show()

python

Results

--------------------------------------------------------------------------- NameError Traceback (most recent call last) File :1 ----> 1 ex=feelpp.expr("sinewave(x,2,0.5):x") 2 x=np.linspace(0,2*math.pi,200) 3 y=ex.evaluate("x",x) NameError: name 'feelpp' is not defined

Manipulate Expressions

1. Create and evaluate expressions

1.1. Scalar expression

1.2. Vectorial expression

1.3. Matricial expression

2. Differentiation

2.1. Derivative of a vectorial expression

3. Special functions

3.1. function with multiple parameters

3.2. clamp

3.3. mapabcd

3.4. step1

3.5. smoothstep

3.6. pulse

3.7. rectangle

3.8. triangle

3.9. sinwave

4. Next steps