309 lines
9.7 KiB
Markdown
309 lines
9.7 KiB
Markdown
# Symbolic Computation
|
|
{:.no_toc}
|
|
|
|
<nav markdown="1" class="toc-class">
|
|
* TOC
|
|
{:toc}
|
|
</nav>
|
|
|
|
## Top
|
|
|
|
Questions to [David Rotermund](mailto:davrot@uni-bremen.de)
|
|
|
|
```shell
|
|
pip install sympy
|
|
```
|
|
|
|
### Overview tutorials
|
|
|
|
||
|
|
|---|
|
|
|[Basic Operations](https://docs.sympy.org/latest/tutorials/intro-tutorial/basic_operations.html)|
|
|
|[Printing](https://docs.sympy.org/latest/tutorials/intro-tutorial/printing.html)|
|
|
|[Simplification](https://docs.sympy.org/latest/tutorials/intro-tutorial/simplification.html)|
|
|
|[Calculus](https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html) |
|
|
|[Solvers](https://docs.sympy.org/latest/tutorials/intro-tutorial/solvers.html)|
|
|
|[Matrices](https://docs.sympy.org/latest/tutorials/intro-tutorial/matrices.html)|
|
|
|
|
### [API Reference](https://docs.sympy.org/latest/reference/index.html)
|
|
|
|
|||
|
|
|---|---|
|
|
|[Basics](https://docs.sympy.org/latest/reference/public/basics/index.html#basic-modules)|Contains a description of operations for the basic modules. Subcategories include: absolute basics, manipulation, assumptions, functions, simplification, calculus, solvers, and some other subcategories.|
|
|
|[Code Generation](https://docs.sympy.org/latest/reference/public/codegeneration/index.html#codegen-module)|Contains a description of methods for the generation of compilable and executable code.|
|
|
|[Logic](https://docs.sympy.org/latest/reference/public/logic/index.html#logic)|Contains method details for the logic and sets modules.|
|
|
|[Matrices](https://docs.sympy.org/latest/reference/public/matrices/index.html#matrices-modules)|Discusses methods for the matrices, tensor and vector modules.|
|
|
|[Number Theory](https://docs.sympy.org/latest/reference/public/numbertheory/index.html#numtheory-module)|Documents methods for the Number theory module.|
|
|
|[Physics](https://docs.sympy.org/latest/reference/public/physics/index.html#physics-docs)|Contains documentation for Physics methods.|
|
|
|[Utilities](https://docs.sympy.org/latest/reference/public/utilities/index.html#utilities)|Contains docstrings for methods of several utility modules. Subcategories include: Interactive, Parsing, Printing, Testing, Utilities.|
|
|
|[Topics](https://docs.sympy.org/latest/reference/public/topics/index.html#topics)|Contains method docstrings for several modules. Subcategories include : Plotting, Polynomials, Geometry, Category Theory, Cryptography, Differential, Holonomic, Lie Algebra, and Stats.|
|
|
|
|
|
|
### [Basics](https://docs.sympy.org/latest/reference/public/basics/index.html#basic-modules)
|
|
|
|
||
|
|
|---|
|
|
|[Assumptions](https://docs.sympy.org/latest/modules/assumptions/index.html)|
|
|
|[Calculus](https://docs.sympy.org/latest/modules/calculus/index.html)|
|
|
|[Combinatorics](https://docs.sympy.org/latest/modules/combinatorics/index.html)|
|
|
|[Functions](https://docs.sympy.org/latest/modules/functions/index.html)|
|
|
|[Integrals](https://docs.sympy.org/latest/modules/integrals/index.html)|
|
|
|[Series](https://docs.sympy.org/latest/modules/series/index.html)|
|
|
|[Simplify](https://docs.sympy.org/latest/modules/simplify/index.html)|
|
|
|[Solvers](https://docs.sympy.org/latest/modules/solvers/index.html)|
|
|
|[abc](https://docs.sympy.org/latest/modules/abc.html)|
|
|
|[Algebras](https://docs.sympy.org/latest/modules/algebras.html)|
|
|
|[Concrete](https://docs.sympy.org/latest/modules/concrete.html)|
|
|
|[Core](https://docs.sympy.org/latest/modules/core.html)|
|
|
|[Discrete](https://docs.sympy.org/latest/modules/discrete.html)|
|
|
|[Numerical Evaluation](https://docs.sympy.org/latest/modules/evalf.html)|
|
|
|[Numeric Computation](https://docs.sympy.org/latest/modules/numeric-computation.html)|
|
|
|[Term Rewriting](https://docs.sympy.org/latest/modules/rewriting.html)|
|
|
|
|
|
|
## [Some examples](https://docs.sympy.org/latest/tutorials/intro-tutorial/intro.html#a-more-interesting-example)
|
|
|
|
## [Substitution](https://docs.sympy.org/latest/tutorials/intro-tutorial/basic_operations.html#substitution)
|
|
```python
|
|
import sympy
|
|
|
|
x, y = sympy.symbols("x y")
|
|
|
|
expr = sympy.cos(x) + 1
|
|
z = expr.subs(x, y**2)
|
|
print(z) # -> cos(y**2) + 1
|
|
```
|
|
|
|
### [Derivatives](https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html#derivatives)
|
|
|
|
```python
|
|
import sympy
|
|
|
|
x, y = sympy.symbols("x y")
|
|
|
|
y = sympy.diff(sympy.sin(x) * sympy.exp(x), x)
|
|
print(y) # -> exp(x)*sin(x) + exp(x)*cos(x)
|
|
```
|
|
|
|
### [Integrals](https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html#integrals)
|
|
|
|
```python
|
|
import sympy
|
|
|
|
x, y = sympy.symbols("x y")
|
|
|
|
y = sympy.integrate(sympy.cos(x), x)
|
|
print(y) # -> sin(x)
|
|
```
|
|
|
|
|
|
### [(Taylor) Series Expansion](https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html#series-expansion)
|
|
|
|
```python
|
|
import sympy
|
|
|
|
x, y, z = sympy.symbols("x y z")
|
|
y = sympy.cos(x)
|
|
z = y.series(x, 0, 8) # around x = 0 , up order 7
|
|
|
|
print(z) # -> 1 - x**2/2 + x**4/24 - x**6/720 + O(x**8)
|
|
```
|
|
|
|
### [simplify](https://docs.sympy.org/latest/tutorials/intro-tutorial/simplification.html#simplify)
|
|
|
|
```python
|
|
import sympy
|
|
|
|
x, y, z = sympy.symbols("x y z")
|
|
y = sympy.simplify(sympy.sin(x) ** 2 + sympy.cos(x) ** 2)
|
|
|
|
print(y) # -> 1
|
|
```
|
|
|
|
### [Solving Equations Algebraically](https://docs.sympy.org/latest/tutorials/intro-tutorial/solvers.html)
|
|
|
|
```python
|
|
solveset(equation, variable=None, domain=S.Complexes)
|
|
```
|
|
|
|
> Recall from the [gotchas section](https://docs.sympy.org/latest/tutorials/intro-tutorial/gotchas.html#tutorial-gotchas-equals) of this tutorial that symbolic equations in SymPy are not represented by = or ==, but by Eq.
|
|
|
|
```python
|
|
import sympy
|
|
|
|
x, y, z = sympy.symbols("x y z")
|
|
|
|
z = sympy.Eq(x, y)
|
|
```
|
|
|
|
Output:
|
|
|
|
$$x=y$$
|
|
|
|
|
|
```python
|
|
import sympy
|
|
|
|
x, y, z = sympy.symbols("x y z")
|
|
|
|
y = sympy.Eq(x**2 - x, 0)
|
|
z = sympy.solveset(y, x)
|
|
print(z) # -> {0, 1}
|
|
```
|
|
|
|
### [Solving Differential Equations](https://docs.sympy.org/latest/tutorials/intro-tutorial/solvers.html#solving-differential-equations)
|
|
|
|
```python
|
|
import sympy
|
|
|
|
# Undefined functions
|
|
f = sympy.symbols("f", cls=sympy.Function)
|
|
|
|
x = sympy.symbols("x")
|
|
|
|
|
|
diffeq = sympy.Eq(f(x).diff(x, x) - 2 * f(x).diff(x) + f(x), sympy.sin(x))
|
|
|
|
print(diffeq) # -> Eq(f(x) - 2*Derivative(f(x), x) + Derivative(f(x), (x, 2)), sin(x))
|
|
|
|
result = sympy.dsolve(diffeq, f(x))
|
|
print(result) # -> Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)
|
|
```
|
|
|
|
Sometimes it is necessary to simplyify the problem for the solver:
|
|
|
|
```python
|
|
import sympy
|
|
|
|
u = sympy.symbols("u", cls=sympy.Function, real=True)
|
|
t, tau, a0, urest, uc, r, i, uthr = sympy.symbols(
|
|
"t tau a0 urest uc r i uthr", real=True
|
|
)
|
|
c0, c1, c2 = sympy.symbols("c0 c1 c2", real=True)
|
|
|
|
diffeq_rhs = (a0 * (u(t) - urest) * (u(t) - uc) + r * i) / tau
|
|
|
|
print("Original")
|
|
print(diffeq_rhs)
|
|
print()
|
|
|
|
diffeq_rhs = sympy.expand(diffeq_rhs)
|
|
diffeq_rhs = sympy.collect(diffeq_rhs, u(t))
|
|
|
|
c0_coeffs = diffeq_rhs.coeff(u(t), 0)
|
|
c1_coeffs = diffeq_rhs.coeff(u(t), 1)
|
|
c2_coeffs = diffeq_rhs.coeff(u(t), 2)
|
|
|
|
diffeq_rhs = diffeq_rhs.subs(c0_coeffs, c0)
|
|
diffeq_rhs = diffeq_rhs.subs(c1_coeffs, c1)
|
|
diffeq_rhs = diffeq_rhs.subs(c2_coeffs, c2)
|
|
|
|
print("Simplified")
|
|
print(diffeq_rhs)
|
|
print()
|
|
|
|
diffeq = sympy.Eq(u(t).diff(t), diffeq_rhs)
|
|
|
|
print("Full equation")
|
|
print(diffeq)
|
|
print()
|
|
|
|
solved_diffeq_rhs = sympy.dsolve(diffeq, u(t), ics={u(0): uc}, simplify=False).rhs
|
|
|
|
solved_diffeq_rhs = solved_diffeq_rhs.subs(c0, c0_coeffs)
|
|
solved_diffeq_rhs = solved_diffeq_rhs.subs(c1, c1_coeffs)
|
|
solved_diffeq_rhs = solved_diffeq_rhs.subs(c2, c2_coeffs)
|
|
|
|
print("Result")
|
|
print(solved_diffeq_rhs)
|
|
print()
|
|
```
|
|
|
|
Output:
|
|
```python
|
|
Original
|
|
(a0*(-uc + u(t))*(-urest + u(t)) + i*r)/tau
|
|
|
|
Simplified
|
|
c0 + c1*u(t) + c2*u(t)**2
|
|
|
|
Full equation
|
|
Eq(Derivative(u(t), t), c0 + c1*u(t) + c2*u(t)**2)
|
|
|
|
Result
|
|
-t + sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*log(uc - 2*sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*(a0*uc*urest/tau + i*r/tau) + tau*sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*(-a0*uc/tau - a0*urest/tau)**2/(2*a0) + tau*(-a0*uc/tau - a0*urest/tau)/(2*a0)) - sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*log(uc + 2*sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*(a0*uc*urest/tau + i*r/tau) - tau*sqrt(-1/(4*a0*(a0*uc*urest/tau + i*r/tau)/tau - (-a0*uc/tau - a0*urest/tau)**2))*(-a0*uc/tau - a0*urest/tau)**2/(2*a0) + tau*(-a0*uc/tau - a0*urest/tau)/(2*a0))
|
|
```
|
|
|
|
## [Numerical Evaluation](https://docs.sympy.org/latest/modules/evalf.html)
|
|
|
|
```python
|
|
import sympy
|
|
|
|
x, y = sympy.symbols("x y")
|
|
|
|
expr = sympy.cos(x) + 1
|
|
print(expr) # -> cos(x) + 1
|
|
expr = expr.subs(x, 0.333 * sympy.pi)
|
|
print(expr) # -> cos(0.333*pi) + 1
|
|
print(sympy.N(expr)) # -> 1.50090662536071
|
|
```
|
|
|
|
## Plot your function
|
|
|
|
|
|
```python
|
|
import sympy
|
|
import inspect
|
|
import numpy as np
|
|
import matplotlib.pyplot as plt
|
|
|
|
f = sympy.symbols("f", cls=sympy.Function)
|
|
x = sympy.symbols("x")
|
|
diffeq = sympy.Eq(f(x).diff(x, x) - 2 * f(x).diff(x) + f(x), sympy.sin(x))
|
|
result = sympy.dsolve(diffeq, f(x))
|
|
|
|
symbols = list(result.rhs.free_symbols)
|
|
|
|
f = sympy.lambdify(symbols, result.rhs, "numpy")
|
|
|
|
|
|
print("The arguments of the result:")
|
|
print(inspect.getfullargspec(f).args)
|
|
print("The source code behind f:")
|
|
print(inspect.getsource(f))
|
|
|
|
np_x = np.linspace(0, 2 * np.pi, 100)
|
|
|
|
plt.plot(f(x=np_x, C1=1.0, C2=1.0))
|
|
plt.xlabel("x")
|
|
plt.ylabel("f(x)")
|
|
plt.show()
|
|
```
|
|
|
|
|
|
|
|
|
|
## Assumptions about the symbols
|
|
|
|
```python
|
|
C1 = sympy.symbols('C1', positive=True)
|
|
```
|
|
|
|
|||
|
|
|---|---|
|
|
|real=True| The symbol represents a real number.|
|
|
|positive=True| The symbol represents a positive number.|
|
|
|negative=True| The symbol represents a negative number.|
|
|
|integer=True| The symbol represents an integer.|
|
|
|prime=True| The symbol represents a prime number.|
|
|
|odd=True| The symbol represents an odd number.|
|
|
|even=True| The symbol represents an even number.|
|
|
|
|
## Limiting the range of a symbol
|
|
|
|
```python
|
|
import sympy
|
|
|
|
x = sympy.symbols("x")
|
|
sympy.solve([x**2 - 1, x >= 0.5, x <= 3], x)
|
|
```
|