238 lines
7.5 KiB
Markdown
238 lines
7.5 KiB
Markdown
# Symbolic Computation
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{:.no_toc}
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<nav markdown="1" class="toc-class">
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* TOC
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{:toc}
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</nav>
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## Top
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Questions to [David Rotermund](mailto:davrot@uni-bremen.de)
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```shell
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pip install sympy
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```
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### Overview tutorials
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|[Basic Operations](https://docs.sympy.org/latest/tutorials/intro-tutorial/basic_operations.html)|
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|[Printing](https://docs.sympy.org/latest/tutorials/intro-tutorial/printing.html)|
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|[Simplification](https://docs.sympy.org/latest/tutorials/intro-tutorial/simplification.html)|
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|[Calculus](https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html) |
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|[Solvers](https://docs.sympy.org/latest/tutorials/intro-tutorial/solvers.html)|
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|[Matrices](https://docs.sympy.org/latest/tutorials/intro-tutorial/matrices.html)|
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### [API Reference](https://docs.sympy.org/latest/reference/index.html)
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|[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.|
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|[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.|
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|[Logic](https://docs.sympy.org/latest/reference/public/logic/index.html#logic)|Contains method details for the logic and sets modules.|
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|[Matrices](https://docs.sympy.org/latest/reference/public/matrices/index.html#matrices-modules)|Discusses methods for the matrices, tensor and vector modules.|
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|[Number Theory](https://docs.sympy.org/latest/reference/public/numbertheory/index.html#numtheory-module)|Documents methods for the Number theory module.|
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|[Physics](https://docs.sympy.org/latest/reference/public/physics/index.html#physics-docs)|Contains documentation for Physics methods.|
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|[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.|
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|[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.|
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### [Basics](https://docs.sympy.org/latest/reference/public/basics/index.html#basic-modules)
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|[Assumptions](https://docs.sympy.org/latest/modules/assumptions/index.html)|
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|[Calculus](https://docs.sympy.org/latest/modules/calculus/index.html)|
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|[Combinatorics](https://docs.sympy.org/latest/modules/combinatorics/index.html)|
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|[Functions](https://docs.sympy.org/latest/modules/functions/index.html)|
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|[Integrals](https://docs.sympy.org/latest/modules/integrals/index.html)|
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|[Series](https://docs.sympy.org/latest/modules/series/index.html)|
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|[Simplify](https://docs.sympy.org/latest/modules/simplify/index.html)|
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|[Solvers](https://docs.sympy.org/latest/modules/solvers/index.html)|
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|[abc](https://docs.sympy.org/latest/modules/abc.html)|
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|[Algebras](https://docs.sympy.org/latest/modules/algebras.html)|
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|[Concrete](https://docs.sympy.org/latest/modules/concrete.html)|
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|[Core](https://docs.sympy.org/latest/modules/core.html)|
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|[Discrete](https://docs.sympy.org/latest/modules/discrete.html)|
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|[Numerical Evaluation](https://docs.sympy.org/latest/modules/evalf.html)|
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|[Numeric Computation](https://docs.sympy.org/latest/modules/numeric-computation.html)|
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|[Term Rewriting](https://docs.sympy.org/latest/modules/rewriting.html)|
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## [Some examples](https://docs.sympy.org/latest/tutorials/intro-tutorial/intro.html#a-more-interesting-example)
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## [Substitution](https://docs.sympy.org/latest/tutorials/intro-tutorial/basic_operations.html#substitution)
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```python
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import sympy
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x, y = sympy.symbols("x y")
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expr = sympy.cos(x) + 1
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z = expr.subs(x, y**2)
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print(z) # -> cos(y**2) + 1
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```
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### [Derivatives](https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html#derivatives)
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```python
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import sympy
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x, y = sympy.symbols("x y")
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y = sympy.diff(sympy.sin(x) * sympy.exp(x), x)
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print(y) # -> exp(x)*sin(x) + exp(x)*cos(x)
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```
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### [Integrals](https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html#integrals)
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```python
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import sympy
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x, y = sympy.symbols("x y")
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y = sympy.integrate(sympy.cos(x), x)
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print(y) # -> sin(x)
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```
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### [(Taylor) Series Expansion](https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html#series-expansion)
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```python
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import sympy
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x, y, z = sympy.symbols("x y z")
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y = sympy.cos(x)
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z = y.series(x, 0, 8) # around x = 0 , up order 7
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print(z) # -> 1 - x**2/2 + x**4/24 - x**6/720 + O(x**8)
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```
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### [simplify](https://docs.sympy.org/latest/tutorials/intro-tutorial/simplification.html#simplify)
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```python
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import sympy
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x, y, z = sympy.symbols("x y z")
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y = sympy.simplify(sympy.sin(x) ** 2 + sympy.cos(x) ** 2)
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print(y) # -> 1
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```
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### [Solving Equations Algebraically](https://docs.sympy.org/latest/tutorials/intro-tutorial/solvers.html)
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```python
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solveset(equation, variable=None, domain=S.Complexes)
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```
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> 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.
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```python
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import sympy
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x, y, z = sympy.symbols("x y z")
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z = sympy.Eq(x, y)
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```
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Output:
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$$x=y$$
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```python
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import sympy
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x, y, z = sympy.symbols("x y z")
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y = sympy.Eq(x**2 - x, 0)
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z = sympy.solveset(y, x)
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print(z) # -> {0, 1}
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```
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### [Solving Differential Equations](https://docs.sympy.org/latest/tutorials/intro-tutorial/solvers.html#solving-differential-equations)
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```python
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import sympy
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# Undefined functions
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f = sympy.symbols("f", cls=sympy.Function)
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x = sympy.symbols("x")
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diffeq = sympy.Eq(f(x).diff(x, x) - 2 * f(x).diff(x) + f(x), sympy.sin(x))
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print(diffeq) # -> Eq(f(x) - 2*Derivative(f(x), x) + Derivative(f(x), (x, 2)), sin(x))
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result = sympy.dsolve(diffeq, f(x))
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print(result) # -> Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)
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```
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## [Numerical Evaluation](https://docs.sympy.org/latest/modules/evalf.html)
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```python
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import sympy
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x, y = sympy.symbols("x y")
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expr = sympy.cos(x) + 1
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print(expr) # -> cos(x) + 1
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expr = expr.subs(x, 0.333 * sympy.pi)
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print(expr) # -> cos(0.333*pi) + 1
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print(sympy.N(expr)) # -> 1.50090662536071
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```
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## Plot your function
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```python
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import sympy
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import inspect
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import numpy as np
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import matplotlib.pyplot as plt
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f = sympy.symbols("f", cls=sympy.Function)
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x = sympy.symbols("x")
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diffeq = sympy.Eq(f(x).diff(x, x) - 2 * f(x).diff(x) + f(x), sympy.sin(x))
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result = sympy.dsolve(diffeq, f(x))
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symbols = list(result.rhs.free_symbols)
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f = sympy.lambdify(symbols, result.rhs, "numpy")
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print("The arguments of the result:")
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print(inspect.getfullargspec(f).args)
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print("The source code behind f:")
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print(inspect.getsource(f))
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np_x = np.linspace(0, 2 * np.pi, 100)
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plt.plot(f(x=np_x, C1=1.0, C2=1.0))
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plt.xlabel("x")
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plt.ylabel("f(x)")
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plt.show()
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```
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## Assumptions about the symbols
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```python
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C1 = sympy.symbols('C1', positive=True)
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```
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|real=True| The symbol represents a real number.|
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|positive=True| The symbol represents a positive number.|
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|negative=True| The symbol represents a negative number.|
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|integer=True| The symbol represents an integer.|
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|prime=True| The symbol represents a prime number.|
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|odd=True| The symbol represents an odd number.|
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|even=True| The symbol represents an even number.|
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