134 lines
3.1 KiB
Markdown
134 lines
3.1 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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|[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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## [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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