# Symbolic Computation {:.no_toc} ## 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() ``` ![image1](image1.png) ## 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) ```