symbolic-computation-guide

# Symbolic Computation Guide A skill for using computer algebra systems (CAS) in mathematical research. Covers symbolic differentiation, integration, equation solving, series expansion, linear algebra, and polynomial arithmetic using SymPy, SageMath, and Mathematica, with practical workflows for research mathematics. ## SymPy Fundamentals ### Symbolic Expressions and Manipulation ```python from sympy import ( symbols, expand, factor, simplify, cancel, apart, sin, cos, exp, log, sqrt, pi, oo, I, Rational, Eq, solve, solveset, S ) x, y, z, t, n, k = symbols("x y z t n k") a, b, c = symbols("a b c", real=True) # Expression manipulation expr = (x + 1) ** 3 expanded = expand(expr) # x**3 + 3*x**2 + 3*x + 1 factored = factor(expanded) # (x + 1)**3 # Trigonometric simplification from sympy import trigsimp trig_expr = sin(x)**2 + cos(x)**2 simplified = trigsimp(trig_expr) # 1 # Partial fraction decomposition rational = (x**2 + 2*x + 3) / ((x + 1) * (x + 2) * (x + 3)) partial = apart(rational, x) # 3/(2*(x + 3)) - 2/(x + 2) + 1/(2*(x + 1)) ``` ### Calculus ```python from sympy import diff, integrate, limit, series, Sum, Product # Differentiation f = x**3 * exp(-x) * sin(x) f_prime = diff(f, x) f_double_prime = diff(f, x, 2) # Integration # Definite integral area = integrate(exp(-x**2), (x, -oo, oo)) # sqrt(pi) # Indefinite integral antideriv = integrate(x * sin(x), x) # -x*cos(x) + sin(x) # Limits lim_result = limit(sin(x) / x, x, 0) # 1 lim_inf = limit((1 + 1/n)**n, n, oo) # E (Euler's number) # Taylor series taylor = series(exp(x) * cos(x), x, 0, n=6) # 1 + x - x**3/3 - x**4/6 + ... # Summation harmonic = Sum(1/k, (k, 1, n)) partial_sum = harmonic.doit() # harmonic(n) -- returns harmonic number geometric = Sum(x**k, (k, 0, oo)) closed_form = geometric.doit() # Piecewise(1/(1 - x), Abs(x) < 1) ``` ### Equation Solving ```python # Algebraic equations solutions = solve(x**3 - 6*x**2 + 11*x - 6, x) # [1, 2, 3] # System of equations system_sol = solve([ 2*x + 3*y - 7, x - y + 1 ], [x, y]) # {x: 4/5, y: 9/5} # Differential equations from sympy import Function, dsolve, Derivative f = Function("f") # f''(x) + f(x) = 0 (simple harmonic oscillator) ode = Eq(f(x).diff(x, 2) + f(x), 0) general_solution = dsolve(ode, f(x)) # f(x) = C1*sin(x) + C2*cos(x) # With initial conditions particular = dsolve(ode, f(x), ics={f(0): 1, f(x).diff(x).subs(x, 0): 0}) # f(x) = cos(x) ``` ## Linear Algebra ### Symbolic Matrix Operations ```python from sympy import Matrix, eye, zeros, det, Rational # Define a symbolic matrix A = Matrix([ [1, 2, 3], [4, 5, 6], [7, 8, 10] ]) # Basic operations print(f"Determinant: {det(A)}") # -3 print(f"Inverse:\n{A.inv()}") print(f"Eigenvalues: {A.eigenvals()}") print(f"Rank: {A.rank()}") # Characteristic polynomial lam = symbols("lambda") char_poly = (A - lam * eye(3)).det() char_poly = expand(char_poly) # Jordan normal form P, J = A.jordan_form() # Null space and column space null = A.nullspace() col_space = A.columnspace() # Symbolic matrix with parameters M = Matrix([ [a, b], [c, a] ]) eigenvals = M.eigenvals() # {a - sqrt(b*c): 1, a + sqrt(b*c): 1} ``` ## SageMath for Research ### Number Theory ```python # SageMath syntax (Python-based, but with enhanced number theory) # Run in SageMath environment or via sage -python """ # Prime factorization factor(2024) # 2^3 * 11 * 23 # Modular arithmetic R = IntegerModRing(17) R(3)^(-1) # multiplicative inverse of 3 mod 17 # Elliptic curves E = EllipticCurve(QQ, [-1, 0]) E.rank() E.torsion_subgroup() E.gens() # Polynomial rings R.<x,y> = PolynomialRing(QQ) I = R.ideal(x^2 + y^2 - 1, x - y) I.groebner_basis() # [y^2 - 1/2, x - y] # Group theory G = SymmetricGroup(4) G.order() # 24 G.center() G.normal_subgroups() """ ``` ### Combinatorics and Graph Theory ```python """ # SageMath combinatorics Partitions(10).cardinality() # 42 # Graph theory G = graphs.PetersenGraph() G.chromatic_number() # 3 G.is_vertex_transitive() # True G.automorphism_group().order() # 120 # Posets and lattices P = posets.BooleanLattice(3) P.is_lattice() P.mobius_function(P.bottom(), P.top()) """ ``` ## Mathematica / Wolfram Language ### Common Research Patterns ```mathematica (* Symbolic integration *) Integrate[x^n * Exp[-x], {x, 0, Infinity}, Assumptions -> n > -1] (* Result: Gamma[1 + n] *) (* Solve a PDE *) DSolve[D[u[x, t], t] == k * D[u[x, t], {x, 2}], u[x, t], {x, t}] (* Asymptotic expansion *) Series[Gamma[n + 1], {n, Infinity, 3}] (* Minimize with constraints *) NMinimize[{x^2 + y^2, x + y >= 1}, {x, y}] (* Compute a sum in closed form *) Sum[1/k^2, {k, 1, Infinity}] (* Pi^2/6 *) ``` ## Practical Workflows ### Verifying Research Computations Common CAS workflow in mathematical research: 1. **Conjecture formulation**: Test conjectures for small cases programmatically 2. **Identity verification**: Verify algebraic identities symbolically 3. **Closed-form discovery**: Use pattern matching and OEIS lookup 4. **Proof assistance**: Compute bounds, verify inequalities 5. **Counterexample search**: Systematically search parameter spaces ```python from sympy import simplify, Abs def verify_identity(lhs, rhs): """Verify a proposed mathematical identity symbolically.""" diff = simplify(lhs - rhs) if diff == 0: return "VERIFIED: identity holds symbolically" else: return f"NOT VERIFIED: difference = {diff}" # Example: verify Cauchy-Schwarz for 2D a1, a2, b1, b2 = symbols("a1 a2 b1 b2", real=True) lhs = (a1*b1 + a2*b2)**2 rhs = (a1**2 + a2**2) * (b1**2 + b2**2) diff = expand(rhs - lhs) # (a1*b2 - a2*b1)**2 >= 0, confirming Cauchy-Schwarz ``` ## Tools and Resources - **SymPy**: Pure Python CAS, integrates with Jupyter and NumPy - **SageMath**: Comprehensive open-source math system (wraps GAP, PARI, Singular, etc.) - **Mathematica / Wolfram Alpha**: Commercial CAS with unmatched integration database - **Maxima**: Free CAS (Lisp-based), used in wxMaxima GUI - **PARI/GP**: Specialized in number theory computations - **Macaulay2**: Commutative algebra and algebraic geometry - **GAP**: Computational group theory - **OEIS (oeis.org)**: Online Encyclopedia of Integer Sequences for pattern identification