math-computation

name math-computation description Mathematical computation including symbolic math, numerical methods, linear algebra, calculus, differential equations, optimization, and mathematical modeling. Uses Python with SymPy, NumPy, SciPy. Use when user asks to solve equations, compute integrals/derivatives, do matrix operations, solve ODEs/PDEs, optimize functions, or build mathematical models. Triggers on "solve equation", "integral", "derivative", "matrix", "eigenvalue", "differential equation", "optimization", "linear algebra", "symbolic math", "proof". Mathematical Computation Symbolic and numerical mathematics. Venv: source /Users/zhangmingda/clawd/.venv/bin/activate Symbolic Math (SymPy) from sympy import * x, y, z, t = symbols( 'x y z t' ) a, b, c = symbols( 'a b c' , real= True ) n, k = symbols( 'n k' , integer= True , positive= True ) # Solve equations solve(x** 2 - 5 *x + 6 , x) # [2, 3] solve([x + y - 5 , x - y - 1 ], [x, y]) # {x: 3, y: 2} # Calculus diff(sin(x)*exp(x), x) # derivative integrate(x** 2 * exp(-x), (x, 0 , oo)) # definite integral limit(sin(x)/x, x, 0 ) # limit series(exp(x), x, 0 , 5 ) # Taylor series # Linear algebra M = Matrix([[ 1 , 2 ], [ 3 , 4 ]]) M.eigenvals() # eigenvalues M.eigenvects() # eigenvectors M.det() # determinant M.inv() # inverse # Differential equations f = Function( 'f' ) dsolve(f(x).diff(x, 2 ) + f(x), f(x)) # y'' + y = 0 # Simplification simplify(sin(x)** 2 + cos(x)** 2 ) # 1 trigsimp(expr) factor(expr) expand(expr) # LaTeX output latex(expr) # for paper-ready equations Numerical Methods (SciPy) from scipy import optimize, integrate, linalg, interpolate import numpy as np # Root finding root = optimize.brentq( lambda x: x** 3 - 2 *x - 5 , 2 , 3 ) # Optimization result = optimize.minimize( lambda x: (x[ 0 ]- 1 )** 2 + (x[ 1 ]- 2.5 )** 2 , x0=[ 0 , 0 ], method= 'Nelder-Mead' ) # Constrained optimization from scipy.optimize import linprog, minimize result = minimize(objective, x0, constraints=constraints, bounds=bounds) # Numerical integration val, err = integrate.quad( lambda x: np.exp(-x** 2 ), -np.inf, np.inf) # √π # ODE solving from scipy.integrate import solve_ivp def lorenz ( t, state, sigma= 10 , rho= 28 , beta= 8 / 3 ): x, y, z = state return [sigma*(y-x), x*(rho-z)-y, x*y-beta*z] sol = solve_ivp(lorenz, [ 0 , 50 ], [ 1 , 1 , 1 ], dense_output= True , max_step= 0.01 ) # Interpolation f_interp = interpolate.interp1d(x_data, y_data, kind= 'cubic' ) # FFT from scipy.fft import fft, fftfreq yf = fft(signal) xf = fftfreq(N, 1 /sample_rate) Linear Algebra # NumPy A = np.array([[ 1 , 2 ], [ 3 , 4 ]]) np.linalg.eig(A) # eigendecomposition np.linalg.svd(A) # SVD np.linalg.solve(A, b) # solve Ax = b np.linalg.norm(A) # matrix norm np.linalg.matrix_rank(A) # Sparse matrices (SciPy) from scipy.sparse import csr_matrix, linalg as sparse_linalg Mathematical Modeling Workflow Define the system and variables Formulate equations (conservation laws, constitutive relations) Non-dimensionalize if appropriate Solve analytically (SymPy) or numerically (SciPy) Validate against known solutions or data Sensitivity analysis on parameters Visualize results Common Models Population dynamics : Lotka-Volterra, SIR/SEIR epidemiological Diffusion : Heat equation, Fick's law Mechanics : Newton's laws, Lagrangian/Hamiltonian Economics : Supply-demand, game theory, optimal control Networks : Graph theory, flow optimization Tips Use SymPy for exact solutions, SciPy for numerical Always verify numerical solutions against analytical when possible Check units and dimensional consistency Use latex() to generate paper-ready equations For large systems, consider sparse matrix methods