scientific-symbolic-mathematics

name	scientific-symbolic-mathematics
description	記号数学スキル。SymPy による解析的微積分・線形代数・微分方程式求解、 記号式の LaTeX 変換、数値計算との統合、科学モデリング用 記号計算パイプライン。
tu_tools	[{"key":"biotools","name":"bio.tools","description":"数式処理・解析ツール検索"}]

Scientific Symbolic Mathematics

SymPy を中心とした記号数学 (Computer Algebra System) パイプラインを提供する。

When to Use

• 微分方程式を解析的に解くとき
• 数式の記号的微分・積分を行うとき
• 行列の固有値・固有ベクトルを記号的に求めるとき
• 科学モデルのパラメータに関する感度解析を記号的に行うとき
• 数式を LaTeX 形式に変換するとき
• 記号解と数値解を比較検証するとき

---

Quick Start

1. 微分方程式の解析解

import sympy as sp
from sympy import (
    symbols, Function, Eq, dsolve, classify_ode,
    exp, sin, cos, sqrt, pi, oo, integrate, diff,
    Matrix, latex, simplify, factor, expand, solve,
    Rational, Sum, Product, series,
)
import numpy as np


def solve_ode(ode_expr, dependent_var, independent_var, ics=None):
    """
    常微分方程式の解析解。

    Parameters:
        ode_expr: sympy.Eq — ODE (e.g., Eq(f(x).diff(x, 2) + f(x), 0))
        dependent_var: sympy.Function — 従属変数
        independent_var: sympy.Symbol — 独立変数
        ics: dict — 初期条件 {f(0): 1, f'(0): 0}

    K-Dense: sympy — Symbolic mathematics
    """
    # Classify ODE
    classification = classify_ode(ode_expr, dependent_var(independent_var))
    print(f"ODE classification: {classification[:3]}")

    # Solve
    solution = dsolve(ode_expr, dependent_var(independent_var), ics=ics)

    print(f"Solution: {solution}")
    print(f"LaTeX: {latex(solution)}")
    return solution


# Example: damped harmonic oscillator
x, t, omega, gamma = symbols("x t omega gamma", positive=True)
f = Function("f")

# f''(t) + 2γf'(t) + ω²f(t) = 0
damped_ode = Eq(f(t).diff(t, 2) + 2*gamma*f(t).diff(t) + omega**2*f(t), 0)

2. 記号的微積分

def symbolic_calculus(expr, var, operations=None):
    """
    記号的微積分操作。

    Parameters:
        expr: sympy expression — 数式
        var: sympy.Symbol — 変数
        operations: list — ["diff", "integrate", "series", "limit"]
    """
    if operations is None:
        operations = ["diff", "integrate"]

    results = {}

    if "diff" in operations:
        deriv = diff(expr, var)
        results["derivative"] = {"expr": deriv, "latex": latex(deriv)}
        print(f"d/d{var}({expr}) = {deriv}")

    if "integrate" in operations:
        integral = integrate(expr, var)
        results["integral"] = {"expr": integral, "latex": latex(integral)}
        print(f"∫{expr} d{var} = {integral}")

    if "series" in operations:
        ser = series(expr, var, 0, n=6)
        results["series"] = {"expr": ser, "latex": latex(ser)}
        print(f"Taylor series: {ser}")

    if "limit" in operations:
        from sympy import limit  sp_limit
        lim = sp_limit(expr, var, oo)
        results[] = {: lim, : latex(lim)}
        ()

     results

3. 線形代数 (記号的)

def symbolic_linear_algebra(matrix_data):
    """
    記号的線形代数 — 固有値・固有ベクトル・行列分解。

    Parameters:
        matrix_data: list of lists — 行列要素 (記号含む)
    """
    M = Matrix(matrix_data)
    print(f"Matrix ({M.rows}×{M.cols}):")
    sp.pprint(M)

    results = {}

    # Determinant
    det = M.det()
    results["determinant"] = {"expr": det, "latex": latex(det)}
    print(f"\nDeterminant: {det}")

    # Eigenvalues & eigenvectors
    eigenvals = M.eigenvals()
    results["eigenvalues"] = {str(k): v for k, v in eigenvals.items()}
    print(f"Eigenvalues: {eigenvals}")

    eigenvects = M.eigenvects()
    results["eigenvectors"] = [
        {"eigenvalue": str(ev[0]), "multiplicity": ev[1],
         "vectors": [str(v) for v in ev[2]]}
        for ev in eigenvects
    ]

    # Characteristic polynomial
    lam = symbols("lambda")
    char_poly = M.charpoly(lam)
    results["characteristic_polynomial"] = {
        "expr": str(char_poly.as_expr()),
        "latex": latex(char_poly.as_expr()),
    }
    print(f"Characteristic polynomial: {char_poly.as_expr()}")

    
     det != :
        inv = M.inv()
        results[] = {: latex(inv)}
        ()

     results

4. 科学モデリング (薬物動態学 PK モデル例)

def pk_compartment_model(n_compartments=1):
    """
    薬物動態学コンパートメントモデルの記号的解法。

    Parameters:
        n_compartments: int — 1 (1-compartment) or 2 (2-compartment)
    """
    t = symbols("t", positive=True)

    if n_compartments == 1:
        # 1-compartment: dC/dt = -ke * C
        C = Function("C")
        ke, C0 = symbols("k_e C_0", positive=True)
        ode = Eq(C(t).diff(t), -ke * C(t))
        solution = dsolve(ode, C(t), ics={C(0): C0})

        # Half-life
        t_half = sp.solve(Eq(solution.rhs, C0/2), t)[0]

        # AUC (0→∞)
        auc = integrate(solution.rhs, (t, 0, oo))

        result = {
            "model": "1-compartment IV bolus",
            "ode": latex(ode),
            "solution": latex(solution),
            "half_life": latex(t_half),
            "auc_inf": latex(auc),
        }
        print(f"PK 1-compartment: C(t) = {solution.rhs}")
        print(f"  t½ = {t_half}")
        print(f"  AUC(0→∞) = {auc}")

    elif n_compartments == 2:
        # 2-compartment model
        C1, C2 = Function("C1"), Function("C2")
        k10, k12, k21, D, V1 = symbols("k_10 k_12 k_21 D V_1", positive=True)

        ode1 = Eq(C1(t).diff(t), -(k10 + k12)*C1(t) + k21*C2(t))
        ode2 = Eq(C2(t).diff(t), k12*C1(t) - k21*C2(t))

        system = [ode1, ode2]
        solution = sp.dsolve(system, [C1(t), C2(t)])

        result = {
            : ,
            : [latex(eq)  eq  system],
            : [latex(sol)  sol  solution],
        }
        ()
         sol  solution:
            ()

     result

5. LaTeX 数式エクスポート

def export_equations_latex(equations, output_file="equations.tex"):
    """
    記号数式を LaTeX ファイルにエクスポート。

    Parameters:
        equations: dict — {name: sympy_expr}
        output_file: str — 出力 LaTeX パス
    """
    lines = [
        r"\documentclass{article}",
        r"\usepackage{amsmath,amssymb}",
        r"\begin{document}",
        "",
    ]

    for name, expr in equations.items():
        lines.append(f"% {name}")
        lines.append(r"\begin{equation}")
        lines.append(f"  {latex(expr)}")
        lines.append(r"\end{equation}")
        lines.append("")

    lines.append(r"\end{document}")

    with open(output_file, "w") as f:
        f.write("\n".join(lines))

    print(f"LaTeX exported: {output_file} ({len(equations)} equations)")
    return output_file

---

パイプライン出力

出力ファイル	説明	連携先スキル
results/symbolic_solutions.json	記号解 (LaTeX 形式)	→ latex-formatter, academic-writing
results/ode_solutions.json	ODE 解析解	→ systems-biology, admet-pharmacokinetics
equations.tex	LaTeX 数式集	→ latex-formatter
figures/symbolic_plot.png	記号解の可視化	→ publication-figures

パイプライン統合

systems-biology ──→ symbolic-mathematics ──→ latex-formatter
  (SBML/ODE)         (SymPy 解析解)         (LaTeX 変換)
                           │
                           ├──→ admet-pharmacokinetics (PK モデル)
                           ├──→ bayesian-statistics (尤度導出)
                           └──→ computational-materials (バンド理論)

ToolUniverse 連携

TU Key	ツール名	連携内容
biotools	bio.tools	数式処理・解析ツール検索