function_generator.py

import sympy as sp

class FunctionGenerator:
    def __init__(self):
        self.x = sp.Symbol('x')
        
    def generate(self, sequence):
        """生成函数表达式
        
        参数:
            sequence (list): 输入数列
            
        返回:
            tuple: (符号表达式, 一般数学表达式, LaTeX表达式)
        """
        try:
            sequence = [float(x) for x in sequence]
            if len(sequence) < 2:
                raise ValueError("至少需要两个数来生成函数")
                
            # 生成符号表达式
            symbol_expr = self._generate_symbolic_expression(sequence)
            
            # 生成一般数学表达式
            general_expr = self._generate_general_expression(symbol_expr)
            
            # 生成LaTeX表达式
            latex_expr = self._generate_latex_expression(symbol_expr)
            
            return symbol_expr, general_expr, latex_expr
            
        except Exception as e:
            raise ValueError(f"生成函数时出错: {str(e)}")
            
    def _generate_symbolic_expression(self, sequence):
        """生成符号表达式"""
        # 使用拉格朗日插值
        terms = []
        for i in range(len(sequence)):
            numerator = 1
            denominator = 1
            for j in range(len(sequence)):
                if j != i:
                    numerator *= (self.x - (j + 1))
                    denominator *= ((i + 1) - (j + 1))
            term = sp.Rational(sequence[i]) * numerator / denominator
            terms.append(term)
            
        return sp.simplify(sum(terms))
        
    def _generate_general_expression(self, symbol_expr):
        """生成一般数学表达式"""
        expr_str = str(symbol_expr)
        # 仅替换非符号表达式中的 ** 为 ^
        expr_str = expr_str.replace('**', '^')
        return f"y = {expr_str}"
        
    def _generate_latex_expression(self, symbol_expr):
        """生成LaTeX表达式"""
        return f"y = {sp.latex(symbol_expr)}"
        
    def _identify_pattern(self, sequence):
        """识别数列模式"""
        # 检查指数增长
        if len(sequence) > 1:
            ratios = [sequence[i+1]/sequence[i] for i in range(len(sequence)-1)]
            if all(abs(r - ratios[0]) < 1e-10 for r in ratios):
                base = sp.Rational(sequence[1]) / sp.Rational(sequence[0])
                return f"{base}**(x)"
                
        # 检查线性增长
        differences = [sequence[i+1] - sequence[i] for i in range(len(sequence)-1)]
        if len(set(round(d, 10) for d in differences)) == 1:
            diff = differences[0]
            return f"{sp.Rational(sequence[0])} + {sp.Rational(diff)}*(x)"
            
        return None

def test_function_generator(test_sequences=None):
    """测试函数生成器"""
    generator = FunctionGenerator()
    test_sequences = test_sequences or [
        [1, 2, 4, 8, 16],  # 指数增长
        [1, 3, 5, 7, 9],   # 线性增长
        [1, 2, 4, 8, 16, 31]  # 复杂序列
    ]
    
    for seq in test_sequences:
        print(f"\n测试序列: {seq}")
        try:
            symbol_expr, general_expr, latex_expr = generator.generate(seq)
            print("符号表达式: y =", symbol_expr)
            print("一般数学表达式:", general_expr)
            print("LaTeX表达式:", latex_expr)
        except Exception as e:
            print(f"错误: {str(e)}")

if __name__ == "__main__":
    test_function_generator()