numerical_integration.hpp

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/**
 * @file numerical_integration.hpp
 * @brief 数值积分算法:辛普森法则和高斯求积(符号-数值混合桥接).
 */
#pragma once
#include "numeric_evaluation.hpp"
#include "symbolic.hpp"
#include <memory>
#include <string>
 
namespace LMCAS {
 
struct QuadratureEstimate {
    double value = 0.0;
    double estimated_error = 0.0;
};
 
/**
 * @brief Checked composite Simpson integration.
 *
 * Endpoints and samples are evaluated through evaluate_numeric(); invalid
 * arguments, odd subinterval counts, unbound variables, domain errors,
 * cancellation, and budget exhaustion are reported explicitly. Equal finite
 * endpoints return exact zero before reserving any integrand-sampling budget.
 * Other finite endpoints are mapped through normalized coordinates, so their
 * mathematical span need not itself be representable as a double when the
 * result is finite.
 * The returned value is the step-doubling Richardson extrapolation
 * `S(2n) + (S(2n) - S(n)) / 15`; estimated_error is the corresponding
 * `abs(S(2n) - S(n)) / 15` estimate, not a certified bound.
 */
LMCAS_API Result<QuadratureEstimate> quadrature_simpson_numeric(
    const std::shared_ptr<SymbolicExpr>& f,
    const std::string& var,
    const std::shared_ptr<SymbolicExpr>& a,
    const std::shared_ptr<SymbolicExpr>& b,
    ComputationContext& context,
    int n = 100
);
 
LMCAS_API Result<QuadratureEstimate> quadrature_simpson_numeric(
    const std::shared_ptr<SymbolicExpr>& f,
    const std::string& var,
    const std::shared_ptr<SymbolicExpr>& a,
    const std::shared_ptr<SymbolicExpr>& b,
    int n = 100
);
/**
 * @brief Checked Gauss-Legendre integration for orders 1 through 20.
 *
 * Every supported order is evaluated by LMMC's Gauss-Legendre implementation.
 * Equal finite endpoints return exact zero before reserving quadrature samples.
 * Other finite extreme endpoints use an overflow-safe affine map to [-1,1].
 * The returned estimated_error is the difference from an adjacent-order rule,
 * not a certified bound.
 */
LMCAS_API Result<QuadratureEstimate> quadrature_gaussian_numeric(
    const std::shared_ptr<SymbolicExpr>& f,
    const std::string& var,
    const std::shared_ptr<SymbolicExpr>& a,
    const std::shared_ptr<SymbolicExpr>& b,
    ComputationContext& context,
    int n = 5
);
 
LMCAS_API Result<QuadratureEstimate> quadrature_gaussian_numeric(
    const std::shared_ptr<SymbolicExpr>& f,
    const std::string& var,
    const std::shared_ptr<SymbolicExpr>& a,
    const std::shared_ptr<SymbolicExpr>& b,
    int n = 5
);
 
/**
 * @brief Checked adaptive Simpson integration.
 *
 * All endpoint and integrand evaluation goes through evaluate_numeric(); missing
 * variables, domain errors, cancellation, and budget exhaustion are represented
 * by CasError in the Result channel. Finite extreme endpoints are bisected and
 * scaled through overflow-safe center and half-length formulas.
 * The returned estimated_error is an absolute estimate, not a certified bound.
 */
LMCAS_API Result<QuadratureEstimate> adaptive_simpson_numeric(
    const std::shared_ptr<SymbolicExpr>& f,
    const std::string& var,
    const std::shared_ptr<SymbolicExpr>& a,
    const std::shared_ptr<SymbolicExpr>& b,
    ComputationContext& context,
    double tol = 1e-10,
    int max_depth = 50
);
 
LMCAS_API Result<QuadratureEstimate> adaptive_simpson_numeric(
    const std::shared_ptr<SymbolicExpr>& f,
    const std::string& var,
    const std::shared_ptr<SymbolicExpr>& a,
    const std::shared_ptr<SymbolicExpr>& b,
    double tol = 1e-10,
    int max_depth = 50
);
 
} // namespace LMCAS