使用表达式模板用C符号微分++(Symbolic differentiation using exp

如何使用C中的表达式模板来实现符号微分++

一般你想要的方式来表示你的符号（即编码例如表达模板3 * x * x + 42 ），和一元函数，可以计算的衍生物。希望你用C ++元编程非常熟悉，知道这意味着什么和需要，而是给你一个想法：

// This should come from the expression templates
template<typename Lhs, typename Rhs>
struct plus_node;

// Metafunction that computes a derivative
template<typename T>
struct derivative;

// derivative<foo>::type is the result of computing the derivative of foo

// Derivative of lhs + rhs
template<typename Lhs, typename Rhs>
struct derivative<plus_node<Lhs, Rhs> > {
    typedef plus_node<
        typename derivative<Lhs>::type
        , typename derivative<Rhs>::type
    > type;
};

// and so on

然后你会占用两个部分（表示和计算），这样，这将是方便使用。例如derivative(3 * x * x + 42)(6)可能意味着'计算的导数3 * x * x + 42 6在X'。

但是即使你不知道如何才能写出表达式模板，它需要用C来写一个元程序是什么++我不建议去一下这种方式。模板元编程需要大量的样板，并可能很乏味。相反，我向您天才Boost.Proto库，而这恰恰是旨在帮助（使用表达式模板）写EDSLs和这些表达式模板进行操作。这也未必容易学习使用，但我发现，学习如何做到同样的事情，而无需使用较困难 。下面介绍了可实际上理解并计算一个样本程序derivative(3 * x * x + 42)(6)

#include <iostream>

#include <boost/proto/proto.hpp>

using namespace boost::proto;

// Assuming derivative of one variable, the 'unknown'
struct unknown {};

// Boost.Proto calls this the expression wrapper
// elements of the EDSL will have this type
template<typename Expr>
struct expression;

// Boost.Proto calls this the domain
struct derived_domain
: domain<generator<expression>> {};

// We will use a context to evaluate expression templates
struct evaluation_context: callable_context<evaluation_context const> {
    double value;

    explicit evaluation_context(double value)
        : value(value)
    {}

    typedef double result_type;

    double operator()(tag::terminal, unknown) const
    { return value; }
};
// And now we can do:
// evalutation_context context(42);
// eval(expr, context);
// to evaluate an expression as though the unknown had value 42

template<typename Expr>
struct expression: extends<Expr, expression<Expr>, derived_domain> {
    typedef extends<Expr, expression<Expr>, derived_domain> base_type;

    expression(Expr const& expr = Expr())
        : base_type(expr)
    {}

    typedef double result_type;

    // We spare ourselves the need to write eval(expr, context)
    // Instead, expr(42) is available
    double operator()(double d) const
    {
        evaluation_context context(d);
        return eval(*this, context);
    }
};

// Boost.Proto calls this a transform -- we use this to operate
// on the expression templates
struct Derivative
: or_<
    when<
        terminal<unknown>
        , boost::mpl::int_<1>()
    >
    , when<
        terminal<_>
        , boost::mpl::int_<0>()
    >
    , when<
        plus<Derivative, Derivative>
        , _make_plus(Derivative(_left), Derivative(_right))
    >
    , when<
        multiplies<Derivative, Derivative>
        , _make_plus(
            _make_multiplies(Derivative(_left), _right)
            , _make_multiplies(_left, Derivative(_right))
        )
    >
    , otherwise<_>
> {};

// x is the unknown
expression<terminal<unknown>::type> const x;

// A transform works as a functor
Derivative const derivative;

int
main()
{
    double d = derivative(3 * x * x + 3)(6);
    std::cout << d << '\n';
}