sadET

sadET is a C++ library for automatic differentiation using expression templates. It provides a simple (and sometimes efficient) way to compute derivatives of mathematical expressions. The good thing about expresison templates is that the mathematical expressions and their derivatives are constructed at compile-time, while their numerical evaluation is occurres at run-time on demand (sort of lazy evaluation).

Current Features

• Supports basic arithmetic operations: addition, subtraction, multiplication, division, and power
• Supports unary operations: negative, exponential, logarithm, and trigonometric functions (sin, cos)
• Allows the creation of variables and constants
• Provides functions to evaluate expressions and compute derivatives

Usage

To use the sadET library, include the sadET.hpp header file in your C++ code:

#include <sadET.hpp>

Use Constant<numType,value> to declare constants and Variable<ID,numType> to decalre variables. Example:

sadET::Constant<long double, 0.4> c;
sadET::Variable<0, double> x;
sadET::Variable<1, double> y;

Due to the powerful type-deduction that C++ offers, you only need to declare types of your cponstants and variables, and the cpmpiler will figure it out. Consider this:

sadET::Constant<long double, 0.4> c;
sadET::Variable<0, double> x;
auto f = pow(x,c);

Here, f automatically is an instance of sadET::Pow<sadET::Variable<0, double>, sadET::Constant<long double, 0.4>>. However, all numerical operations will return long double, since it is their "biggest" common type.

You can simply use the evaluate function to get the value of an expression, and derivative to get its derivative. For example:

#include<iostream>
#include<sadET.hpp>


using std::cout;
using std::map;

using sadET::IDType;
using sadET::Constant;
using sadET::Variable;

int main(){
    Constant<long double,0.4> c;
    Variable<0,double> x;
    Variable<1,double> y;

    map<IDType,long double> at= {{x.getID(),0.2},{y.getID(),0.8}} ;

    //note that c++ can figure out that evaluate and derivative belong to sadET!
    cout<<evaluate( derivative( sin(x) * cos(y) + pow( x, sin(y+c) ) ,  x,y ) , at )<<"\n";

    return 0;
}

For full examples, see the Examples directory. I hope to continue adding new features, so keep an eye on this repo.

Work in progress... What I want to add in the near future:

• Sort variables wrt ids's in operators. This may help you simplify things.
• Define Sum and Product for many variables (e.g. Sum<expr, restExpr...>) recursively, but with expressions ordered in some way.
• Simplification rules. This feels harder, due to the heuristic nature of the subject.

If you know how to do these things, and like sadET, let me know at dkaramit@yahoo.com.