Newton's method in Java

This project provides a very simple implementation of the Newton-Raphson method for solving bivariate non-linear equation systems.

Example

As an example, we solve the following equation system:

3 * x^2 + 2 * y^2 - 35 = 0
4 * x^2 - 3 * y^2 - 24 = 0

This system has four solutions: (+-3, +-2)

Basic solution

Implementing the object functions is straightforward. The first function:

Function2D object1 = new Function2D() {
	public Double evaluate(Vector2D input) {
		return 3d * input.x * input.x + 2d * input.y * input.y - 35d;
	}
}

And the second function:

Function2D object2 = new Function2D() {
	public Double evaluate(Vector2D input) {
		return 4d * input.x * input.x - 3d * input.y * input.y - 24d;
	}
}

We may now solve the equation system with:

Vector2D result = new NewtonRaphson2D(object1, object2)
                                     .accuracy(1e-6)
                                     .iterationsPerTry(1000)
                                     .iterationsTotal(100000)
                                     .solve();

This very simple variant of the solver will use the finite difference method for approximating the derivatives. The result is:

Measure	Value
Time	0.0006 [ms]
Tries	1
Iterations	6
Quality	0.999999

Without further constraints, the solver will return the solution (3,2). We may specify constraints to find a solution with negative parameters:

Constraint2D constraint = new Constraint2D(){ 
	public Boolean evaluate(Vector2D input) { return input.x < 0 && input.y < 0; } 
};
                    
solver = new NewtonRaphson2D(object1, object2, constraint)...

First enhanced solution

We can compute the partial derivatives of our object functions:

d/dx(f1) =   6 * x
d/dy(f1) =   4 * y
d/dx(f2) =   8 * x
d/dy(f2) = - 6 * y

And provide instances of Function2D to implement these derivatives:

Function2D derivative11 = new Function2D() {
	public Double evaluate(Vector2D input) {
		return 6d * input.x;
	}
}

We can also run a simple check, to compare our explicit forms with the results of the finite difference method to make sure that we didn't make any mistakes:

Function2DUtil util = new Function2DUtil(1e-6);
util.isDerivativeFunction1(object1, derivative11, 0.01, 100, 0.001, 0.1d, 0.01d);
util.isDerivativeFunction2(object1, derivative12, 0.01, 100, 0.001, 0.1d, 0.01d);

Finally, we run the solver:

solver = new NewtonRaphson2D(object1, object2, derivative11, derivative12, derivative21, derivative22);

Using the explicit forms of the derivatives will speed up our computations, in this simple example by a factor of about 20%:

Measure	Value
Time	0.005 [ms]
Tries	1
Iterations	6
Quality	0.999999

Second enhanced solution

We can further speed up the solving process by realizing that our object functions share some code. We can therefore implement a "master" function that evaluates the object functions and the partial derivatives at the same time:

return new Function<Vector2D, Pair<Vector2D, SquareMatrix2D>>() {

	// Prepare result objects
	Vector2D                       object       = new Vector2D();
	SquareMatrix2D                 derivatives  = new SquareMatrix2D();
	Pair<Vector2D, SquareMatrix2D> result       = new Pair<Vector2D, SquareMatrix2D>(object, derivatives);

	/**
	 * Eval
	 * @param input
	 * @return
	 */
	public Pair<Vector2D, SquareMatrix2D> evaluate(Vector2D input) {
	
  		// Prepare
  		double xSquare = input.x * input.x;
		double ySquare = input.y * input.y;
                
		// Compute
		object.x = 3d * xSquare + 2d * ySquare - 35d;
		object.y = 4d * xSquare - 3d * ySquare - 24d;
		derivatives.x1 = + 6d * input.x;
		derivatives.x2 = + 4d * input.y;
		derivatives.y1 = + 8d * input.x;
		derivatives.y2 = - 6d * input.y;
                
		// Return
		return result;
	}
};

This will speed up our computations by an additional 20%:

Measure	Value
Time	0.0004 [ms]
Tries	1
Iterations	6
Quality	0.999999

The complete implementation of this example can be found here

Download

A binary version (JAR file) is available for download here.

The according Javadoc is available for download here.

Documentation

Online documentation is here.

License

Apache License 2.0