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## Methods that use the Derivative | Extreme Optimization Numerical Libraries for .NET Professional |

Newton devised a method to solve equations using values of the function and its first derivative. It is implemented by the NewtonRaphsonSolver class.

The NewtonRaphsonSolver class has two constructors. The first takes no arguments. The function
to solve and its derivative must be set using the TargetFunction and DerivativeOfTargetFunction
properties, which are both Func

The second constructor takes these three properties as its three parameters.

// Use the parameterless constructor: NewtonRaphsonSolver solver = new NewtonRaphsonSolver(); // Then set the target function and its derivative: solver.TargetFunction = Math.Sin; solver.DerivativeOfTargetFunction = Math.Cos; // and the initial guess: solver.InitialGuess = 4; // Or specify all three directly in the constructor: NewtonRaphsonSolver solver2 = new NewtonRaphsonSolver(Math.Sin, Math.Cos, 4);

The iteration can be controlled as outlined before by setting the AbsoluteTolerance, RelativeTolerance, ConvergenceCriterion, MaxIterations properties. The Solve method calculates the zero of the target function:

Console.WriteLine("Newton-Raphson Solver: sin(x) = 0"); Console.WriteLine(" Initial guess: 4"); double result = solver.Solve(); Console.WriteLine(" Result: {0}", solver.Status); Console.WriteLine(" Solution: {0}", solver.Result); Console.WriteLine(" Estimated error: {0}", solver.EstimatedError); Console.WriteLine(" # iterations: {0}", solver.IterationsNeeded);

You can still use this class if you don't have the derivative of the target function.
In this case, use the static GetNumericalDifferentiator method of the
FunctionMath class
to create a Func

It should be noted, however, that this technique requires significantly more function evaluations. If it is possible to find an interval that contains the root, the preferred method is to use one of the root bracketing solvers from the previous section.

A better alternative, if the function is simple enough to express in a single expression,
is to use automatic differentiation to have the derivative calculated for you.
To do this, you pass a lambda expression that evaluates the function to the
GetDerivative(Expression

Both techniques are illustrated below:

solver.TargetFunction = Special.BesselJ0; var derivative1 = FunctionMath.GetNumericalDifferentiator(Special.BesselJ0); var derivative2 = SymbolicMath.GetDerivative(x => Special.BesselJ0(x)); solver.DerivativeOfTargetFunction = derivative2; solver.InitialGuess = 5; Console.WriteLine("Zero of Bessel function near x=5:"); result = solver.Solve(); Console.WriteLine(" Solution: {0}", solver.Result); Console.WriteLine(" # iterations: {0}", solver.IterationsNeeded);

Newton, I. *Methodus fluxionum et serierum infinitarum.* 1664-1671.

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