Nonlinear Systems in C# QuickStart Sample

Illustrates the use of the NewtonRaphsonSystemSolver and DoglegSystemSolver classes for solving systems of nonlinear equations in C#.

This sample is also available in: Visual Basic, F#, IronPython.

Overview

This QuickStart sample demonstrates how to solve systems of nonlinear equations using Numerics.NET’s equation solver classes.

The sample illustrates solving a system of two nonlinear equations using both the Newton-Raphson method and Powell’s dogleg method. It shows:

• How to define systems of equations using delegates
• How to provide derivatives (the Jacobian matrix) for better performance
• How to use automatic differentiation when derivatives are not available
• How to control convergence criteria and iteration limits
• How to interpret solver status and results
• How to use Powell’s dogleg method for more robust convergence
• How to solve systems without providing derivatives

The sample includes extensive comments explaining each step and demonstrates various solver options and parameters that control the solution process. It shows how to handle both simple cases and more complex scenarios where additional control over the solution process is needed.

The code

using System;
using Numerics.NET;
// The equation solver classes reside in the
// Numerics.NET.EquationSolvers namespace.
using Numerics.NET.EquationSolvers;
using Numerics.NET.Algorithms;

// Illustrates solving systems of non-linear equations using
// classes in the Numerics.NET.EquationSolvers namespace
// of Numerics.NET.

// The license is verified at runtime. We're using
// a 30 day trial key here. For more information, see
//     https://numerics.net/trial-key
Numerics.NET.License.Verify("your-trial-key-here");

//
// Target function
//

// The function we are trying to solve can be provided
// on one of two ways. The first is as an array of
// Func<Vector<double>, double> delegates. See the end of this
// sample for definitions of the methods that are referenced here.
Func<Vector<double>, double>[] f =
{
    x => Math.Exp(x[0])*Math.Cos(x[1]) - x[0]*x[0] + x[1]*x[1],
    x => Math.Exp(x[0])*Math.Sin(x[1]) - 2*x[0]*x[1]
};
// We can also supply the Jacobian, which is the matrix of partial
// derivatives. We do so by providing the gradient of each target
// function as a delegate that takes a second argument:
// the vector that is to hold the return value. This avoids unnecessary
// creation of new var instances.
Func<Vector<double>, Vector<double>, Vector<double>>[] df =
{
    (x,y) => {
        y[0] = Math.Exp(x[0])*Math.Cos(x[1]) - 2*x[0];
        y[1] = -Math.Exp(x[0])*Math.Sin(x[1]) + 2*x[1];
        return y;
    },
    (x,y) => {
        y[0] = Math.Exp(x[0])*Math.Sin(x[1]) - 2*x[1];
        y[1] = Math.Exp(x[0])*Math.Cos(x[1]) - 2*x[0];
        return y;
    }
};

// The initial values are supplied as a vector:
var initialGuess = Vector.Create(0.5, 0.5);

//
// Newton-Raphson Method
//

// The Newton-Raphson method is implemented by
// the NewtonRaphsonSystemSolver class.
var solver = new NewtonRaphsonSystemSolver(f, df, initialGuess);

// and call the Solve method to obtain the solution:
var solution = solver.Solve();

Console.WriteLine("N-dimensional Newton-Raphson Solver:");
Console.WriteLine("exp(x)*cos(y) - x^2 + y^2 = 0");
Console.WriteLine("exp(x)*sin(y) - 2xy = 0");
Console.WriteLine($"  Initial guess: {initialGuess:F2}");
// The Status property indicates
// the result of running the algorithm.
Console.WriteLine($"  Status: {solver.Status}");
// The result is also available through the
// Result property.
Console.WriteLine($"  Solution: {solver.Result}");
Console.WriteLine($"  Function value: {solver.ValueTest.Error}");
// You can find out the estimated error of the result
// through the EstimatedError property:
Console.WriteLine($"  Estimated error: {solver.EstimatedError}");
Console.WriteLine($"  # iterations: {solver.IterationsNeeded}");
Console.WriteLine($"  # evaluations: {solver.EvaluationsNeeded}");

// You can use Automatic Differentiation to compute the Jacobian.
// To do so, set the target functions using the SetSymbolicTargetFunctions
// method:
solver = new NewtonRaphsonSystemSolver();
solver.InitialGuess = initialGuess;

solver.SetSymbolicTargetFunctions(
    x => Math.Exp(x[0]) * Math.Cos(x[1]) - x[0] * x[0] + x[1] * x[1],
    x => Math.Exp(x[0]) * Math.Sin(x[1]) - 2 * x[0] * x[1]);

solution = solver.Solve();
Console.WriteLine("Using Automatic Differentiation:");
Console.WriteLine($"  Solution: {solver.Result}");
Console.WriteLine($"  Function value: {solver.ValueTest.Error}");
Console.WriteLine($"  # iterations: {solver.IterationsNeeded}");

// When you don't have the Jacobian for the target functions
// and you don't use Automatic Differentiation, the equation solver
// will use a numerical approximation.

//
// Controlling the process
//

Console.WriteLine("Same with modified parameters:");
// You can set the maximum # of iterations:
// If the solution cannot be found in time, the
// Status will return a value of
// IterationStatus.IterationLimitExceeded
solver.MaxIterations = 10;

// The ValueTest property returns the convergence
// test based on the function value. We can set
// its tolerance property:
solver.ValueTest.Tolerance = 1e-10;
// Its Norm property determines how the error
// is calculated. Here, we choose the maximum
// of the function values:
solver.ValueTest.Norm = VectorConvergenceNorm.Maximum;

// The SolutionTest property returns the test
// on the change in location of the solution.
solver.SolutionTest.Tolerance = 1e-8;
// You can specify how convergence is to be tested
// through the ConvergenceCriterion property:
solver.SolutionTest.ConvergenceCriterion = ConvergenceCriterion.WithinRelativeTolerance;

solver.InitialGuess = initialGuess;
solution = solver.Solve();
Console.WriteLine($"  Status: {solver.Status}");
Console.WriteLine($"  Solution: {solver.Result}");
// The estimated error will be less than 5e-14
Console.WriteLine($"  Estimated error: {solver.SolutionTest.Error}");
Console.WriteLine($"  # iterations: {solver.IterationsNeeded}");
Console.WriteLine($"  # evaluations: {solver.EvaluationsNeeded}");

//
// Powell's dogleg method
//

// The dogleg method is more robust than Newton's method.
// It will converge often when Newton's method fails.
DoglegSystemSolver dogleg = new DoglegSystemSolver(f, df, initialGuess);

// Unique to the dogleg method is the TrustRegionRadius property.
// Any step of the algorithm is not larger than this value.
// It is adjusted at each iteration.
dogleg.TrustRegionRadius = 0.5;

// Call the Solve method to obtain the solution:
solution = dogleg.Solve();

Console.WriteLine("Powell's Dogleg Solver:");
Console.WriteLine($"  Initial guess: {initialGuess:F2}");
Console.WriteLine($"  Status: {dogleg.Status}");
Console.WriteLine($"  Solution: {dogleg.Result}");
Console.WriteLine($"  Estimated error: {dogleg.EstimatedError}");
Console.WriteLine($"  # iterations: {dogleg.IterationsNeeded}");
Console.WriteLine($"  # evaluations: {dogleg.EvaluationsNeeded}");

// The dogleg method can work without derivatives. Care is taken
// to keep the number of evaluations down to a minimum.
dogleg.JacobianFunction = null;
// Call the Solve method to obtain the solution:
solution = dogleg.Solve();

Console.WriteLine("Powell's Dogleg Solver (no derivatives):");
Console.WriteLine($"  Initial guess: {initialGuess:F2}");
Console.WriteLine($"  Status: {dogleg.Status}");
Console.WriteLine($"  Solution: {dogleg.Result}");
Console.WriteLine($"  Estimated error: {dogleg.EstimatedError}");
Console.WriteLine($"  # iterations: {dogleg.IterationsNeeded}");
Console.WriteLine($"  # evaluations: {dogleg.EvaluationsNeeded}");

Console.Write("Press Enter key to exit...");
Console.ReadLine();

// First set of functions.
static double f1(Vector<double> x)
{
    return Math.Exp(x[0])*Math.Cos(x[1]) - x[0]*x[0] + x[1]*x[1];
}
static double f2(Vector<double> x)
{
    return Math.Exp(x[0])*Math.Sin(x[1]) - 2*x[0]*x[1];
}
// Gradient of the first set of functions.
static Vector<double> df1(Vector<double> x, Vector<double> df)
{
    df[0] = Math.Exp(x[0])*Math.Cos(x[1]) - 2*x[0];
    df[1] = -Math.Exp(x[0])*Math.Sin(x[1]) + 2*x[1];
    return df;
}
static Vector<double> df2(Vector<double> x, Vector<double> df)
{
    df[0] = Math.Exp(x[0])*Math.Sin(x[1]) - 2*x[1];
    df[1] = Math.Exp(x[0])*Math.Cos(x[1]) - 2*x[0];
    return df;
}