If (variables = null || variables.Length = 0) return false įor ( int i = 0 i /// Creates formulae for equations and their derivatives. / /// /// protected bool AssignVariables(string variables) / /// Creates translator and adds Real variables to it. / public class AnalyticalSystem: ConstructedSystem
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Using this library the following descendant class for NonlinearSystem realized:Ĭopy Code /// /// System of Nonlinear Analytical Equations. This library provides not only analytical expression evaluation, but also analytical derivative calculation. The ANALYTICS C# library used for that purpose. Realizationįor realization of the approach with analytical derivative calculation, an analytical evaluation engine is required. The NewtonRaphsonSolver class used in all examples as the solver, requiring the derivative (Jacobian) calculations. The MATHEMATICS library realizes several final solver classes. / public override SolutionResult Solve(NonlinearSystem system, double x0, / /// Common Iterative Solution method.
/ /// /// Current x values /// Next (calculated) x values protected abstract bool Iteration(NonlinearSystem system, double x0, ref double x1) / public abstract class IterativeSolver : NonlinearSolver / /// /// /// /// /// public abstract SolutionResult Solve(NonlinearSystem system, double x0,Īs for common nonlinear solvers are iterative ones, the inherited class IterativeSolver provided and it overrides Solve method in the form of iteration step sequence:Ĭopy Code /// /// Base Abstract Iterative Solver. / public abstract class NonlinearSolver / /// /// /// public virtual bool Jacobian(double x, ref double J) / (must be overridden in inherited classes if derivative calculation supported) / /// Calculates the system's Jacobian.
/ /// /// /// public abstract bool Calculate(double x, ref double y) / /// Calculates equation values for given variable values. / /// Dimension (the number of equations). / public abstract class NonlinearSystem The presented realization of the classes implemented in the MATHEMATICS library as the following:Ĭopy Code /// /// Base class for Nonlinear Equation system. In general case there must be 2 main classes: NonlinearSystem – represents nonlinear equations system and NonlinearSolver – represents nonlinear solver. Backgroundįor understanding analytical realization of nonlinear equations system solution, let us consider the general programming environment for the problem. Calculations of analytical expressions and derivation process carried out with ANALYTICS C# library. This article introduces alternative approach, when the nonlinear system is set up in the form of analytical expressions and derivative expressions also provided in analytical form. Common realizations of computational libraries uses numerical calculation of the derivatives – finite difference. The algorithms require calculation not only values of system equations at some point, but also the values of gradient, Jacobian or Hessian of the system, that are all some type of derivatives of the system. They realize well-known mathematical algorithms – Newton-Raphson method, Levenberg-Marquardt method, Powell’s Dog Leg method and so on. There are many libraries for solving nonlinear equation systems.
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