Root Bracketing Solvers in Visual Basic QuickStart Sample
Illustrates the use of the root bracketing solvers for solving equations in one variable in Visual Basic.
This sample is also available in: C#, F#, IronPython.
Overview
This QuickStart sample demonstrates how to use root bracketing solvers to find solutions to nonlinear equations in one variable using Numerics.NET.
Root bracketing solvers are numerical algorithms that find solutions to equations of the form f(x)=0 by repeatedly narrowing down an interval known to contain a root. The sample shows three different methods:
- The Bisection method, which simply halves the interval at each step
- The Regula Falsi (False Position) method, which uses linear interpolation but can be slow to converge in some cases
- The Dekker-Brent method, which combines the reliability of bisection with faster convergence techniques
The sample demonstrates how to:
- Set up and configure different root bracketing solvers
- Specify the target function to solve
- Control convergence criteria and iteration limits
- Handle the results and check solution status
- Access error estimates and iteration counts
Examples use simple trigonometric functions to illustrate the behavior and relative performance of each method. The code shows how to handle both well-behaved and more challenging cases.
The code
Option Infer On
' The RootBracketingSolver and derived classes reside in the
' Numerics.NET.EquationSolvers namespace.
Imports Numerics.NET.EquationSolvers
' Function delegates reside in the Numerics.NET
' namespace.
Imports Numerics.NET.Algorithms
Imports Numerics.NET
' Illustrates the use of the root bracketing solvers
' in the Numerics.NET.EquationSolvers namespace of Numerics.NET.
Module RootBracketingSolvers
Sub Main()
' 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")
' Root bracketing solvers are used to solve
' non-linear equations in one variable.
'
' Root bracketing solvers start with an interval
' which is known to contain a root. This interval
' is made smaller and smaller in successive
' iterations until a certain tolerance is reached,
' or the maximum number of iterations has been
' exceeded.
'
' The properties and methods that give you control
' over the iteration are shared by all classes
' that implement iterative algorithms.
'
' Target function
'
' The function we are trying to solve must be
' provided as a Func(Of Double, Double). For more
' information about this delegate, see the
' Functions QuickStart sample.
Dim f As Func(Of Double, Double) = AddressOf Math.Cos
' All root bracketing solvers inherit from
' RootBracketingSolver, an abstract base class.
Dim solver As RootBracketingSolver
'
' Bisection method
'
' The bisection method halves the interval during
' each iteration. It is implemented by the
' BisectionSolver class.
Console.WriteLine("BisectionSolver: cos(x) = 0 over [1,2]")
solver = New BisectionSolver()
solver.LowerBound = 1
solver.UpperBound = 2
solver.TargetFunction = f
Dim result As Double = solver.Solve()
' The Status property indicates
' the result of running the algorithm.
Console.WriteLine(" Result: {0}",
solver.Status)
' The result is also available through the
' Result property.
Console.WriteLine($" Solution: {solver.Result}")
' You can find out the estimated error of the result
' through the EstimatedError property:
Console.WriteLine(" Estimated error: {0}",
solver.EstimatedError)
Console.WriteLine(" # iterations: {0}",
solver.IterationsNeeded)
'
' Regula Falsi method
'
' The Regula Falsi method optimizes the selection
' of the next interval. Unfortunately, the
' optimization breaks down in some cases.
' Here is an example:
Console.WriteLine("RegulaFalsiSolver: cos(x) = 0 over [1,2]")
solver = New RegulaFalsiSolver()
solver.LowerBound = 1
solver.UpperBound = 2
solver.MaxIterations = 1000
solver.TargetFunction = f
result = solver.Solve()
Console.WriteLine(" Result: {0}",
solver.Status)
Console.WriteLine($" Solution: {solver.Result}")
Console.WriteLine(" Estimated error: {0}",
solver.EstimatedError)
Console.WriteLine(" # iterations: {0}",
solver.IterationsNeeded)
' However, for sin(x) = 0, everything is fine:
Console.WriteLine("RegulaFalsiSolver: sin(x) = 0 over [-0.5,1]")
solver = New RegulaFalsiSolver()
solver.LowerBound = -0.5
solver.UpperBound = 1
solver.TargetFunction = AddressOf Math.Sin
result = solver.Solve()
Console.WriteLine(" Result: {0}",
solver.Status)
Console.WriteLine($" Solution: {solver.Result}")
Console.WriteLine(" Estimated error: {0}",
solver.EstimatedError)
Console.WriteLine(" # iterations: {0}",
solver.IterationsNeeded)
'
' Dekker-Brent method
'
' The Dekker-Brent method combines the best of
' both worlds. It is the most robust and, on average,
' the fastest method.
Console.WriteLine("DekkerBrentSolver: cos(x) = 0 over [1,2]")
solver = New DekkerBrentSolver()
solver.LowerBound = 1
solver.UpperBound = 2
solver.TargetFunction = f
result = solver.Solve()
Console.WriteLine(" Result: {0}",
solver.Status)
Console.WriteLine($" Solution: {solver.Result}")
Console.WriteLine(" Estimated error: {0}",
solver.EstimatedError)
Console.WriteLine(" # iterations: {0}",
solver.IterationsNeeded)
'
' 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 = 20
' You can specify how convergence is to be tested
' through the ConvergenceCriterion property:
solver.ConvergenceCriterion =
ConvergenceCriterion.WithinRelativeTolerance
' And, of course, you can set the absolute or
' relative tolerance.
solver.RelativeTolerance = 0.00001
' In this example, the absolute tolerance will be
' ignored.
solver.AbsoluteTolerance = 1.0E-24
solver.LowerBound = 157081
solver.UpperBound = 157082
solver.TargetFunction = f
result = solver.Solve()
Console.WriteLine(" Result: {0}",
solver.Status)
Console.WriteLine($" Solution: {solver.Result}")
' The estimated error will be less than 0.157
Console.WriteLine(" Estimated error: {0}",
solver.EstimatedError)
Console.WriteLine(" # iterations: {0}",
solver.IterationsNeeded)
Console.Write("Press Enter key to exit...")
Console.ReadLine()
End Sub
End Module