Root Bracketing Solvers in IronPython QuickStart Sample

Illustrates the use of the root bracketing solvers for solving equations in one variable in IronPython.

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

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

import numerics

from math import *

# The RootBracketingSolver and derived classes reside in the 
# Extreme.Mathematics.EquationSolvers namespace.
from Extreme.Mathematics.EquationSolvers import *
# Function delegates reside in the Extreme.Mathematics
# namespace.
from Extreme.Mathematics import *
from Extreme.Mathematics.Algorithms import *

# Illustrates the use of the root bracketing solvers 
# in the Extreme.Mathematics.EquationSolvers namespace of the Extreme
# Optimization Mathematics Library for .NET.

# 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<double, double>. For more
# information about this delegate, see the
# FunctionDelegates QuickStart sample.
f = cos

# All root bracketing solvers inherit from
# RootBracketingSolver, an abstract base class.

#
# Bisection method
#

# The bisection method halves the interval during
# each iteration. It is implemented by the
# BisectionSolver class.
print "BisectionSolver: cos(x) = 0 over [1,2]"
solver = BisectionSolver()
solver.LowerBound = 1
solver.UpperBound = 2
# The target function is a Func<double, double>.
# See above.
solver.TargetFunction = f
result = solver.Solve()
# The Status property indicates
# the result of running the algorithm.
print "  Result:", solver.Status
# The result is also available through the
# Result property.
print "  Solution:", solver.Result
# You can find out the estimated error of the result
# through the EstimatedError property:
print "  Estimated error:", solver.EstimatedError
print "  # iterations:", 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:
print "RegulaFalsiSolver: cos(x) = 0 over [1,2]"
solver = RegulaFalsiSolver()
solver.LowerBound = 1
solver.UpperBound = 2
solver.MaxIterations = 1000
solver.TargetFunction = f
result = solver.Solve()
print "  Result:", solver.Status
print "  Solution:", solver.Result
print "  Estimated error:", solver.EstimatedError
print "  # iterations:", solver.IterationsNeeded
			
# However, for sin(x) = 0, everything is fine:
print "RegulaFalsiSolver: sin(x) = 0 over [-0.5,1]"
solver = RegulaFalsiSolver()
solver.LowerBound = -0.5
solver.UpperBound = 1
solver.TargetFunction = sin
result = solver.Solve()
print "  Result:", solver.Status
print "  Solution:", solver.Result
print "  Estimated error:", solver.EstimatedError
print "  # iterations:", 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.
print "DekkerBrentSolver: cos(x) = 0 over [1,2]"
solver = DekkerBrentSolver()
solver.LowerBound = 1
solver.UpperBound = 2
solver.TargetFunction = f
result = solver.Solve()
print "  Result:", solver.Status
print "  Solution:", solver.Result
print "  Estimated error:", solver.EstimatedError
print "  # iterations:", solver.IterationsNeeded

#
# Controlling the process
#
print "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 = 1e-6
# In this example, the absolute tolerance will be 
# ignored.
solver.AbsoluteTolerance = 1e-24
solver.LowerBound = 157081
solver.UpperBound = 157082
solver.TargetFunction = f
result = solver.Solve()
print "  Result:", solver.Status
print "  Solution:", solver.Result
# The estimated error will be less than 0.157
print "  Estimated error:", solver.EstimatedError
print "  # iterations:", solver.IterationsNeeded