package edu.northwestern.at.utils.math.rootfinders;
/* Please see the license information at the end of this file. */
import edu.northwestern.at.utils.math.*;
/** Find roots of equations using Bisection.
*
* <p>
* The Method of Bisection is a root-finding method which requires an
* initial interval [x0,x1] bracketing a root and that the function be
* continuous in that interval.
* </p>
*
* <p>
* An updated estimate of the root value is computed by
* using the midpoint of the two previous values. Depending
* upon the sign of the function at the interval midpoint,
* the midpoint replaces either the lower interval value
* (if f(midpoint) < 0) or the upper interval value
* (if f(midpoint) > 0). This bisection process halves the
* search interval on each iteration
* </p>
*
* <p>
* If the function whose root is being sought has a derivative
* at each point in the interval, the Method of Secants or
* Brent's Method is a better choice.
* </p>
*/
public class Bisection implements MonadicFunctionRootFinder
{
/** Find root using the Method of Bisection.
*
* @param x0 First approximation to root value.
* @param x1 Second approximation to root value.
* @param tol Desired accuracy for root value.
* @param maxIter Maximum number of iterations.
* @param function Class implementing MonadicFunction
* interface to provide function values.
* @param convergenceTest RootFinderConvergenceTest which
* tests for convergence of the root-finding
* process.
* @param iterationInformation Class implementing
* RootFinderIterationInformation
* for retrieving information about
* each iteration of root finding
* process. Set to null if you don't
* want this information.
*
* @return Approximation to root of function.
*
* @throws IllegalArgumentException
* if [x0,x1] cannot be expanded
* to bracket a root or function
* is null.
*
* <p>
* This implementation always starts by attempting to expand the root
* bracketing interval to enclose a root.
* </p>
*/
public static double bisection
(
double x0 ,
double x1 ,
double tol ,
int maxIter ,
MonadicFunction function ,
RootFinderConvergenceTest convergenceTest ,
RootFinderIterationInformation iterationInformation
)
throws IllegalArgumentException
{
/* Calculated value of x at each iteration. */
double x;
/* Function value at x0 . */
double f0;
/* Function value at x1 . */
double f1;
/* Function value at calculated value of x . */
double fx;
/* Ratio of function values at two successive approximants. */
double r;
/* Root, if within desired tolerance. */
double root;
// Make sure function is not null.
if ( function == null )
{
throw new IllegalArgumentException(
"Function cannot be null" );
}
// Set initial function values.
f0 = function.f( x0 );
f1 = function.f( x1 );
// Test if there is a root in the
// provided interval.
// For this to be true, the function values
// at the left and right end of the interval
// must have different signs. If the signs
// are the same, try expanding the interval
// geometrically and see if we can find a
// new interval bracketing the root.
if ( ( ( f0 > 0.0 ) && ( f1 > 0.0 ) ) ||
( ( f0 < 0.0 ) && ( f1 < 0.0 ) ) )
{
double[] bracket = new double[]{ x0 , x1 };
if ( !BracketRoot.bracketRoot( bracket, function, maxIter, 1.6 ) )
{
// Give up if we can't find a new interval
// bracketing a root.
throw new IllegalArgumentException(
"Cannot expand interval [x0,x1] to contain root." );
}
// Use new bracketing interval.
else
{
x0 = bracket[ 0 ];
x1 = bracket[ 1 ];
f0 = function.f( x0 );
f1 = function.f( x1 );
}
}
// Begin method of secants loop.
x = 0.0D;
for( int iter = 0; iter < maxIter; iter++ )
{
// Compute new approximant at midpoint of
// previous two approximants.
x = ( x0 + x1 ) / 2.0D;
fx = function.f( x );
// Post updated iteration information.
if ( iterationInformation != null )
{
iterationInformation.iterationInformation(
x , fx , Double.NaN , iter );
}
// Check if new approximant is accurate enough.
if ( convergenceTest.converged( x1 , x0 , fx , tol , tol ) ) break;
// Update root estimate if convergence
// not yet achieved.
if ( ( fx * f0 ) > 0.0D )
{
x0 = x;
f0 = fx;
}
else
{
x1 = x;
f1 = fx;
}
}
return x;
}
/** Find root using the Method of Bisection.
*
* @param x0 First approximation to root value.
* @param x1 Second approximation to root value.
* @param tol Desired accuracy for root value.
* @param maxIter Maximum number of iterations.
* @param function Class implementing MonadicFunction
* interface to provide function values.
*
* @return Approximation to root of function.
*
* @throws IllegalArgumentException
* if [x0,x1] cannot be expanded
* to bracket a root or function
* is null.
*
* <p>
* This implementation always starts by attempting to expand the root
* bracketing interval to enclose a root.
* </p>
*/
public static double bisection
(
double x0 ,
double x1 ,
double tol ,
int maxIter ,
MonadicFunction function
)
throws IllegalArgumentException
{
return bisection(
x0 , x1 , tol , maxIter , function ,
new StandardRootFinderConvergenceTest() ,
null );
}
/** Find root using the Method of Bisection.
*
* @param x0 First approximation to root value.
* @param x1 Second approximation to root value.
* @param function Class implementing MonadicFunction
* interface to provide function values.
*
* @return Approximation to root of function.
*
* @throws IllegalArgumentException
* if [x0,x1] cannot be expanded
* to bracket a root or function
* is null.
*
* <p>
* This implementation always starts by attempting to expand the root
* bracketing interval to enclose a root. Up to 250 iterations are
* attempted with the convergence tolerance set to Constants.MACHEPS .
* </p>
*/
public static double bisection
(
double x0 ,
double x1 ,
MonadicFunction function
)
throws IllegalArgumentException
{
return bisection(
x0 , x1 , Constants.MACHEPS , 250 , function ,
new StandardRootFinderConvergenceTest() ,
null );
}
/** Implementation for {@link MonadicFunctionRootFinder} interface.
*/
public double findRoot
(
double x0 ,
double x1 ,
double tol ,
int maxIter ,
MonadicFunction function ,
MonadicFunction derivativeFunction ,
RootFinderConvergenceTest convergenceTest ,
RootFinderIterationInformation iterationInformation
)
throws IllegalArgumentException
{
return bisection(
x0 , x1 , tol , maxIter , function , convergenceTest ,
iterationInformation );
}
/** Constructor if RootFinder interface used.
*/
public Bisection()
{
}
}
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