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package bisection;
import java.util.function.DoubleUnaryOperator;
import bjc.utils.math.Dual;
import bjc.utils.math.DualExpr;
/*
* Benjamin Culkin
* 1/16/2018
* CS 479
* Bisection
*
* Use the bisection method to find bracketed roots of arbitrary real-valued functions.
*/
/**
* Use the bisection method to find bracketed roots of arbitrary real-valued
* functions.
*
* @author student
*
*/
public class NeoBisection {
/**
* Maximum number of iterations to attempt.
*/
public static final int MAXITR = 500;
/**
* Main method
*
* @param args
* Unused CLI args
*/
public static void main(String[] args) {
/* Bisection Method. */
/* The functions we're approximating a root for. */
DoubleUnaryOperator functionA = x -> Math.cos(x) - x;
DoubleUnaryOperator functionB = x -> {
double valA = Math.pow(x, 5);
double valB = 3 * Math.pow(x, 4);
double valC = 4 * Math.pow(x, 3);
return valA - valB + valC - 1;
};
DoubleUnaryOperator functionC = x -> (x * x) - 3;
/* The approximated roots. */
double answerA = bisect(functionA, 0, 1, 0.0001);
double answerB = bisect(functionB, 0, 1, 0.0001);
double answerC = bisect(functionC, 1, 2, 0.0000001);
/* Print out the answers + their function. */
System.out.printf("Bisection Method:\n");
System.out.printf("\tx = cos(x) => %.4f\n", answerA);
System.out.printf("\tx^5 - 3x^4 + 4x^2 - 1 => %.4f\n", answerB);
System.out.printf("\tx^2 - 3 => %.7f\n\n", answerC);
/* Newton's / Secant Method. */
/* The variable to manipulate in the expressions. */
Dual varX = new Dual();
DualExpr varXExpr = new DualExpr(varX);
/* The functions to find the roots of. */
DualExpr dualA = new DualExpr(DualExpr.ExprType.SUBTRACTION, new DualExpr(DualExpr.ExprType.COS, varXExpr),
varXExpr);
DualExpr dualB;
{
/* Construct the second function. */
DualExpr tempA = new DualExpr(varXExpr, 5);
DualExpr tempB = new DualExpr(DualExpr.ExprType.MULTIPLICATION, new DualExpr(new Dual(3)),
new DualExpr(varXExpr, 4));
DualExpr tempC = new DualExpr(DualExpr.ExprType.MULTIPLICATION, new DualExpr(new Dual(4)),
new DualExpr(varXExpr, 3));
dualB = new DualExpr(DualExpr.ExprType.ADDITION, new DualExpr(DualExpr.ExprType.SUBTRACTION, tempA, tempB),
new DualExpr(DualExpr.ExprType.SUBTRACTION, tempC, new DualExpr(new Dual(1))));
}
DualExpr dualC = new DualExpr(DualExpr.ExprType.SUBTRACTION, new DualExpr(varXExpr, 2),
new DualExpr(new Dual(3)));
/* Print out dualized expressions. */
System.out.printf("Expressions:\n");
System.out.printf("\t%s\n", dualA);
System.out.printf("\t%s\n", dualB);
System.out.printf("\t%s\n", dualC);
/* The approximated roots, using Newton's method. */
double newtonA = newton(dualA, varX, 0, 1, 0.0001);
double newtonB = newton(dualB, varX, 0, 1, 0.0001);
double newtonC = newton(dualC, varX, 1, 2, 0.0000001);
/* Print out the answers + their function. */
System.out.printf("Newton's Method:\n");
System.out.printf("\tx = cos(x) => %.4f\n", newtonA);
System.out.printf("\tx^5 - 3x^4 + 4x^2 - 1 => %.4f\n", newtonB);
System.out.printf("\tx^2 - 3 => %.7f\n\n", newtonC);
/* The approximated roots, using the secant method. */
double secantA = secant(dualA, varX, 0, 1, 0.0001);
double secantB = secant(dualB, varX, 0, 1, 0.0001);
double secantC = secant(dualC, varX, 1, 2, 0.0000001);
/* Print out the answers + their function. */
System.out.printf("Secant Method:\n");
System.out.printf("\tx = cos(x) => %.4f\n", secantA);
System.out.printf("\tx^5 - 3x^4 + 4x^2 - 1 => %.4f\n", secantB);
System.out.printf("\tx^2 - 3 => %.7f\n", secantC);
}
/*
* Calculate the number of iterations to approximate to a specified tolerance.
*/
private static double iterCount(double a, double b, double tol) {
double inside = Math.log((b - a) / tol);
return Math.ceil(inside / Math.log(2));
}
/* Calculate a root for an equation by the bisection method. */
private static double bisect(DoubleUnaryOperator func, double a, double b, double tol) {
/* Calculate the number of iterations. */
double N = iterCount(a, b, tol);
double val = 0.0;
/* Set our values. */
double newa = a;
double newb = b;
double newc = (a + b) / 2;
/*
* Continue onwards until we get the approximation to within the right
* tolerance.
*/
for (int i = 0; i < N; i++) {
newc = (newa + newb) / 2;
val = func.applyAsDouble(newc);
/* Pick the right direction to bisect in. */
if (func.applyAsDouble(newa) * val < 0) {
newb = newc;
} else {
newa = newc;
}
}
/* Return the right value. */
return newc;
}
/**
* Use Newton's method to find the root of an equation.
*
* @param func
* The function to find a root for.
*
* @param var
* The variable in the function.
*
* @param lo
* The lower bound for the root
*
* @param hi
* The higher bound for the root
*
* @param tol
* The tolerance for the answer
*
* @return The estimated value for the equation.
*/
public static double newton(DualExpr func, Dual var, double lo, double hi, double tol) {
/* Initial guess for root. */
double newmid = (lo + hi) / 2;
for (int i = 0; i < MAXITR; i++) {
/* Previous root guess. */
double prevmid = newmid;
/* Set the variable properly. */
var.real = newmid;
var.dual = 1;
/* Evaluate the function and its derivative. */
Dual res = func.evaluate();
/* Use Newton's method to refine our solution. */
newmid = newmid - (res.real / res.dual);
System.out.printf("\tTRACE: prevmid: %f\t\tnewmid: %f\n", prevmid, newmid);
/* Hand back the solution if it's good enough. */
if (Math.abs(newmid - prevmid) < tol) {
return newmid;
}
}
System.out.println("Newton's method iteration limit reached.");
/* Give back the solution. */
return newmid;
}
/**
* Calculate a root using the secant method.
*
* @param func
* The function to calculate a root for.
* @param var
* The variable in the function.
* @param lo
* The lower bound of the root.
* @param hi
* The upper bound of the root.
* @param tol
* The tolerance to find the root to.
* @return The root, to within the desired tolerance.
*/
public static double secant(DualExpr func, Dual var, double lo, double hi, double tol) {
/* Initial guesses for root. */
double guess1 = (lo + hi) / 3; // 1/3 into the range
double guess2 = ((lo + hi) / 3) * 2; // 2/3 into the range
for (int i = 0; i < MAXITR; i++) {
var.real = guess1;
var.dual = 1;
/* Evaluate the first guess. */
Dual res1 = func.evaluate();
var.real = guess2;
var.dual = 1;
/* Evaluate the first guess. */
Dual res2 = func.evaluate();
double oldGuess1 = guess1;
/* Use the secant method to refine our guesses. */
guess1 = guess2;
guess2 = ((oldGuess1 * res2.real) - (guess1 * res1.real)) / (res1.real - res2.real);
System.out.printf("\tTRACE: guess1: %f\t\tguess2: %f\n", guess1, guess2);
/* Hand back the solution if it's good enough. */
if (Math.abs(guess1 - guess2) < tol) {
return guess2;
}
}
System.out.println("Secant method iteration limit reached.");
/* Give back the solution. */
return guess2;
}
}
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