## Παράδειγμα: Υλοποίηση √2 με NR

Το παρακάτω παράδειγμα υπολογίζει τη √2 δημιουργώντας μια άπειρου μήκους ακολουθία με διαδοχικά καλύτερες προσεγγίσεις.

```import java.util.Optional;
import java.util.function.Predicate;
import java.util.function.DoubleFunction;
import java.util.stream.Stream;

/** Find a square root using the Newton-Raphson approximation */
class SquareRoot {

/** Obtain successive approximations of a function's root using the
* Newton-Raphson method. */
static class NewtonRaphson {
/** f(x) and f'(x) */
DoubleFunction fx, fdx;

NewtonRaphson(DoubleFunction fx, DoubleFunction fdx) {
this.fx = fx;
this.fdx = fdx;
}

/** Return next approximation, given the previous one */
Double nextApproximation(double previous) {
// xₙ₊₁ = xₙ - f(xₙ) / f′(xₙ)
return previous - (double)fx.apply(previous) / (double)fdx.apply(previous);
}
}

/** Test whether successive parts of a series differ more than a value */
static class NotWithin implements Predicate<Double> {
/** Previous value in series */
Optional<Double> previous = Optional.empty();
/** Difference value above which the test method returns true */
Double epsilon;

NotWithin(double d) {
epsilon = d;
}

/**
* Return true if successive parts of the series do not differ by
* less than the specified epsilon.
*/
@Override
public boolean test(Double d) {
boolean r;

if (previous.isPresent())
r = (Math.abs(previous.get() - d) > epsilon);
else
r = true;
previous = Optional.of(d);
return r;
}
}

public static void main(String args[]) {
final double SQRT_TO_FIND = 2;
DoubleFunction fx = (x -> x * x - SQRT_TO_FIND); // f(x) = x² - α
DoubleFunction fdx = (x -> 2 * x); // f'(x) = 2x
var rootTwo = new NewtonRaphson(fx, fdx);
var greaterThanEpsilon = new NotWithin(1e-15);

// SQRT_TO_FIND is also our first approximation
System.out.println(Stream.iterate(SQRT_TO_FIND, rootTwo::nextApproximation)
.dropWhile(greaterThanEpsilon)
.findFirst()
.get());
}
}
```