/**
* Find the smallest real root of p within lowerBound and upperBound (bounds may or may not be
* included). If no real root exists in this interval, Double.NaN ist returned.
*
* @param p
* @param lowerBound
* @param upperBound
* @return
*/
public double findFirstRootIn(UnivariatePolynomial p, double lowerBound, double upperBound) {
if (makeSquarefree) {
// make p squarefree
UnivariatePolynomial gcd = UnivariatePolynomial.gcd(p, p.derive());
if (gcd.degree() > 0)
// Polynomial not squarefree!
p = p.div(gcd);
}
return EVAL(p, lowerBound, upperBound, p.evaluateAt(lowerBound), p.evaluateAt(upperBound));
}
public double[] findAllRootsIn(UnivariatePolynomial p, double lowerBound, double upperBound) {
p = p.shrink();
if (makeSquarefree) {
// make p squarefree
UnivariatePolynomial gcd = UnivariatePolynomial.gcd(p, p.derive());
if (gcd.degree() > 0)
// Polynomial not squarefree!
p = p.div(gcd);
}
return null;
}
private static double bisect(
UnivariatePolynomial p, double lowerBound, double upperBound, double fl, double fu) {
double[] a = p.getCoeffs();
assert fl * fu < 0.0 : "tried bisection on interval without sign change";
while (Math.abs(upperBound - lowerBound) > EPSILON) {
double center = 0.5 * (lowerBound + upperBound);
double fc = a[a.length - 1];
for (int i = a.length - 2; i >= 0; i--) fc = fc * center + a[i];
if (fc * fl < 0.0) {
upperBound = center;
fu = fc;
} else if (fc == 0.0) {
return center;
} else {
lowerBound = center;
fl = fc;
}
}
return lowerBound;
}
public double EVAL(
UnivariatePolynomial p, double lowerBound, double upperBound, double fl, double fu) {
// System.out.println( "EVAL on (" + lowerBound + "," + upperBound + ") |-> (" + fl + "," + fu +
// ")" );
double m = (lowerBound + upperBound) / 2.0;
double r = (upperBound - lowerBound) / 2.0;
// System.out.println( "(m,r)=("+m+","+r+")" );
if (upperBound - lowerBound < EPSILON) {
// System.out.println( "epsilon reached" );
// prevent infinite loop (e.g. for non-squarefree polynomials)
if (fl * fu < 0.0) return m;
else return m; // java.lang.Double.NaN;
}
// test for exclusion predicate: no real root in (m-r,m+r)
UnivariatePolynomial t = p.shift(m);
// System.out.println( "Taylor exp. of f(x-" + m + "): " + t );
double fm = t.getCoeff(0);
double ta[] = t.getCoeffs();
ta[0] = -Math.abs(ta[0]);
for (int i = 1; i < ta.length; ++i) ta[i] = Math.abs(ta[i]);
if (0.0 > new UnivariatePolynomial(ta).evaluateAt(r)) return java.lang.Double.NaN;
// System.out.println( "Exclusion predicate false" );
// test for inclusion predicate: exactly one real root in (m-r,m+r)
UnivariatePolynomial t_der = p.derive().shift(m);
// System.out.println( "Taylor exp. of f'(x-" + m + "): " + t_der );
double ta_der[] = t_der.getCoeffs();
ta_der[0] = -Math.abs(ta_der[0]);
for (int i = 1; i < ta_der.length; ++i) ta_der[i] = Math.abs(ta_der[i]);
if (fl * fu <= 0.0 && 0.0 > new UnivariatePolynomial(ta_der).evaluateAt(r))
return bisect(p, lowerBound, upperBound, fl, fu);
// System.out.println( "Inclusion predicate false" ); System.out.println();
// bisect
double result_left = EVAL(p, lowerBound, m, fl, fm);
if (!java.lang.Double.isNaN(result_left)) return result_left;
else return EVAL(p, m, upperBound, fm, fu);
}
/**
* Find all real roots of p.
*
* @param p
* @return
*/
public double[] findAllRoots(UnivariatePolynomial p) {
return findAllRootsIn(
p,
-p.stretch(-1.0).maxPositiveRootBound() * (1.0 + 2.220446049250313E-16),
p.maxPositiveRootBound() * (1.0 + 2.220446049250313E-16));
}
private static double bisect(UnivariatePolynomial p, double lowerBound, double upperBound) {
return bisect(p, lowerBound, upperBound, p.evaluateAt(lowerBound), p.evaluateAt(upperBound));
}