/**
* Test if x is an approximate real root.
*
* @param x approximate real root.
* @param f univariate polynomial, non-zero.
* @param fp univariate polynomial, non-zero, deriviative of f.
* @param eps requested interval length.
* @return true if x is a decimal approximation of a real v with f(v) = 0 with |d-v| < eps,
* else false.
*/
public boolean isApproximateRoot(
BigDecimal x, GenPolynomial<BigDecimal> f, GenPolynomial<BigDecimal> fp, BigDecimal eps) {
if (x == null) {
throw new IllegalArgumentException("null root not allowed");
}
if (f == null || f.isZERO() || f.isConstant() || eps == null) {
return true;
}
BigDecimal dc = BigDecimal.ONE; // only for clarity
// f(x)
BigDecimal fx = PolyUtil.<BigDecimal>evaluateMain(dc, f, x);
// System.out.println("fx = " + fx);
if (fx.isZERO()) {
return true;
}
// f'(x)
BigDecimal fpx = PolyUtil.<BigDecimal>evaluateMain(dc, fp, x); // f'(d)
// System.out.println("fpx = " + fpx);
if (fpx.isZERO()) {
return false;
}
BigDecimal d = fx.divide(fpx);
if (d.isZERO()) {
return true;
}
if (d.abs().compareTo(eps) <= 0) {
return true;
}
System.out.println("x = " + x);
System.out.println("d = " + d);
return false;
}
public static void example13() {
BigRational fac = new BigRational();
UnivPowerSeriesRing<BigRational> pfac = new UnivPowerSeriesRing<BigRational>(fac, 11, "y");
UnivPowerSeries<BigRational> exp = pfac.getEXP();
System.out.println("exp = " + exp);
System.out.println("exp(1) = " + exp.evaluate(fac.getONE()));
System.out.println("exp(0) = " + exp.evaluate(fac.getZERO()));
BigDecimal dfac = new BigDecimal();
UnivPowerSeriesRing<BigDecimal> dpfac = new UnivPowerSeriesRing<BigDecimal>(dfac, 11, "z");
UnivPowerSeries<BigDecimal> dexp = dpfac.getEXP();
System.out.println("exp = " + dexp);
System.out.println("exp(1) = " + dexp.evaluate(dfac.getONE()));
System.out.println("exp(0) = " + dexp.evaluate(dfac.getZERO()));
}
/**
* Test if x is an approximate real root.
*
* @param x approximate real root.
* @param f univariate polynomial, non-zero.
* @param eps requested interval length.
* @return true if x is a decimal approximation of a real v with f(v) = 0 with |d-v| < eps,
* else false.
*/
public boolean isApproximateRoot(BigDecimal x, GenPolynomial<C> f, C eps) {
if (x == null) {
throw new IllegalArgumentException("null root not allowed");
}
if (f == null || f.isZERO() || f.isConstant() || eps == null) {
return true;
}
BigDecimal e = new BigDecimal(eps.getRational());
e = e.multiply(new BigDecimal("1000")); // relax
BigDecimal dc = BigDecimal.ONE;
// polynomials with decimal coefficients
GenPolynomialRing<BigDecimal> dfac = new GenPolynomialRing<BigDecimal>(dc, f.ring);
GenPolynomial<BigDecimal> df = PolyUtil.<C>decimalFromRational(dfac, f);
GenPolynomial<C> fp = PolyUtil.<C>baseDeriviative(f);
GenPolynomial<BigDecimal> dfp = PolyUtil.<C>decimalFromRational(dfac, fp);
//
return isApproximateRoot(x, df, dfp, e);
}
/**
* Approximate real root.
*
* @param iv real root isolating interval with f(left) * f(right) < 0.
* @param f univariate polynomial, non-zero.
* @param eps requested interval length.
* @return a decimal approximation d such that |d-v| < eps, for f(v) = 0, v real.
*/
public BigDecimal approximateRoot(Interval<C> iv, GenPolynomial<C> f, C eps)
throws NoConvergenceException {
if (iv == null) {
throw new IllegalArgumentException("null interval not allowed");
}
BigDecimal d = iv.toDecimal();
if (f == null || f.isZERO() || f.isConstant() || eps == null) {
return d;
}
if (iv.length().compareTo(eps) < 0) {
return d;
}
BigDecimal left = new BigDecimal(iv.left.getRational());
BigDecimal right = new BigDecimal(iv.right.getRational());
BigDecimal e = new BigDecimal(eps.getRational());
BigDecimal q = new BigDecimal("0.25");
// System.out.println("left = " + left);
// System.out.println("right = " + right);
e = e.multiply(d); // relative error
// System.out.println("e = " + e);
BigDecimal dc = BigDecimal.ONE;
// polynomials with decimal coefficients
GenPolynomialRing<BigDecimal> dfac = new GenPolynomialRing<BigDecimal>(dc, f.ring);
GenPolynomial<BigDecimal> df = PolyUtil.<C>decimalFromRational(dfac, f);
GenPolynomial<C> fp = PolyUtil.<C>baseDeriviative(f);
GenPolynomial<BigDecimal> dfp = PolyUtil.<C>decimalFromRational(dfac, fp);
// Newton Raphson iteration: x_{n+1} = x_n - f(x_n)/f'(x_n)
int i = 0;
final int MITER = 50;
int dir = 0;
while (i++ < MITER) {
BigDecimal fx = PolyUtil.<BigDecimal>evaluateMain(dc, df, d); // f(d)
if (fx.isZERO()) {
return d;
}
BigDecimal fpx = PolyUtil.<BigDecimal>evaluateMain(dc, dfp, d); // f'(d)
if (fpx.isZERO()) {
throw new NoConvergenceException("zero deriviative should not happen");
}
BigDecimal x = fx.divide(fpx);
BigDecimal dx = d.subtract(x);
// System.out.println("dx = " + dx);
if (d.subtract(dx).abs().compareTo(e) <= 0) {
return dx;
}
while (dx.compareTo(left) < 0 || dx.compareTo(right) > 0) { // dx < left: dx - left < 0
// dx > right: dx - right > 0
// System.out.println("trying to leave interval");
if (i++ > MITER) { // dx > right: dx - right > 0
throw new NoConvergenceException("no convergence after " + i + " steps");
}
if (i > MITER / 2 && dir == 0) {
BigDecimal sd = new BigDecimal(iv.randomPoint().getRational());
d = sd;
x = sd.getZERO();
logger.info("trying new random starting point " + d);
i = 0;
dir = 1;
}
if (i > MITER / 2 && dir == 1) {
BigDecimal sd = new BigDecimal(iv.randomPoint().getRational());
d = sd;
x = sd.getZERO();
logger.info("trying new random starting point " + d);
// i = 0;
dir = 2; // end
}
x = x.multiply(q); // x * 1/4
dx = d.subtract(x);
// System.out.println(" x = " + x);
// System.out.println("dx = " + dx);
}
d = dx;
}
throw new NoConvergenceException("no convergence after " + i + " steps");
}
