/**
* Real root bound. With f(M) * f(-M) != 0.
*
* @param f univariate polynomial.
* @return M such that -M < root(f) < M.
*/
public C realRootBound(GenPolynomial<C> f) {
if (f == null) {
return null;
}
RingFactory<C> cfac = f.ring.coFac;
C M = cfac.getONE();
if (f.isZERO() || f.isConstant()) {
return M;
}
C a = f.leadingBaseCoefficient().abs();
for (C c : f.getMap().values()) {
C d = c.abs().divide(a);
if (M.compareTo(d) < 0) {
M = d;
}
}
// works also without this case, only for optimization
// to use rational number interval end points
// can fail if real root is in interval [r,r+1]
// for too low precision or too big r, since r is approximation
if ((Object) M instanceof RealAlgebraicNumber) {
RealAlgebraicNumber Mr = (RealAlgebraicNumber) M;
BigRational r = Mr.magnitude();
M = cfac.fromInteger(r.numerator()).divide(cfac.fromInteger(r.denominator()));
}
M = M.sum(f.ring.coFac.getONE());
// System.out.println("M = " + M);
return M;
}
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()));
}
/**
* Magnitude bound.
*
* @param iv interval.
* @param f univariate polynomial.
* @return B such that |f(c)| < B for c in iv.
*/
public C magnitudeBound(Interval<C> iv, GenPolynomial<C> f) {
if (f == null) {
return null;
}
if (f.isZERO()) {
return f.ring.coFac.getONE();
}
if (f.isConstant()) {
return f.leadingBaseCoefficient().abs();
}
GenPolynomial<C> fa =
f.map(
new UnaryFunctor<C, C>() {
public C eval(C a) {
return a.abs();
}
});
// System.out.println("fa = " + fa);
C M = iv.left.abs();
if (M.compareTo(iv.right.abs()) < 0) {
M = iv.right.abs();
}
// System.out.println("M = " + M);
RingFactory<C> cfac = f.ring.coFac;
C B = PolyUtil.<C>evaluateMain(cfac, fa, M);
// works also without this case, only for optimization
// to use rational number interval end points
// can fail if real root is in interval [r,r+1]
// for too low precision or too big r, since r is approximation
if ((Object) B instanceof RealAlgebraicNumber) {
RealAlgebraicNumber Br = (RealAlgebraicNumber) B;
BigRational r = Br.magnitude();
B = cfac.fromInteger(r.numerator()).divide(cfac.fromInteger(r.denominator()));
}
// System.out.println("B = " + B);
return B;
}
/**
* ComplexAlgebraicNumber magnitude.
*
* @return |this| as complex rational number.
*/
public Complex<BigRational> magnitude() {
try {
Rectangle<C> v =
ring.engine.invariantMagnitudeRectangle(
ring.root, ring.algebraic.modul, number.val, ring.getEps());
ring.setRoot(v);
// System.out.println("new v = " + v);
Complex<C> ev =
ring.engine.complexRectangleMagnitude(
v, ring.algebraic.modul, number.val); // , ring.eps);
// C re = ev.getRe();
// if ( (Object) re instanceof Rational) { // true by type parameter
BigRational er = ev.getRe().getRational();
BigRational ei = ev.getIm().getRational();
ComplexRing<BigRational> cr = new ComplexRing<BigRational>(er.factory());
return new Complex<BigRational>(cr, er, ei);
// } else {
// throw new RuntimeException("Rational expected, but was " + ev.getClass());
// }
} catch (InvalidBoundaryException e) { // should not happen
e.printStackTrace();
throw new RuntimeException(e);
}
}
/** Test rational coefficient reduction. */
public void testRatReduction() {
do {
a = fac.random(kl, ll, el);
} while (a.isZERO());
do {
b = fac.random(kl, ll, el);
} while (b.isZERO());
// System.out.println("a = " + a);
// System.out.println("b = " + b);
L = new ArrayList<GenWordPolynomial<BigRational>>();
L.add(a);
e = red.normalform(L, a);
// System.out.println("e = " + e);
assertTrue("isZERO( e )", e.isZERO());
L.add(b);
e = red.normalform(L, a);
// System.out.println("e = " + e);
assertTrue("isZERO( e ) some times", e.isZERO());
L = new ArrayList<GenWordPolynomial<BigRational>>();
L.add(a);
assertTrue("isTopRed( a )", red.isTopReducible(L, a));
assertTrue("isRed( a )", red.isReducible(L, a));
L.add(b);
assertTrue("isTopRed( b )", red.isTopReducible(L, b));
assertTrue("isRed( b )", red.isReducible(L, b));
c = fac.random(kl, ll, el);
// System.out.println("c = " + c);
e = red.normalform(L, c);
// System.out.println("e = " + e);
assertTrue("isNF( e )", red.isNormalform(L, e));
Word u = new Word(wfac, "f");
Word v = new Word(wfac, "abc");
c = a.multiply(cfac.getONE(), u, v);
// System.out.println("c = " + c);
e = red.normalform(L, c);
// System.out.println("e = " + e);
assertTrue("isNF( e )", red.isNormalform(L, e));
assertTrue("e == 0", e.isZERO());
}
public static void example7() {
final BigRational fac = new BigRational();
final UnivPowerSeriesRing<BigRational> pfac =
new UnivPowerSeriesRing<BigRational>(fac, 11, "y");
UnivPowerSeries<BigRational> exp =
pfac.fixPoint(
new UnivPowerSeriesMap<BigRational>() {
public UnivPowerSeries<BigRational> map(UnivPowerSeries<BigRational> e) {
return e.integrate(fac.getONE());
}
});
System.out.println("exp = " + exp);
UnivPowerSeries<BigRational> tan =
pfac.fixPoint(
new UnivPowerSeriesMap<BigRational>() {
public UnivPowerSeries<BigRational> map(UnivPowerSeries<BigRational> t) {
return t.multiply(t).sum(pfac.getONE()).integrate(fac.getZERO());
}
});
System.out.println("tan = " + tan);
UnivPowerSeries<BigRational> sin =
new UnivPowerSeries<BigRational>(
pfac,
new Coefficients<BigRational>() {
@Override
public BigRational generate(int i) {
BigRational c;
if (i == 0) {
c = fac.getZERO();
} else if (i == 1) {
c = fac.getONE();
} else {
c = get(i - 2).negate();
c = c.divide(fac.fromInteger(i)).divide(fac.fromInteger(i - 1));
}
return c;
}
});
System.out.println("sin = " + sin);
UnivPowerSeries<BigRational> sin1 =
pfac.fixPoint(
new UnivPowerSeriesMap<BigRational>() {
public UnivPowerSeries<BigRational> map(UnivPowerSeries<BigRational> e) {
return e.negate().integrate(fac.getONE()).integrate(fac.getZERO());
}
});
System.out.println("sin1 = " + sin1);
UnivPowerSeries<BigRational> cos =
new UnivPowerSeries<BigRational>(
pfac,
new Coefficients<BigRational>() {
@Override
public BigRational generate(int i) {
BigRational c;
if (i == 0) {
c = fac.getONE();
} else if (i == 1) {
c = fac.getZERO();
} else {
c = get(i - 2).negate();
c = c.divide(fac.fromInteger(i)).divide(fac.fromInteger(i - 1));
}
return c;
}
});
System.out.println("cos = " + cos);
UnivPowerSeries<BigRational> cos1 =
pfac.fixPoint(
new UnivPowerSeriesMap<BigRational>() {
public UnivPowerSeries<BigRational> map(UnivPowerSeries<BigRational> e) {
return e.negate().integrate(fac.getZERO()).integrate(fac.getONE());
}
});
System.out.println("cos1 = " + cos1);
UnivPowerSeries<BigRational> cos2 = pfac.solveODE(sin1.negate(), fac.getONE());
System.out.println("cos2 = " + cos2);
UnivPowerSeries<BigRational> sin2 = pfac.solveODE(cos1, fac.getZERO());
System.out.println("sin2 = " + sin2);
UnivPowerSeries<BigRational> sinh =
new UnivPowerSeries<BigRational>(
pfac,
new Coefficients<BigRational>() {
@Override
public BigRational generate(int i) {
BigRational c;
if (i == 0) {
c = fac.getZERO();
} else if (i == 1) {
c = fac.getONE();
} else {
c = get(i - 2);
c = c.divide(fac.fromInteger(i)).divide(fac.fromInteger(i - 1));
}
return c;
}
});
System.out.println("sinh = " + sinh);
UnivPowerSeries<BigRational> cosh =
new UnivPowerSeries<BigRational>(
pfac,
new Coefficients<BigRational>() {
@Override
public BigRational generate(int i) {
BigRational c;
if (i == 0) {
c = fac.getONE();
} else if (i == 1) {
c = fac.getZERO();
} else {
c = get(i - 2);
c = c.divide(fac.fromInteger(i)).divide(fac.fromInteger(i - 1));
}
return c;
}
} // , null
);
System.out.println("cosh = " + cosh);
UnivPowerSeries<BigRational> sinhcosh = sinh.sum(cosh);
System.out.println("sinh+cosh = " + sinhcosh);
System.out.println("sinh+cosh == exp: " + sinhcosh.equals(exp));
}
/** Test scalar multiplication. */
public void testMultiplication() {
BigRational cfac = new BigRational(1);
GenVectorModul<BigRational> mfac = new GenVectorModul<BigRational>(cfac, ll);
BigRational r, s, t;
GenVector<BigRational> a, b, c, d, e;
r = cfac.random(kl);
// System.out.println("r = " + r);
s = r.inverse();
// System.out.println("s = " + s);
a = mfac.random(kl, q);
// System.out.println("a = " + a);
c = a.scalarMultiply(r);
d = c.scalarMultiply(s);
// System.out.println("c = " + c);
// System.out.println("d = " + d);
assertEquals("a*b*(1/b) = a", a, d);
b = mfac.random(kl, q);
// System.out.println("b = " + b);
t = cfac.getONE();
// System.out.println("t = " + t);
c = a.linearCombination(b, t);
d = b.linearCombination(a, t);
// System.out.println("c = " + c);
// System.out.println("d = " + d);
assertEquals("a+1*b = b+1*a", c, d);
c = a.linearCombination(b, t);
d = a.sum(b);
// System.out.println("c = " + c);
// System.out.println("d = " + d);
assertEquals("a+1*b = b+1*a", c, d);
s = t.negate();
// System.out.println("s = " + s);
c = a.linearCombination(b, t);
d = c.linearCombination(b, s);
// System.out.println("c = " + c);
// System.out.println("d = " + d);
assertEquals("a+1*b+(-1)*b = a", a, d);
c = a.linearCombination(t, b, t);
d = c.linearCombination(t, b, s);
// System.out.println("c = " + c);
// System.out.println("d = " + d);
assertEquals("a*1+b*1+b*(-1) = a", a, d);
t = cfac.getZERO();
// System.out.println("t = " + t);
c = a.linearCombination(b, t);
// System.out.println("c = " + c);
assertEquals("a+0*b = a", a, c);
d = a.linearCombination(t, b, t);
// System.out.println("d = " + d);
assertEquals("0*a+0*b = 0", mfac.getZERO(), d);
r = a.scalarProduct(b);
s = b.scalarProduct(a);
// System.out.println("r = " + r);
// System.out.println("s = " + s);
assertEquals("a.b = b.a", r, s);
}
/** Test rational coefficient reduction with recording. */
public void testRatReductionRecording() {
List<GenWordPolynomial<BigRational>> lrow, rrow = null;
do {
a = fac.random(kl, ll, el);
} while (a.isZERO());
do {
b = fac.random(kl, ll, el);
} while (b.isZERO());
c = fac.random(kl, ll, el);
d = fac.random(kl, ll, el);
// System.out.println("a = " + a);
// System.out.println("b = " + b);
// System.out.println("c = " + c);
// System.out.println("d = " + d);
L = new ArrayList<GenWordPolynomial<BigRational>>();
L.add(a);
lrow = new ArrayList<GenWordPolynomial<BigRational>>(L.size());
rrow = new ArrayList<GenWordPolynomial<BigRational>>(L.size());
e = fac.getZERO();
for (int m = 0; m < L.size(); m++) {
lrow.add(e);
rrow.add(e);
}
e = red.normalform(lrow, rrow, L, a);
// System.out.println("e = " + e);
// System.out.println("lrow = " + lrow);
// System.out.println("rrow = " + rrow);
assertTrue("isZERO( e )", e.isZERO());
assertTrue("is Reduction ", red.isReductionNF(lrow, rrow, L, a, e));
L.add(b);
lrow = new ArrayList<GenWordPolynomial<BigRational>>(L.size());
rrow = new ArrayList<GenWordPolynomial<BigRational>>(L.size());
e = fac.getZERO();
for (int m = 0; m < L.size(); m++) {
lrow.add(e);
rrow.add(e);
}
e = red.normalform(lrow, rrow, L, b);
// System.out.println("e = " + e);
// System.out.println("lrow = " + lrow);
// System.out.println("rrow = " + rrow);
assertTrue("is Reduction ", red.isReductionNF(lrow, rrow, L, b, e));
L.add(c);
lrow = new ArrayList<GenWordPolynomial<BigRational>>(L.size());
rrow = new ArrayList<GenWordPolynomial<BigRational>>(L.size());
e = fac.getZERO();
for (int m = 0; m < L.size(); m++) {
lrow.add(e);
rrow.add(e);
}
e = red.normalform(lrow, rrow, L, c);
// System.out.println("e = " + e);
// System.out.println("lrow = " + lrow);
// System.out.println("rrow = " + rrow);
assertTrue("is Reduction ", red.isReductionNF(lrow, rrow, L, c, e));
L.add(d);
lrow = new ArrayList<GenWordPolynomial<BigRational>>(L.size());
rrow = new ArrayList<GenWordPolynomial<BigRational>>(L.size());
e = fac.getZERO();
for (int m = 0; m < L.size(); m++) {
lrow.add(e);
rrow.add(e);
}
Word u = new Word(wfac, "f");
Word v = new Word(wfac, "abc");
d = a.multiply(cfac.random(3), u, v);
// System.out.println("d = " + d);
e = red.normalform(lrow, rrow, L, d);
// System.out.println("e = " + e);
// System.out.println("lrow = " + lrow);
// System.out.println("rrow = " + rrow);
assertTrue("is Reduction ", red.isReductionNF(lrow, rrow, L, d, e));
}