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/*
* $Id: Examples.java 4057 2012-07-26 20:35:44Z kredel $
*/
package edu.jas.poly;
import java.util.ArrayList;
import java.util.List;
import edu.jas.arith.BigInteger;
import edu.jas.arith.BigRational;
import edu.jas.arith.ModInteger;
import edu.jas.arith.ModIntegerRing;
/**
* Examples for polynomials usage.
* @author Heinz Kredel.
*/
public class Examples {
/**
* main.
*/
public static void main(String[] args) {
//example0();
/*
example1();
example2();
example3();
example4();
example5();
*/
//example6();
example7();
example7();
//example8();
//example9();
//example10();
//example11();
//example12();
}
/**
* example0. for PPPJ 2006.
*/
public static void example0() {
BigInteger z = new BigInteger();
TermOrder to = new TermOrder();
String[] vars = new String[] { "x1", "x2", "x3" };
GenPolynomialRing ring;
ring = new GenPolynomialRing(z, 3, to, vars);
System.out.println("ring = " + ring);
GenPolynomial pol;
pol = ring.parse("3 x1^2 x3^4 + 7 x2^5 - 61");
System.out.println("pol = " + pol);
System.out.println("pol = " + pol.toString(ring.getVars()));
GenPolynomial one;
one = ring.parse("1");
System.out.println("one = " + one);
System.out.println("one = " + one.toString(ring.getVars()));
GenPolynomial p;
p = pol.subtract(pol);
System.out.println("p = " + p);
System.out.println("p = " + p.toString(ring.getVars()));
p = pol.multiply(pol);
System.out.println("p = " + p);
System.out.println("p = " + p.toString(ring.getVars()));
}
/**
* example1. random polynomial with rational coefficients. Q[x_1,...x_7]
*/
public static void example1() {
System.out.println("\n\n example 1");
BigRational cfac = new BigRational();
System.out.println("cfac = " + cfac);
GenPolynomialRing fac;
fac = new GenPolynomialRing(cfac, 7);
//System.out.println("fac = " + fac);
System.out.println("fac = " + fac);
GenPolynomial a = fac.random(10);
System.out.println("a = " + a);
}
/**
* example2. random polynomial with coefficients of rational polynomials.
* Q[x_1,...x_7][y_1,...,y_3]
*/
public static void example2() {
System.out.println("\n\n example 2");
BigRational cfac = new BigRational();
System.out.println("cfac = " + cfac);
GenPolynomialRing fac;
fac = new GenPolynomialRing(cfac, 7);
System.out.println("fac = " + fac);
GenPolynomialRing> gfac;
gfac = new GenPolynomialRing>(fac, 3);
System.out.println("gfac = " + gfac);
GenPolynomial> a = gfac.random(10);
System.out.println("a = " + a);
}
/**
* example3. random rational algebraic number. Q(alpha)
*/
public static void example3() {
System.out.println("\n\n example 3");
BigRational cfac = new BigRational();
System.out.println("cfac = " + cfac);
GenPolynomialRing mfac;
mfac = new GenPolynomialRing(cfac, 1);
System.out.println("mfac = " + mfac);
GenPolynomial modul = mfac.random(8).monic();
// assume !mo.isUnit()
System.out.println("modul = " + modul);
AlgebraicNumberRing fac;
fac = new AlgebraicNumberRing(modul);
System.out.println("fac = " + fac);
AlgebraicNumber a = fac.random(15);
System.out.println("a = " + a);
}
protected static long getPrime() {
long prime = 2; //2^60-93; // 2^30-35; //19; knuth (2,390)
for (int i = 1; i < 60; i++) {
prime *= 2;
}
prime -= 93;
//System.out.println("prime = " + prime);
return prime;
}
/**
* example4. random modular algebraic number. Z_p(alpha)
*/
public static void example4() {
System.out.println("\n\n example 4");
long prime = getPrime();
ModIntegerRing cfac = new ModIntegerRing(prime);
System.out.println("cfac = " + cfac);
GenPolynomialRing mfac;
mfac = new GenPolynomialRing(cfac, 1);
System.out.println("mfac = " + mfac);
GenPolynomial modul = mfac.random(8).monic();
// assume !modul.isUnit()
System.out.println("modul = " + modul);
AlgebraicNumberRing fac;
fac = new AlgebraicNumberRing(modul);
System.out.println("fac = " + fac);
AlgebraicNumber a = fac.random(12);
System.out.println("a = " + a);
}
/**
* example5. random solvable polynomial with rational coefficients.
* Q{x_1,...x_6, {x_2 * x_1 = x_1 x_2 +1, ...} }
*/
public static void example5() {
System.out.println("\n\n example 5");
BigRational cfac = new BigRational();
System.out.println("cfac = " + cfac);
GenSolvablePolynomialRing sfac;
sfac = new GenSolvablePolynomialRing(cfac, 6);
//System.out.println("sfac = " + sfac);
WeylRelations wl = new WeylRelations(sfac);
wl.generate();
System.out.println("sfac = " + sfac);
GenSolvablePolynomial a = sfac.random(5);
System.out.println("a = " + a);
System.out.println("a = " + a.toString(sfac.vars));
GenSolvablePolynomial b = a.multiply(a);
System.out.println("b = " + b);
System.out.println("b = " + b.toString(sfac.vars));
System.out.println("sfac = " + sfac);
}
/**
* example6. Fateman benchmark: p = (x+y+z)^20; q = p * (p+1) Z[z,y,x]
*/
public static void example6() {
System.out.println("\n\n example 6");
BigInteger cfac = new BigInteger();
System.out.println("cfac = " + cfac);
TermOrder to = new TermOrder(TermOrder.INVLEX);
System.out.println("to = " + to);
GenPolynomialRing fac;
fac = new GenPolynomialRing(cfac, 3, to);
System.out.println("fac = " + fac);
fac.setVars(new String[] { "z", "y", "x" });
System.out.println("fac = " + fac);
GenPolynomial x = fac.univariate(0);
GenPolynomial y = fac.univariate(1);
GenPolynomial z = fac.univariate(2);
System.out.println("x = " + x);
System.out.println("x = " + x.toString(fac.vars));
System.out.println("y = " + y);
System.out.println("y = " + y.toString(fac.vars));
System.out.println("z = " + z);
System.out.println("z = " + z.toString(fac.vars));
GenPolynomial p = x.sum(y).sum(z).sum(fac.getONE());
//BigInteger f = cfac.fromInteger(10000000001L);
// p = p.multiply( f );
System.out.println("p = " + p);
System.out.println("p = " + p.toString(fac.vars));
GenPolynomial q = p;
for (int i = 1; i < 20; i++) {
q = q.multiply(p);
}
//System.out.println("q = " + q.toString( fac.vars ) );
System.out.println("q = " + q.length());
GenPolynomial q1 = q.sum(fac.getONE());
GenPolynomial q2;
long t = System.currentTimeMillis();
q2 = q.multiply(q1);
t = System.currentTimeMillis() - t;
System.out.println("q2 = " + q2.length());
System.out.println("time = " + t + " ms");
}
/**
* example7. Fateman benchmark: p = (x+y+z)^20; q = p * (p+1) Q[z,y,x]
*/
public static void example7() {
System.out.println("\n\n example 7");
BigRational cfac = new BigRational();
System.out.println("cfac = " + cfac);
TermOrder to = new TermOrder(TermOrder.INVLEX);
System.out.println("to = " + to);
GenPolynomialRing fac;
fac = new GenPolynomialRing(cfac, 3, to);
System.out.println("fac = " + fac);
fac.setVars(new String[] { "z", "y", "x" });
System.out.println("fac = " + fac);
long mi = 1L;
//long mi = Integer.MAX_VALUE;
GenPolynomial x = fac.univariate(0, mi);
GenPolynomial y = fac.univariate(1, mi);
GenPolynomial z = fac.univariate(2, mi);
// System.out.println("x = " + x);
System.out.println("x = " + x.toString(fac.vars));
// System.out.println("y = " + y);
System.out.println("y = " + y.toString(fac.vars));
// System.out.println("z = " + z);
System.out.println("z = " + z.toString(fac.vars));
GenPolynomial p = x.sum(y).sum(z).sum(fac.getONE());
//BigRational f = cfac.fromInteger(10000000001L);
// f = f.multiply( f );
//p = p.multiply( f );
// System.out.println("p = " + p);
System.out.println("p = " + p.toString(fac.vars));
int mpow = 20;
System.out.println("mpow = " + mpow);
GenPolynomial q = p;
for (int i = 1; i < mpow; i++) {
q = q.multiply(p);
}
//System.out.println("q = " + q.toString( fac.vars ) );
System.out.println("len(q) = " + q.length());
System.out.println("deg(q) = " + q.degree());
GenPolynomial q1 = q.sum(fac.getONE());
GenPolynomial q2;
long t = System.currentTimeMillis();
q2 = q.multiply(q1);
t = System.currentTimeMillis() - t;
System.out.println("len(q2) = " + q2.length());
System.out.println("deg(q2) = " + q2.degree());
System.out.println("LeadEV(q2) = " + q2.leadingExpVector());
System.out.println("time = " + t + " ms");
}
/**
* example8. Chebyshev polynomials
*
* T(0) = 1 T(1) = x T(n) = 2x * T(n-1) - T(n-2)
*/
public static void example8() {
int m = 10;
BigInteger fac = new BigInteger();
String[] var = new String[] { "x" };
GenPolynomialRing ring = new GenPolynomialRing(fac, 1, var);
List> T = new ArrayList>(m);
GenPolynomial t, one, x, x2, x2b;
one = ring.getONE();
x = ring.univariate(0);
//x2 = ring.parse("2 x");
//x2a = x.multiply( fac.fromInteger(2) );
x2b = x.multiply(new BigInteger(2));
x2 = x2b;
T.add(one);
T.add(x);
for (int n = 2; n < m; n++) {
t = x2.multiply(T.get(n - 1)).subtract(T.get(n - 2));
T.add(t);
}
for (int n = 0 /*m-2*/; n < m; n++) {
System.out.println("T[" + n + "] = " + T.get(n)); //.toString(var) );
}
}
/**
* example9. Legendre polynomials
*
* P(0) = 1 P(1) = x P(n) = 1/n [ (2n-1) * x * P(n-1) - (n-1) * P(n-2) ]
*/
// P(n+1) = 1/(n+1) [ (2n+1) * x * P(n) - n * P(n-1) ]
public static void example9() {
int n = 10;
BigRational fac = new BigRational();
String[] var = new String[] { "x" };
GenPolynomialRing ring = new GenPolynomialRing(fac, 1, var);
List> P = new ArrayList>(n);
GenPolynomial t, one, x, xc;
BigRational n21, nn;
one = ring.getONE();
x = ring.univariate(0);
P.add(one);
P.add(x);
for (int i = 2; i < n; i++) {
n21 = new BigRational(2 * i - 1);
xc = x.multiply(n21);
t = xc.multiply(P.get(i - 1));
nn = new BigRational(i - 1);
xc = P.get(i - 2).multiply(nn);
t = t.subtract(xc);
nn = new BigRational(1, i);
t = t.multiply(nn);
P.add(t);
}
for (int i = 0; i < n; i++) {
System.out.println("P[" + i + "] = " + P.get(i).toString(var));
System.out.println();
}
}
/**
* example10. Hermite polynomials
*
* H(0) = 1 H(1) = 2 x H(n) = 2 * x * H(n-1) - 2 * (n-1) * H(n-2)
*/
// H(n+1) = 2 * x * H(n) - 2 * n * H(n-1)
public static void example10() {
int n = 100;
BigInteger fac = new BigInteger();
String[] var = new String[] { "x" };
GenPolynomialRing ring = new GenPolynomialRing(fac, 1, var);
List> H = new ArrayList>(n);
GenPolynomial t, one, x2, xc, x;
BigInteger n2, nn;
one = ring.getONE();
x = ring.univariate(0);
n2 = new BigInteger(2);
x2 = x.multiply(n2);
H.add(one);
H.add(x2);
for (int i = 2; i < n; i++) {
t = x2.multiply(H.get(i - 1));
nn = new BigInteger(2 * (i - 1));
xc = H.get(i - 2).multiply(nn);
t = t.subtract(xc);
H.add(t);
}
for (int i = n - 1; i < n; i++) {
System.out.println("H[" + i + "] = " + H.get(i).toString(var));
System.out.println();
}
}
/**
* example11. degree matrix;
*
*/
public static void example11() {
int n = 50;
BigRational fac = new BigRational();
GenPolynomialRing ring = new GenPolynomialRing(fac, n);
System.out.println("ring = " + ring + "\n");
GenPolynomial p = ring.random(5, 3, 6, 0.5f);
System.out.println("p = " + p + "\n");
List> dem = TermOrderOptimization. degreeMatrix(p);
System.out.println("dem = " + dem + "\n");
List> polys = new ArrayList>();
polys.add(p);
for (int i = 0; i < 5; i++) {
polys.add(ring.random(5, 3, 6, 0.1f));
}
System.out.println("polys = " + polys + "\n");
dem = TermOrderOptimization. degreeMatrix(polys);
System.out.println("dem = " + dem + "\n");
List perm;
perm = TermOrderOptimization.optimalPermutation(dem);
System.out.println("perm = " + perm + "\n");
List> pdem;
pdem = TermOrderOptimization.> listPermutation(perm, dem);
System.out.println("pdem = " + pdem + "\n");
GenPolynomialRing pring;
pring = TermOrderOptimization. permutation(perm, ring);
System.out.println("ring = " + ring);
System.out.println("pring = " + pring + "\n");
List> ppolys;
ppolys = TermOrderOptimization. permutation(perm, pring, polys);
System.out.println("ppolys = " + ppolys + "\n");
dem = TermOrderOptimization. degreeMatrix(ppolys);
//System.out.println("pdem = " + dem + "\n");
perm = TermOrderOptimization.optimalPermutation(dem);
//System.out.println("pperm = " + perm + "\n");
int i = 0;
for (Integer j : perm) {
if (i != (int) j) {
System.out.println("error = " + i + " != " + j + "\n");
}
i++;
}
OptimizedPolynomialList op;
op = TermOrderOptimization. optimizeTermOrder(ring, polys);
System.out.println("op:\n" + op);
if (!op.equals(new PolynomialList(pring, ppolys))) {
System.out.println("error = " + "\n" + op);
}
}
/**
* example12. type games.
*/
public static void example12() {
System.out.println("\n\n example 12");
BigRational t1 = new BigRational();
System.out.println("t1 = " + t1);
BigInteger t2 = new BigInteger();
System.out.println("t2 = " + t2);
System.out.println("t1.isAssignableFrom(t2) = " + t1.getClass().isAssignableFrom(t2.getClass()));
System.out.println("t2.isAssignableFrom(t1) = " + t2.getClass().isAssignableFrom(t1.getClass()));
GenPolynomialRing t3 = new GenPolynomialRing(t2, 3);
System.out.println("t3 = " + t3);
GenSolvablePolynomialRing t4 = new GenSolvablePolynomialRing(t2, 3);
System.out.println("t4 = " + t4);
System.out.println("t3.isAssignableFrom(t4) = " + t3.getClass().isAssignableFrom(t4.getClass()));
System.out.println("t4.isAssignableFrom(t3) = " + t4.getClass().isAssignableFrom(t3.getClass()));
GenPolynomialRing t5 = new GenPolynomialRing(t1, 3);
System.out.println("t5 = " + t5);
System.out.println("t3.isAssignableFrom(t5) = " + t3.getClass().isAssignableFrom(t5.getClass()));
System.out.println("t5.isAssignableFrom(t3) = " + t5.getClass().isAssignableFrom(t3.getClass()));
}
}
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