public Algebraic cc() throws JasymcaException {
Polynomial xn = new Polynomial(var.cc());
Algebraic r = Zahl.ZERO;
for (int i = coef.length - 1; i > 0; i--) r = r.add(coef[i].cc()).mult(xn);
if (coef.length > 0) r = r.add(coef[0].cc());
return r;
}
// Divide through main coefficient; works only for numbers
public Polynomial monic() throws JasymcaException {
Algebraic cm = coef[coef.length - 1];
if (cm.equals(Zahl.ONE)) return this;
if (cm.equals(Zahl.ZERO) || !(cm instanceof Zahl))
throw new JasymcaException("Ill conditioned polynomial: main coefficient Zero or not number");
return (Polynomial) div(cm);
}
// Evaluate Polynomial for var = x
public Algebraic value(Variable var, Algebraic x) throws JasymcaException {
Algebraic r = Zahl.ZERO;
Algebraic v = this.var.value(var, x);
for (int i = coef.length - 1; i > 0; i--) { // Horner
r = r.add(coef[i].value(var, x)).mult(v);
}
if (coef.length > 0) r = r.add(coef[0].value(var, x));
return r;
}
// Map f to coefficients and arg
public Algebraic map(LambdaAlgebraic f) throws JasymcaException {
Algebraic x =
var instanceof SimpleVariable
? new Polynomial(var)
: FunctionVariable.create(
((FunctionVariable) var).fname, f.f_exakt(((FunctionVariable) var).arg));
Algebraic r = Zahl.ZERO;
for (int i = coef.length - 1; i > 0; i--) {
r = r.add(f.f_exakt(coef[i])).mult(x);
}
if (coef.length > 0) r = r.add(f.f_exakt(coef[0]));
return r;
}
public Algebraic add(Algebraic p) throws JasymcaException {
if (p instanceof Rational) return p.add(this);
if (p instanceof Polynomial) {
if (var.equals(((Polynomial) p).var)) {
int len = Math.max(coef.length, ((Polynomial) p).coef.length);
Algebraic[] csum = new Algebraic[len];
for (int i = 0; i < len; i++) csum[i] = coefficient(i).add(((Polynomial) p).coefficient(i));
return (new Polynomial(var, csum)).reduce();
} else if (var.smaller(((Polynomial) p).var)) {
return p.add(this);
}
}
Algebraic[] csum = clone(coef);
csum[0] = coef[0].add(p);
return (new Polynomial(var, csum)).reduce();
}
// square -free-decomposition of p=p1*p2^2*p3^3*...returns [p1,p2,p3..]
public Algebraic[] square_free_dec(Variable var) throws JasymcaException {
if (!dependsmult(var)) return null;
Algebraic dp = deriv(var);
Algebraic gcd_pdp = poly_gcd(dp);
Algebraic q = polydiv(gcd_pdp);
Algebraic p1 = q.polydiv(q.poly_gcd(gcd_pdp));
if (gcd_pdp instanceof Polynomial && ((Polynomial) gcd_pdp).dependsmult(var)) {
Algebraic sq[] = ((Polynomial) gcd_pdp).square_free_dec(var);
Algebraic result[] = new Algebraic[sq.length + 1];
result[0] = p1;
for (int i = 0; i < sq.length; i++) result[i + 1] = sq[i];
return result;
} else {
Algebraic result[] = {p1};
return result;
}
}
public Algebraic mult(Algebraic p) throws JasymcaException {
if (p instanceof Rational) return p.mult(this);
if (p instanceof Polynomial) {
if (var.equals(((Polynomial) p).var)) {
int len = coef.length + ((Polynomial) p).coef.length - 1;
Algebraic[] cprod = new Algebraic[len];
for (int i = 0; i < len; i++) cprod[i] = Zahl.ZERO;
for (int i = 0; i < coef.length; i++)
for (int k = 0; k < ((Polynomial) p).coef.length; k++)
cprod[i + k] = cprod[i + k].add(coef[i].mult(((Polynomial) p).coef[k]));
return new Polynomial(var, cprod).reduce();
} else if (var.smaller(((Polynomial) p).var)) {
return p.mult(this);
}
}
Algebraic[] cprod = new Algebraic[coef.length];
for (int i = 0; i < coef.length; i++) cprod[i] = coef[i].mult(p);
return new Polynomial(var, cprod).reduce();
}
public Algebraic deriv(Variable var) throws JasymcaException {
Algebraic r1 = Zahl.ZERO, r2 = Zahl.ZERO;
Polynomial x = new Polynomial(this.var);
for (int i = coef.length - 1; i > 1; i--) { // Horner
r1 = r1.add(coef[i].mult(new Unexakt(i))).mult(x); // diff Variable
}
if (coef.length > 1) r1 = r1.add(coef[1]);
for (int i = coef.length - 1; i > 0; i--) { // Horner
r2 = r2.add(coef[i].deriv(var)).mult(x); // diff coef
}
if (coef.length > 0) r2 = r2.add(coef[0].deriv(var));
return r1.mult(this.var.deriv(var)).add(r2).reduce();
}
// return a vektor of all different solutions
public Vektor solvepoly() throws JasymcaException {
Vector s = new Vector();
switch (degree()) {
case 0:
break;
case 1:
s.addElement(Zahl.MINUS.mult(coef[0].div(coef[1])));
break;
case 2:
Algebraic p = coef[1].div(coef[2]);
Algebraic q = coef[0].div(coef[2]);
p = Zahl.MINUS.mult(p).div(Zahl.TWO);
q = p.mult(p).sub(q);
if (q.equals(Zahl.ZERO)) {
s.addElement(p);
break;
}
q = FunctionVariable.create("sqrt", q);
s.addElement(p.add(q));
s.addElement(p.sub(q));
break;
/* case 3:
Algebraic a = r.coef[2];
Algebraic b = r.coef[1];
Algebraic c = r.coef[0];
Zahl drei = new Exakt(3.0);
// p=(3b-a^2)/3
p = drei.mult(b).sub(a.mult(a)).div(drei);
// q = c + 2a^3/27 -ab/3
q = c.add(Zahl.TWO.mult(a.pow_n(3)).div(new Exakt(27.))).
sub(a.mult(b).div(drei));
// D^2 = (p/3)^3 + (q/2)^2
Algebraic D = new Polynomial(new FunctionVariable("sqrt",
p.div(drei).pow_n(3).add(q.div(Zahl.TWO).pow_n(2))));
Algebraic u = new Polynomial(new FunctionVariable("csqrt",
Zahl.MINUS.mult(q).div(Zahl.TWO).add(D)));
Algebraic v = new Polynomial(new FunctionVariable("csqrt",
Zahl.MINUS.mult(q).div(Zahl.TWO).sub(D)));
Algebraic r0 = Zahl.MINUS.mult(a).div(drei).add(u).add(v);
Algebraic r1 = Zahl.MINUS.mult(a).div(drei).sub(u.add(v).div(Zahl.TWO));
Algebraic w = new Unexakt(Math.sqrt(3.)).mult(u.sub(v)).
div(Zahl.TWO).mult(Zahl.IONE);
Algebraic r2 = r1.sub(w);
r1 = r1.add(w);
s.addElement(r0);
s.addElement(r1);
s.addElement(r2);
break;*/
default:
// Maybe biquadratic/bicubic etc
// Calculate gcd of exponents for nonzero coefficients
int gcd = -1;
for (int i = 1; i < coef.length; i++) {
if (!coef[i].equals(Zahl.ZERO)) {
if (gcd < 0) gcd = i;
else gcd = gcd(i, gcd);
}
}
int deg = degree() / gcd;
if (deg < 3) { // Solveable
Algebraic cn[] = new Algebraic[deg + 1];
for (int i = 0; i < cn.length; i++) cn[i] = coef[i * gcd];
Polynomial pr = new Polynomial(var, cn);
Vektor sn = pr.solvepoly();
if (gcd == 2) { // sol = +/-sqrt(sn)
cn = new Algebraic[sn.coord.length * 2];
for (int i = 0; i < sn.coord.length; i++) {
cn[2 * i] = FunctionVariable.create("sqrt", sn.coord[i]);
cn[2 * i + 1] = cn[2 * i].mult(Zahl.MINUS);
}
} else { // sol = sn^(1/gcd);
cn = new Algebraic[sn.coord.length];
Zahl wx = new Unexakt(1. / gcd);
for (int i = 0; i < sn.coord.length; i++) {
Algebraic exp = FunctionVariable.create("log", sn.coord[i]);
cn[i] = FunctionVariable.create("exp", exp.mult(wx));
}
}
return new Vektor(cn);
}
throw new JasymcaException("Can't solve expression " + this);
}
return Vektor.create(s);
}
public Algebraic integrate(Variable var) throws JasymcaException {
Algebraic in = Zahl.ZERO;
for (int i = 1; i < coef.length; i++) {
if (!coef[i].depends(var))
if (var.equals(this.var))
// c*x^n -->1/(n+1)*x^(n+1)
in = in.add(coef[i].mult(new Polynomial(var).pow_n(i + 1).div(new Unexakt(i + 1))));
else if (this.var instanceof FunctionVariable
&& ((FunctionVariable) this.var).arg.depends(var))
// f(x)
if (i == 1) in = in.add(((FunctionVariable) this.var).integrate(var).mult(coef[1]));
// (f(x))^2, (f(x))^3 etc
// give up here but try again after exponential normalization
else throw new JasymcaException("Integral not supported.");
else
// Constant: c --> c*x
in = in.add(coef[i].mult(new Polynomial(var).mult(new Polynomial(this.var).pow_n(i))));
else if (var.equals(this.var))
// c(x)*x^n , should not happen if this is canonical
throw new JasymcaException("Integral not supported.");
else if (this.var instanceof FunctionVariable
&& ((FunctionVariable) this.var).arg.depends(var)) {
if (i == 1 && coef[i] instanceof Polynomial && ((Polynomial) coef[i]).var.equals(var)) {
// poly(x)*f(x)
// First attempt: try to isolate inner derivative
// poly(x)*f(w(x)) --> check poly(x)/w' == q : const?
// yes --> Int f dw * q
p("Trying to isolate inner derivative " + this);
try {
FunctionVariable f = (FunctionVariable) this.var;
Algebraic w = f.arg; // Innere Funktion
Algebraic q = coef[i].div(w.deriv(var));
if (q.deriv(var).equals(Zahl.ZERO)) { // q - constant
SimpleVariable v = new SimpleVariable("v");
Algebraic p = FunctionVariable.create(f.fname, new Polynomial(v));
Algebraic r = p.integrate(v).value(v, w).mult(q);
in = in.add(r);
continue;
}
} catch (JasymcaException je) {
// Didn't work, try more methods
}
p("Failed.");
// Some partial integrations follow. To
// avoid endless loops, we flag this section
// Coefficients of coef[i] must not depend on var
for (int k = 0; k < ((Polynomial) coef[i]).coef.length; k++)
if (((Polynomial) coef[i]).coef[k].depends(var))
throw new JasymcaException("Function not supported by this method");
if (loopPartial) {
loopPartial = false;
p("Partial Integration Loop detected.");
throw new JasymcaException("Partial Integration Loop: " + this);
}
// First attempt: x^n*f(x) , n-times diff!
// works for exp,sin,cos
p("Trying partial integration: x^n*f(x) , n-times diff " + this);
try {
loopPartial = true;
Algebraic p = coef[i];
Algebraic f = ((FunctionVariable) this.var).integrate(var);
Algebraic r = f.mult(p);
while (!(p = p.deriv(var)).equals(Zahl.ZERO)) {
f = f.integrate(var).mult(Zahl.MINUS);
r = r.add(f.mult(p));
}
loopPartial = false;
in = in.add(r);
continue;
} catch (JasymcaException je) {
loopPartial = false;
}
p("Failed.");
// Second attempt: x^n*f(x) , 1-times int!
// works for log, atan
p("Trying partial integration: x^n*f(x) , 1-times int " + this);
try {
loopPartial = true;
Algebraic p = coef[i].integrate(var);
Algebraic f = new Polynomial((FunctionVariable) this.var);
Algebraic r = p.mult(f).sub(p.mult(f.deriv(var)).integrate(var));
loopPartial = false;
in = in.add(r);
continue;
} catch (JasymcaException je3) {
loopPartial = false;
}
p("Failed");
// Add more attempts....
throw new JasymcaException("Function not supported by this method");
} else throw new JasymcaException("Integral not supported.");
} else // mainvar independend of var, treat as constant and integrate coef
in = in.add(coef[i].integrate(var).mult(new Polynomial(this.var).pow_n(i)));
}
if (coef.length > 0) in = in.add(coef[0].integrate(var));
return in;
}
