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5.7.3 Degree and coefficients
The degree and low degree of a polynomial can be obtained using the two methods
int ex::degree(const ex & s); int ex::ldegree(const ex & s); |
which also work reliably on non-expanded input polynomials (they even work on rational functions, returning the asymptotic degree). By definition, the degree of zero is zero. To extract a coefficient with a certain power from an expanded polynomial you use
ex ex::coeff(const ex & s, int n); |
You can also obtain the leading and trailing coefficients with the methods
ex ex::lcoeff(const ex & s); ex ex::tcoeff(const ex & s); |
which are equivalent to coeff(s, degree(s)) and coeff(s, ldegree(s)),
respectively.
An application is illustrated in the next example, where a multivariate polynomial is analyzed:
{
symbol x("x"), y("y");
ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
- pow(x+y,2) + 2*pow(y+2,2) - 8;
ex Poly = PolyInp.expand();
for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) {
cout << "The x^" << i << "-coefficient is "
<< Poly.coeff(x,i) << endl;
}
cout << "As polynomial in y: "
<< Poly.collect(y) << endl;
}
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When run, it returns an output in the following fashion:
The x^0-coefficient is y^2+11*y The x^1-coefficient is 5*y^2-2*y The x^2-coefficient is -1 The x^3-coefficient is 4*y As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y |
As always, the exact output may vary between different versions of GiNaC or even from run to run since the internal canonical ordering is not within the user's sphere of influence.
degree(), ldegree(), coeff(), lcoeff(),
tcoeff() and collect() can also be used to a certain degree
with non-polynomial expressions as they not only work with symbols but with
constants, functions and indexed objects as well:
{
symbol a("a"), b("b"), c("c"), x("x");
idx i(symbol("i"), 3);
ex e = pow(sin(x) - cos(x), 4);
cout << e.degree(cos(x)) << endl;
// -> 4
cout << e.expand().coeff(sin(x), 3) << endl;
// -> -4*cos(x)
e = indexed(a+b, i) * indexed(b+c, i);
e = e.expand(expand_options::expand_indexed);
cout << e.collect(indexed(b, i)) << endl;
// -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
}
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