`bit::polynomial` — Monic

We have a method to query if this polynomial is monic and another that makes it monic if that is possible.

1constexpr bool        monic() const;
2constexpr polynomial& make_monic();

1: Returns true if this is a monic polynomial.
2: Attempts to shrink the polynomial to be monic. This does not change the memory footprint of the polynomial.

A monic polynomial has no trailing zero coefficients.

A monic polynomial of degree \(n\) will have exactly \(n+1\) coefficients \(p_0, p_1, \ldots, p_{n-1}\) where \[ p(x) = p_0 + p_q x + \cdots + p_{n-1} x^{n-1}, \] and \(p_{n-1} = 1\).

After a call to make_monic() any nonzero polynomial will have size() = degree() + 1.

The zero polynomial has no degree and is not monic. Polynomial methods usually need to treat the zero-polynomial as a special, generally trivial, edge case. The make_monic() method does nothing to a zero polynomial.

Example

#include <bit/bit.h>
int main()
{
    bit::polynomial<> p;
    std::cout << std::format("p(x) = {} has coefficients = {:p}.\n", p, p.coefficients());
    std::cout << std::format("Size: {}, monic: {}.\n\n", p.size(), p.monic());

    p.resize(7);
    std::cout << std::format("p(x) = {} has coefficients = {:p}.\n", p, p.coefficients());
    std::cout << std::format("Size: {}, monic: {}.\n\n", p.size(), p.monic());

    p[1] = p[3] = 1;
    std::cout << std::format("p(x) = {} has coefficients = {:p}.\n", p, p.coefficients());
    std::cout << std::format("Size: {}, monic: {}.\n\n", p.size(), p.monic());

    p.make_monic();
    std::cout << std::format("p(x) = {} has coefficients = {:p}.\n", p, p.coefficients());
    std::cout << std::format("Size: {}, monic: {}.\n", p.size(), p.monic());
}

Output

p(x) = 0 has coefficients = [].
Size: 0, monic: false.

p(x) = 0 has coefficients = [0 0 0 0 0 0 0].
Size: 7, monic: false.

p(x) = x^1 + x^3 has coefficients = [0 1 0 1 0 0 0].
Size: 7, monic: false.

p(x) = x^1 + x^3 has coefficients = [0 1 0 1].
Size: 4, monic: true.

See Also