A Tour of NTL: Examples: Polynomials

NTL provides extensive support for very fast polynomial arithmetic.
In fact, this was the main motivation for creating NTL in the first place,
because existing computer algebra systems and software
libraries had very slow polynomial arithmetic.
The class `ZZX` represents univariate polynomials
with integer coefficients.
The following program reads a polynomial,
factors it, and prints the factorization.

#include <NTL/ZZXFactoring.h> int main() { ZZX f; cin >> f; vec_pair_ZZX_long factors; ZZ c; factor(c, factors, f); cout << c << "\n"; cout << factors << "\n"; }

When this program is compiled an run on input

[2 10 14 6]which represents the polynomial

2 [[[1 3] 1] [[1 1] 2]]The first line of output is the content of the polynomial, which is 2 in this case as each coefficient of the input polynomial is divisible by 2. The second line is a vector of pairs, the first member of each pair is an irreducible factor of the input, and the second is the exponent to which is appears in the factorization. Thus, all of the above simply means that

2 + 10*X + 14*x^2 +6*X^3 = 2 * (1 + 3*X) * (1 + X)^2

Admittedly, I/O in NTL is not exactly user friendly, but then NTL has no pretensions about being an interactive computer algebra system: it is a library for programmers.

In this example, the type `vec_pair_long_ZZ`
is an NTL vector whose base type is `pair_long_ZZ`.
The type `pair_long_ZZ` is a type created by
another template-like macro mechanism.
In general, for types `S` and `T`,
one can create a type `pair_S_T` which is
a class with a field `a` of type `S`
and a field `b` of type `T`.
See `pair.txt` for more details.

Here is another example. The following program prints out the first 100 cyclotomic polynomials.

#include <NTL/ZZX.h> int main() { vec_ZZX phi(INIT_SIZE, 100); for (long i = 1; i <= 100; i++) { ZZX t; t = 1; for (long j = 1; j <= i-1; j++) if (i % j == 0) t *= phi(j); phi(i) = (ZZX(i, 1) - 1)/t; // ZZX(i, a) == X^i * a cout << phi(i) << "\n"; } }

To illustrate more of the NTL interface, let's look at alternative ways this routine could have been written.

First, instead of

vec_ZZX phi(INIT_SIZE, 100);one can write

vec_ZZX phi; phi.SetLength(100);

Second, instead of

t *= phi(j);one can write this as

mul(t, t, phi(j));or

t = t * phi(j);Also, one can write

Third, instead of

phi(i) = (ZZX(i, 1) - 1)/t;one can write

ZZX t1; SetCoeff(t1, i, 1); SetCoeff(t1, 0, -1); div(phi(i), t1, t);Alternatively, one could directly access the coefficient vector:

ZZX t1; t1.rep.SetLength(i+1); // all vector elements are initialized to zero t1.rep[i] = 1; t1.rep[0] = -1; t1.normalize(); // not necessary here, but good practice in general div(phi(i), t1, t);The coefficient vector of a polynomial is always an NTL vector over the ground ring: in this case

... f.rep[i] == 1 ...is equivalent to

... coeff(f, i) == 1 ...except that in the latter case, a read-only reference to zero is returned if the index

NTL provides a full compliment of operations for polynomials
over the integers, in both operator and procedural form.
All of the basic operations support a "promotion logic" similar
to that for `ZZ`, except that inputs of *both* types
`long` and `ZZ` are promoted to `ZZX`.
See `ZZX.txt` for details,
and see `ZZXFactoring.txt` for details
on the polynomial factoring routines.