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<html><head><title>A Tour of NTL: Examples: Polynomials </title></head><body bgcolor="#fff9e6"><center><a href="tour-ex2.html"><img src="arrow1.gif" alt="[Previous]" align=bottom></a> <a href="tour-examples.html"><img src="arrow2.gif" alt="[Up]" align=bottom></a> <a href="tour-ex4.html"> <img src="arrow3.gif" alt="[Next]" align=bottom></a></center><h1> <p align=center>A Tour of NTL: Examples: Polynomials</p></h1><p> <hr> <p>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 softwarelibraries had very slow polynomial arithmetic.The class <tt>ZZX</tt> represents univariate polynomialswith integer coefficients.The following program reads a polynomial,factors it, and prints the factorization.<p><pre>#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";}</pre><p>When this program is compiled an run on input<pre> [2 10 14 6]</pre>which represents the polynomial <tt>2 + 10*X + 14*x^2 +6*X^3</tt>,the output is<pre> 2 [[[1 3] 1] [[1 1] 2]]</pre>The first line of output is the content of the polynomial, whichis 2 in this case as each coefficient of the input polynomialis 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<pre>2 + 10*X + 14*x^2 +6*X^3 = 2 * (1 + 3*X) * (1 + X)^2 </pre><p>Admittedly, I/O in NTL is not exactly user friendly,but then NTL has no pretensions about being an interactivecomputer algebra system: it is a library for programmers.<p>In this example, the type <tt>vec_pair_long_ZZ</tt>is an NTL vector whose base type is <tt>pair_long_ZZ</tt>.The type <tt>pair_long_ZZ</tt> is a type created byanother template-like macro mechanism.In general, for types <tt>S</tt> and <tt>T</tt>,one can create a type <tt>pair_S_T</tt> which isa class with a field <tt>a</tt> of type <tt>S</tt>and a field <tt>b</tt> of type <tt>T</tt>.See <a href="pair.txt"><tt>pair.txt</tt></a> for more details.<p> <hr> <p>Here is another example.The following program prints out the first 100 cyclotomic polynomials.<pre>#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"; }}</pre><p>To illustrate more of the NTL interface, let's look at alternative ways this routine could have been written.<p>First, instead of<pre> vec_ZZX phi(INIT_SIZE, 100); </pre>one can write<pre> vec_ZZX phi; phi.SetLength(100);</pre><p>Second,instead of<pre> t *= phi(j);</pre>one can write this as<pre> mul(t, t, phi(j));</pre>or<pre> t = t * phi(j);</pre>Also, one can write <tt>phi[j-1]</tt> in place of <tt>phi(j)</tt>.<p>Third, instead of<pre> phi(i) = (ZZX(i, 1) - 1)/t; </pre>one can write<pre> ZZX t1; SetCoeff(t1, i, 1); SetCoeff(t1, 0, -1); div(phi(i), t1, t);</pre>Alternatively, one could directly access the coefficient vector:<pre> 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);</pre>The coefficient vector of a polynomial is always an NTL vectorover the ground ring: in this case <tt>vec_ZZ</tt>.NTL does not try to be a dictator: it gives you free accessto the coefficient vector.However, after fiddling with this vector, you should "normalize"the polynomial, so that the leading coefficient in non-zero:this is an invariant which all routines that work with polynomialsexpect to hold.Of course, if you can avoid directly accessing thecoefficient vector, you should do so.You can always use the <tt>SetCoeff</tt> routine above to set orchange coefficients, and you can always read the value of a coefficientusing the routine <tt>coeff</tt>, e.g., <pre> ... f.rep[i] == 1 ...</pre>is equivalent to<pre> ... coeff(f, i) == 1 ...</pre>except that in the latter case, a read-only reference to zero is returnedif the index <tt>i</tt> is out of range.There are also special-purpose read-only access routines <tt>LeadCoeff(f)</tt>and <tt>ConstTerm(f)</tt>. <p>NTL provides a full compliment of operations for polynomialsover the integers, in both operator and procedural form.All of the basic operations support a "promotion logic" similarto that for <tt>ZZ</tt>, except that inputs of <i>both</i> types <tt>long</tt> and <tt>ZZ</tt> are promoted to <tt>ZZX</tt>.See <a href="ZZX.txt"><tt>ZZX.txt</tt></a> for details,and see <a href="ZZXFactoring.txt"><tt>ZZXFactoring.txt</tt></a> for detailson the polynomial factoring routines.<p><center><a href="tour-ex2.html"><img src="arrow1.gif" alt="[Previous]" align=bottom></a> <a href="tour-examples.html"><img src="arrow2.gif" alt="[Up]" align=bottom></a> <a href="tour-ex4.html"> <img src="arrow3.gif" alt="[Next]" align=bottom></a></center></body></html>?? 快捷鍵說明
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