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75 lines
1.8 KiB
75 lines
1.8 KiB
// Compute the Legendre polynomials.
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#include <cln/number.h>
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#include <cln/integer.h>
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#include <cln/rational.h>
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#include <cln/univpoly.h>
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#include <cln/modinteger.h>
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#include <cln/univpoly_rational.h>
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#include <cln/univpoly_modint.h>
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#include <cln/io.h>
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#include <stdlib.h>
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using namespace std;
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using namespace cln;
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// Computes the n-th Legendre polynomial in R[x], using the formula
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// P_n(x) = 1/(2^n n!) * (d/dx)^n (x^2-1)^n. (Assume n >= 0.)
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const cl_UP_RA legendre (const cl_rational_ring& R, int n)
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{
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cl_univpoly_rational_ring PR = find_univpoly_ring(R);
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cl_UP_RA b = PR->create(2);
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b.set_coeff(2,1);
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b.set_coeff(1,0);
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b.set_coeff(0,-1);
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b.finalize(); // b is now x^2-1
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cl_UP_RA p = (n==0 ? PR->one() : expt_pos(b,n));
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for (int i = 0; i < n; i++)
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p = deriv(p);
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cl_RA factor = recip(factorial(n)*ash(1,n));
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for (int j = degree(p); j >= 0; j--)
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p.set_coeff(j, coeff(p,j) * factor);
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p.finalize();
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return p;
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}
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const cl_UP_MI legendre (const cl_modint_ring& R, int n)
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{
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cl_univpoly_modint_ring PR = find_univpoly_ring(R);
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cl_UP_MI b = PR->create(2);
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b.set_coeff(2,R->canonhom(1));
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b.set_coeff(1,R->canonhom(0));
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b.set_coeff(0,R->canonhom(-1));
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b.finalize(); // b is now x^2-1
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cl_UP_MI p = (n==0 ? PR->one() : expt_pos(b,n));
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for (int i = 0; i < n; i++)
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p = deriv(p);
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cl_MI factor = recip(R->canonhom(factorial(n)*ash(1,n)));
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for (int j = degree(p); j >= 0; j--)
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p.set_coeff(j, coeff(p,j) * factor);
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p.finalize();
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return p;
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}
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int main (int argc, char* argv[])
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{
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if (!(argc == 2 || argc == 3)) {
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cerr << "Usage: legendre n [m]" << endl;
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exit(1);
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}
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int n = atoi(argv[1]);
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if (!(n >= 0)) {
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cerr << "Usage: legendre n [m] with n >= 0" << endl;
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exit(1);
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}
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if (argc == 2) {
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cl_UP p = legendre(cl_RA_ring,n);
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cout << p << endl;
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} else {
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cl_I m = argv[2];
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cl_UP p = legendre(find_modint_ring(m),n);
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cout << p << endl;
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}
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return 0;
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}
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