336 lines
8.1 KiB
C
336 lines
8.1 KiB
C
#include "enumerate_triangle_group.h"
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#include "linalg.h"
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int solve_characteristic_polynomial(mps_context *solv, mps_monomial_poly *poly, mpq_t tr, mpq_t trinv, double *eigenvalues)
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{
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mpq_t coeff1, coeff2, zero;
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cplx_t *roots;
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double radii[3];
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double *radii_p[3];
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mps_boolean errors;
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int result = 0;
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mpq_inits(coeff1, coeff2, zero, NULL);
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mpq_set(coeff1, trinv);
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mpq_sub(coeff2, zero, tr);
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mps_monomial_poly_set_coefficient_int(solv, poly, 0, -1, 0);
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mps_monomial_poly_set_coefficient_q(solv, poly, 1, coeff1, zero);
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mps_monomial_poly_set_coefficient_q(solv, poly, 2, coeff2, zero);
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mps_monomial_poly_set_coefficient_int(solv, poly, 3, 1, 0);
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mps_context_set_input_poly(solv, (mps_polynomial*)poly);
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mps_mpsolve(solv);
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roots = cplx_valloc(3);
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for(int i = 0; i < 3; i++)
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radii_p[i] = &(radii[i]);
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mps_context_get_roots_d(solv, &roots, radii_p);
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errors = mps_context_has_errors(solv);
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if(errors) {
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result = 1;
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} else {
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for(int i = 0; i < 3; i++) {
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eigenvalues[i] = cplx_Re(roots[i]);
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if(fabs(cplx_Im(roots[i])) > radii[i]) // non-real root
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result = 2;
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}
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}
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cplx_vfree(roots);
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mpq_clears(coeff1, coeff2, zero, NULL);
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return result;
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}
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void continued_fraction_approximation(mpq_t out, double in, int level)
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{
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mpq_t tmp;
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if(in < 0) {
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mpq_init(tmp);
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mpq_set_ui(tmp, 0, 1);
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continued_fraction_approximation(out, -in, level);
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mpq_sub(out, tmp, out);
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mpq_clear(tmp);
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return;
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}
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if(level == 0) {
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mpq_set_si(out, (signed long int)round(in), 1); // floor(in)
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} else {
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continued_fraction_approximation(out, 1/(in - floor(in)), level - 1);
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mpq_init(tmp);
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mpq_set_ui(tmp, 1, 1);
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mpq_div(out, tmp, out); // out -> 1/out
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mpq_set_si(tmp, (signed long int)in, 1); // floor(in)
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mpq_add(out, out, tmp);
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mpq_clear(tmp);
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}
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}
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void quartic(NUMBER out, NUMBER in, int a, int b, int c, int d, int e)
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{
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NUMBER tmp;
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INIT(tmp, GETTYPE(in));
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SET_INT(out, a);
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MUL(out, out, in);
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SET_INT(tmp, b);
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ADD(out, out, tmp);
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MUL(out, out, in);
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SET_INT(tmp, c);
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ADD(out, out, tmp);
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MUL(out, out, in);
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SET_INT(tmp, d);
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ADD(out, out, tmp);
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MUL(out, out, in);
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SET_INT(tmp, e);
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ADD(out, out, tmp);
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CLEAR(tmp);
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}
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// discriminant of the polynomial z^3 - x z^2 + y z - 1
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// that is the function x^2 y^2 - 4 x^3 - 4 y^3 - 27 + 18 xy
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void discriminant(NUMBER out, NUMBER x, NUMBER y)
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{
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TYPE type = GETTYPE(out);
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NUMBER x2, x3, y2, y3, tmp;
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INIT(x2, type);INIT(x3, type);INIT(y2, type);INIT(y3, type);INIT(tmp, type);
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MUL(x2, x, x);
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MUL(x3, x2, x);
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MUL(y2, y, y);
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MUL(y3, y2, y);
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MUL(out, x2, y2);
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SET_INT(tmp, -4);
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MUL(tmp, tmp, x3);
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ADD(out, out, tmp);
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SET_INT(tmp, -4);
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MUL(tmp, tmp, y3);
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ADD(out, out, tmp);
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SET_INT(tmp, -27);
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ADD(out, out, tmp);
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SET_INT(tmp, 18);
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MUL(tmp, tmp, x);
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MUL(tmp, tmp, y);
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ADD(out, out, tmp);
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CLEAR(x2);CLEAR(x3);CLEAR(y2);CLEAR(y3);CLEAR(tmp);
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}
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void generators_triangle_rotation_generic(mat *gen, NUMBER rho1, NUMBER rho2, NUMBER rho3, mpq_t s, mpq_t q)
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{
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mat_workspace *ws;
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mat r1,r2,r3;
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NUMBER b1,c1,a2,c2,a3,b3;
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TYPE type = GETTYPE(rho1);
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mpq_t sinv, qinv;
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ws = mat_workspace_init(3, type);
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INIT(b1, type);INIT(c1, type);INIT(a2, type);INIT(c2, type);INIT(a3, type);INIT(b3, type);
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mpq_init(sinv);
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mpq_init(qinv);
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mat_init(r1, 3, type);
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mat_init(r2, 3, type);
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mat_init(r3, 3, type);
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// sinv = s^{-1}
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mpq_inv(sinv, s);
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mpq_inv(qinv, q);
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// c1 = rho2 q, a2 = rho3 q, b3 = rho1 q, b1 = c2 = a3 = 1/q
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SET_Q(c1, q);
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SET_Q(a2, q);
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SET_Q(b3, q);
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MUL(c1, c1, rho2);
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MUL(a2, a2, rho3);
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MUL(b3, b3, rho1);
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SET_INT(b1, 1);
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SET_INT(c2, 1);
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SET_INT(a3, 1);
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SET_Q(b1, qinv);
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SET_Q(c2, qinv);
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SET_Q(a3, qinv);
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mat_zero(r1);
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mat_zero(r2);
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mat_zero(r3);
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SET_INT(*mat_ref(r1, 0, 0), -1);
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SET_INT(*mat_ref(r1, 1, 1), 1);
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SET_INT(*mat_ref(r1, 2, 2), 1);
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SET_INT(*mat_ref(r2, 0, 0), 1);
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SET_INT(*mat_ref(r2, 1, 1), -1);
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SET_INT(*mat_ref(r2, 2, 2), 1);
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SET_INT(*mat_ref(r3, 0, 0), 1);
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SET_INT(*mat_ref(r3, 1, 1), 1);
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SET_INT(*mat_ref(r3, 2, 2), -1);
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SET(*mat_ref(r1, 1, 0), b1);
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SET(*mat_ref(r1, 2, 0), c1);
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SET(*mat_ref(r2, 0, 1), a2);
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SET(*mat_ref(r2, 2, 1), c2);
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SET(*mat_ref(r3, 0, 2), a3);
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SET(*mat_ref(r3, 1, 2), b3);
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mat_zero(gen[0]);
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mat_zero(gen[1]);
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mat_zero(gen[2]);
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// gen[0] = diag(1,s^{-1},s)
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SET_INT(*mat_ref(gen[0], 0, 0), 1);
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SET_Q (*mat_ref(gen[0], 1, 1), sinv);
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SET_Q (*mat_ref(gen[0], 2, 2), s);
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// gen[1] = diag(s,1,s^{-1})
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SET_Q (*mat_ref(gen[1], 0, 0), s);
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SET_INT(*mat_ref(gen[1], 1, 1), 1);
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SET_Q (*mat_ref(gen[1], 2, 2), sinv);
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// gen[2] = diag(s^{-1},s,1)
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SET_Q (*mat_ref(gen[2], 0, 0), sinv);
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SET_Q (*mat_ref(gen[2], 1, 1), s);
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SET_INT(*mat_ref(gen[2], 2, 2), 1);
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// gen[0] = r2 * gen[0] * r3
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// gen[1] = r3 * gen[1] * r1
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// gen[2] = r1 * gen[2] * r2
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mat_multiply(ws, gen[0], r2, gen[0]);
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mat_multiply(ws, gen[0], gen[0], r3);
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mat_multiply(ws, gen[1], r3, gen[1]);
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mat_multiply(ws, gen[1], gen[1], r1);
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mat_multiply(ws, gen[2], r1, gen[2]);
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mat_multiply(ws, gen[2], gen[2], r2);
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// gen[3] = gen[0]^{-1}
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// gen[4] = gen[1]^{-1}
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// gen[5] = gen[2]^{-1}
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mat_pseudoinverse(ws, gen[3], gen[0]);
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mat_pseudoinverse(ws, gen[4], gen[1]);
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mat_pseudoinverse(ws, gen[5], gen[2]);
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mat_workspace_clear(ws);
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CLEAR(b1);CLEAR(c1);CLEAR(a2);CLEAR(c2);CLEAR(a3);CLEAR(b3);
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mpq_clear(sinv);
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mpq_clear(qinv);
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mat_clear(r1);
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mat_clear(r2);
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mat_clear(r3);
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}
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#ifdef QEXT_TRIVIAL
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// p1,p2,p3 are only allowed to be 2,3,4,6
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void generators_triangle_rotation_2346_rational(mat *gen, int p1, int p2, int p3, mpq_t s, mpq_t q)
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{
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mpq_t rho1, rho2, rho3;
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mpq_inits(rho1,rho2,rho3,NULL);
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// rho_i = s^2 + 2s cos(2 pi / p_i) + 1
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// coefficient 2 is the value for p=infinity, not sure if that would even work
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quartic(rho1, s, 0, 0, 1, p1 == 2 ? -2 : p1 == 3 ? -1 : p1 == 4 ? 0 : p1 == 6 ? 1 : 2, 1);
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quartic(rho2, s, 0, 0, 1, p2 == 2 ? -2 : p2 == 3 ? -1 : p2 == 4 ? 0 : p2 == 6 ? 1 : 2, 1);
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quartic(rho3, s, 0, 0, 1, p3 == 2 ? -2 : p3 == 3 ? -1 : p3 == 4 ? 0 : p3 == 6 ? 1 : 2, 1);
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generators_triangle_rotation_generic(gen, rho1, rho2, rho3, s, q);
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mpq_clears(rho1,rho2,rho3,NULL);
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}
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#endif
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#ifdef QEXT
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void generators_triangle_rotation_555_barbot(mat *gen, mpq_t s_, mpq_t q_)
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{
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NUMBER rho, c, one, s;
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INIT(rho, QT_SQRT5);INIT(c, QT_SQRT5);INIT(one, QT_SQRT5);INIT(s, QT_SQRT5);
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// c = - (1+sqrt(5))/2
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mpq_set_si(c->a[0], -1, 2);
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mpq_set_si(c->a[1], -1, 2);
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SET_ONE(one);
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SET_Q(s, s_);
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// rho = s^2 + cs + 1
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SET(rho, one);
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MUL(rho, rho, s);
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ADD(rho, rho, c);
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MUL(rho, rho, s);
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ADD(rho, rho, one);
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generators_triangle_rotation_generic(gen, rho, rho, rho, s_, q_);
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CLEAR(rho);CLEAR(c);CLEAR(one);CLEAR(s);
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}
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#endif
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char *print_word(groupelement_t *g, char *str)
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{
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int i = g->length - 1;
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str[g->length] = 0;
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while(g->parent) {
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str[i--] = 'a' + g->letter;
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g = g->parent;
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}
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return str;
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}
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void enumerate_triangle_rotation_subgroup(group_t *group, mat *gen, mat *matrices)
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{
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mat_workspace *ws;
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mat tmp;
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char buf[100], buf2[100], buf3[100];
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// allocate stuff
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TYPE type = GETTYPE(gen[0]->x[0]);
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ws = mat_workspace_init(3, type);
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mat_init(tmp, 3, type);
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// initialize_triangle_generators(ws, gen, p1, p2, p3, s, q);
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mat_identity(matrices[0]);
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for(int i = 1; i < group->size; i++) {
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if(group->elements[i].length % 2 != 0)
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continue;
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if(!group->elements[i].inverse)
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continue;
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if(!group->elements[i].need_to_compute)
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continue;
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int parent = group->elements[i].parent->id;
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int grandparent = group->elements[i].parent->parent->id;
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int letter;
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if(group->elements[parent].letter == 1 && group->elements[i].letter == 2)
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letter = 0; // p = bc
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else if(group->elements[parent].letter == 2 && group->elements[i].letter == 0)
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letter = 1; // q = ca
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else if(group->elements[parent].letter == 0 && group->elements[i].letter == 1)
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letter = 2; // r = ab
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if(group->elements[parent].letter == 2 && group->elements[i].letter == 1)
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letter = 3; // p^{-1} = cb
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else if(group->elements[parent].letter == 0 && group->elements[i].letter == 2)
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letter = 4; // q^{-1} = ac
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else if(group->elements[parent].letter == 1 && group->elements[i].letter == 0)
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letter = 5; // r^{-1} = ba
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mat_multiply(ws, matrices[i], matrices[grandparent], gen[letter]);
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}
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// free stuff
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mat_clear(tmp);
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mat_workspace_clear(ws);
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}
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