add special element program
This commit is contained in:
Florian Stecker
277
special_element.c
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277
special_element.c
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#include "coxeter.h"
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#include "linalg.h"
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#include "mat.h"
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#include <gsl/gsl_poly.h>
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#include <mps/mps.h>
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#define SWAP(t,x,y) do { t _tmp = (x); (x) = (y); (y) = _tmp; } while (0);
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#define DEBUG(msg, ...)
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int solve_characteristic_polynomial(mps_context *solv, 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_monomial_poly *poly;
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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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poly = mps_monomial_poly_new(solv, 3);
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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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// this version is only for the (4,4,4) group
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void initialize_triangle_generators(mat_workspace *ws, mat *gen, mpq_t m, mpq_t t)
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{
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mpq_t s,sinv,q,x,y;
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mpq_t zero, one, two;
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mpq_t tmp;
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mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
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mpq_set_ui(zero, 0, 1);
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mpq_set_ui(one, 1, 1);
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mpq_set_ui(two, 2, 1);
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// s = (1-m^2)/2m
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mpq_mul(s, m, m);
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mpq_sub(s, one, s);
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mpq_div(s, s, m);
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mpq_div(s, s, two);
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mpq_div(sinv, one, s);
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// q = (1+m^2)/(1-m^2) = 2/(1-m^2) - 1
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mpq_mul(q, m, m);
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mpq_sub(q, one, q);
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mpq_div(q, two, q);
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mpq_sub(q, q, one);
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// x = -tq, y = -q/t
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mpq_mul(x, q, t);
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mpq_sub(x, zero, x);
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mpq_div(y, q, t);
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mpq_sub(y, zero, y);
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// q^2 = xy = 1 + 1/s^2
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// [ -s s*y 0]
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// [ -s*x s*x*y - 1/s 0]
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// [ -s*y s*y^2 - x 1]
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LOOP(i,3) {
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mat_zero(gen[i]);
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mpq_sub(tmp, zero, s);
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mat_set(gen[i%3], i%3, i%3, tmp);
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mpq_mul(tmp, s, y);
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mat_set(gen[i%3], i%3, (i+1)%3, tmp);
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mpq_mul(tmp, s, x);
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mpq_sub(tmp, zero, tmp);
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mat_set(gen[i%3], (i+1)%3, i%3, tmp);
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mpq_mul(tmp, s, x);
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mpq_mul(tmp, tmp, y);
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mpq_sub(tmp, tmp, sinv);
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mat_set(gen[i%3], (i+1)%3, (i+1)%3, tmp);
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mpq_mul(tmp, s, y);
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mpq_sub(tmp, zero, tmp);
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mat_set(gen[i%3], (i+2)%3, i%3, tmp);
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mpq_mul(tmp, s, y);
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mpq_mul(tmp, tmp, y);
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mpq_sub(tmp, tmp, x);
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mat_set(gen[i%3], (i+2)%3, (i+1)%3, tmp);
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mat_set(gen[i%3], (i+2)%3, (i+2)%3, one);
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}
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LOOP(i,3) mat_pseudoinverse(ws, gen[i+3], gen[i]);
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mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
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}
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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(group_t *group, mat *matrices, mpq_t m, mpq_t t)
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{
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mat_workspace *ws;
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mat tmp;
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mat gen[6];
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char buf[100], buf2[100], buf3[100];
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// allocate stuff
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ws = mat_workspace_init(3);
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for(int i = 0; i < 6; i++)
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mat_init(gen[i], 3);
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mat_init(tmp, 3);
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initialize_triangle_generators(ws, gen, m, t);
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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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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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for(int i = 0; i < 6; i++)
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mat_clear(gen[i]);
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mat_clear(tmp);
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mat_workspace_clear(ws);
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}
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int main(int argc, char *argv[])
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{
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mpq_t m, t, tmp;
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double s;
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mat_workspace *ws;
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mat gen[6];
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mps_context *solver;
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mat element, inverse;
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int letter1, letter2, letter;
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mpq_t tr, trinv;
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double x, y;
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int retval;
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double evs[3];
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char buf[100];
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DEBUG("Allocate\n");
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mpq_inits(m, t, tmp, tr, trinv, NULL);
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ws = mat_workspace_init(3);
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for(int i = 0; i < 6; i++)
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mat_init(gen[i], 3);
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mat_init(element, 3);
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mat_init(inverse, 3);
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solver = mps_context_new();
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mps_context_set_output_prec(solver, 20); // relative precision
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mps_context_set_output_goal(solver, MPS_OUTPUT_GOAL_APPROXIMATE);
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for(int i = 1; i <= 99; i++) {
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for(int j = 1; j <= 100; j++) {
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mpq_set_ui(t, j, 100);
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mpq_set_ui(m, i, 100); // 414/1000 ~ sqrt(2)-1 <-> s=1
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s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m));
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DEBUG("Compute matrix\n");
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initialize_triangle_generators(ws, gen, m, t);
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mat_identity(element);
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mat_identity(inverse);
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for(int i = 0; i < strlen(argv[1]); i+=2) {
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letter1 = argv[1][i] - 'a';
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letter2 = argv[1][i+1] - 'a';
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if(letter1 == 1 && letter2 == 2)
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letter = 0; // p = bc
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else if(letter1 == 2 && letter2 == 0)
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letter = 1; // q = ca
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else if(letter1 == 0 && letter2 == 1)
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letter = 2; // r = ab
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else if(letter1 == 2 && letter2 == 1)
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letter = 3; // p^{-1} = cb
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else if(letter1 == 0 && letter2 == 2)
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letter = 4; // q^{-1} = ac
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else if(letter1 == 1 && letter2 == 0)
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letter = 5; // r^{-1} = ba
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mat_multiply(ws, element, element, gen[letter]);
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mat_multiply(ws, inverse, gen[(letter+3)%6], inverse);
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}
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DEBUG("Compute traces\n");
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mat_trace(tr, element);
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mat_trace(trinv, inverse);
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DEBUG("Solve characteristic polynomials\n");
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retval = solve_characteristic_polynomial(solver, tr, trinv, evs);
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if(retval == 1) {
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fprintf(stderr, "Error! Could not solve polynomial.\n");
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return 1;
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}
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if(fabs(evs[0]) < fabs(evs[1]))
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SWAP(double, evs[0], evs[1]);
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if(fabs(evs[1]) < fabs(evs[2]))
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SWAP(double, evs[1], evs[2]);
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if(fabs(evs[0]) < fabs(evs[1]))
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SWAP(double, evs[0], evs[1]);
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x = log(fabs(evs[0]));
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y = -log(fabs(evs[2]));
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// gmp_printf("%Qd %Qd %f %f %f\n", tr, trinv, x, y, y/x);
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gmp_printf("%.5f %.5f %.5f %.5f\n", mpq_get_d(t), mpq_get_d(m), s, y/x);
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}
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}
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DEBUG("Clean up\n");
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mpq_clears(m, t, tmp, tr, trinv, NULL);
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mat_workspace_clear(ws);
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for(int i = 0; i < 6; i++)
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mat_clear(gen[i]);
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mat_clear(element);
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mat_clear(inverse);
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mps_context_free(solver);
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}
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