512 lines
13 KiB
C
512 lines
13 KiB
C
#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, ...) do { print_time(); fprintf(stderr, msg, ##__VA_ARGS__); } while (0);
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//#define DEBUG(msg, ...)
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//#define OUTPUT_POINTS
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#define OUTPUT_SUMMARY
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struct timespec starttime;
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void print_time()
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{
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double diff;
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struct timespec current;
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clock_gettime(CLOCK_REALTIME, ¤t);
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diff = (current.tv_sec - starttime.tv_sec) + (current.tv_nsec - starttime.tv_nsec)*1e-9;
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fprintf(stderr, "[%.3f] ", diff);
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}
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struct result {
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mpq_t tr;
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mpq_t trinv;
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};
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static int compare_result(const void *a_, const void *b_)
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{
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int d = 0;
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struct result **a = (struct result **)a_;
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struct result **b = (struct result **)b_;
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d = mpq_cmp((*a)->tr,(*b)->tr);
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if(d == 0)
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d = mpq_cmp((*a)->trinv, (*b)->trinv);
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return d;
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}
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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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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(mpq_t out, mpq_t in, int a, int b, int c, int d, int e)
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{
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mpq_t tmp;
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mpq_init(tmp);
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mpq_set_si(out, a, 1);
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mpq_mul(out, out, in);
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mpq_set_si(tmp, b, 1);
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mpq_add(out, out, tmp);
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mpq_mul(out, out, in);
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mpq_set_si(tmp, c, 1);
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mpq_add(out, out, tmp);
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mpq_mul(out, out, in);
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mpq_set_si(tmp, d, 1);
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mpq_add(out, out, tmp);
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mpq_mul(out, out, in);
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mpq_set_si(tmp, e, 1);
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mpq_add(out, out, tmp);
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mpq_clear(tmp);
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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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// debug output
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/*
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gmp_printf("m = %Qd, s = %Qd, t = %Qd, q = %Qd, x = %Qd, y = %Qd\n", m, s, t, q, x, y);
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mat_print(gen[0]);
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mat_print(gen[1]);
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mat_print(gen[2]);
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*/
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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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void output_invariants(group_t *group, mat *matrices, mpq_t s, mpq_t q, mps_context *solver)
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{
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mpq_t tr, trinv;
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char buf[100];
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double evs[3];
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int retval;
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mpq_inits(tr, trinv, NULL);
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for(int i = 0; i < group->size; i++) {
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if(group->elements[i].length % 2 != 0 || !group->elements[i].inverse)
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continue;
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mat_trace(tr, matrices[i]);
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mat_trace(trinv, matrices[group->elements[i].inverse->id]);
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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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continue;
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} else if(retval == 2) {
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continue;
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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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gmp_printf("%d %d %s %Qd %Qd %f %f\n", i, group->elements[i].length, print_word(&group->elements[i], buf), tr, trinv, log(evs[0]), -log(evs[2]));
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}
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mpq_clears(tr, trinv, NULL);
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}
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/*
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double max_slope(groupelement_t *group, mat *matrices, mpq_t s, mpq_t t, int *index)
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{
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double max = 0;
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double slope;
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mpq_t tr, trinv;
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char buf[100];
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mpq_inits(tr, trinv, NULL);
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for(int i = 0; i < MAX_ELEMENTS; i++) {
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if(group[i].length % 2 != 0 || !group[i].inverse)
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continue;
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mat_trace(tr, matrices[i]);
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mat_trace(trinv, matrices[group[i].inverse->id]);
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slope = log(mpq_get_d(trinv))/log(mpq_get_d(tr));
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if(slope > max)
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{
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*index = i;
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max = slope;
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}
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}
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mpq_clears(tr, trinv, NULL);
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return max;
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}
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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 *matrices;
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group_t *group;
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int index;
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mps_context *solver;
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int acc = 100;
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int n, nuniq, nmax;
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int retval;
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double evs[3];
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double max_slope;
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char buf[100];
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char buf2[100];
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struct result *invariants;
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struct result **distinct_invariants;
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clock_gettime(CLOCK_REALTIME, &starttime);
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nmax = atoi(argv[1]);
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DEBUG("Allocate\n");
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mpq_inits(m, t, tmp, NULL);
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matrices = malloc(nmax*sizeof(mat));
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for(int i = 0; i < nmax; i++)
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mat_init(matrices[i], 3);
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invariants = malloc(nmax*sizeof(struct result));
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distinct_invariants = malloc(nmax*sizeof(struct result));
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for(int i = 0; i < nmax; i++) {
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mpq_init(invariants[i].tr);
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mpq_init(invariants[i].trinv);
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distinct_invariants[i] = &invariants[i];
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}
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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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/*
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DEBUG("Approximate parameters\n");
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// get approximate s and q values
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sapprox = atof(argv[2]);
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tapprox = atof(argv[3]);
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tqfactor = pow((sapprox*sapprox-sapprox+1)*(sapprox*sapprox-sapprox+1)*(sapprox*sapprox+1), 1/6.0);
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qapprox = tapprox/tqfactor;
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for(int i = 0; ; i++) {
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continued_fraction_approximation(tmp, sapprox, i);
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if(fabs(mpq_get_d(t)-sapprox) < 1e-10
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|| (mpz_cmpabs_ui(mpq_numref(tmp),acc) > 0 && mpz_cmpabs_ui(mpq_denref(tmp),acc) > 0))
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break;
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mpq_set(s, tmp);
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}
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mpq_canonicalize(s);
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for(int i = 0; ; i++) {
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continued_fraction_approximation(tmp, qapprox, i);
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if(fabs(mpq_get_d(t)-qapprox) < 1e-10
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|| (mpz_cmpabs_ui(mpq_numref(tmp),acc) > 0 && mpz_cmpabs_ui(mpq_denref(tmp),acc) > 0))
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break;
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mpq_set(q, tmp);
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}
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mpq_canonicalize(q);
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tqfactor = pow((mpq_get_d(s)*mpq_get_d(s)-mpq_get_d(s)+1)*(mpq_get_d(s)*mpq_get_d(s)-mpq_get_d(s)+1)*(mpq_get_d(s)*mpq_get_d(s)+1), 1/6.0);
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#ifdef OUTPUT_POINTS
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gmp_fprintf(stdout, "\"s = %Qd = %.3f, q = %Qd, t = %.3f\"\n", s, mpq_get_d(s), q, mpq_get_d(q)*tqfactor);
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#endif
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*/
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// group
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DEBUG("Generate group\n");
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group = coxeter_init_triangle(4, 4, 4, nmax);
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fprintf(stderr, "max word length = %d\n", group->elements[nmax-1].length);
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for(int ttick = 45; ttick <= 65; ttick++) {
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for(int mtick = 45; mtick < 65; mtick++) {
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mpq_set_ui(t, ttick, 100);
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mpq_set_ui(m, mtick, 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 matrices\n");
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enumerate(group, matrices, m, t);
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n = 0;
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DEBUG("Compute traces\n");
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for(int i = 0; i < nmax; i++) {
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if(group->elements[i].length % 2 != 0 || !group->elements[i].inverse)
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continue;
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mat_trace(invariants[i].tr, matrices[i]);
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mat_trace(invariants[i].trinv, matrices[group->elements[i].inverse->id]);
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distinct_invariants[n++] = &invariants[i];
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// gmp_printf("%Qd %Qd %d %s\n", invariants[i].tr, invariants[i].trinv, i, print_word(&group->elements[i], buf));
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}
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DEBUG("Get unique traces\n");
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qsort(distinct_invariants, n, sizeof(struct result*), compare_result);
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nuniq = 0;
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for(int i = 0; i < n; i++) {
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if(i == 0 || compare_result(&distinct_invariants[i], &distinct_invariants[nuniq-1]) != 0)
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distinct_invariants[nuniq++] = distinct_invariants[i];
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}
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max_slope = 0;
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int max_slope_index;
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DEBUG("Solve characteristic polynomials\n");
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for(int i = 0; i < nuniq; i++) {
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retval = solve_characteristic_polynomial(solver, distinct_invariants[i]->tr, distinct_invariants[i]->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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continue;
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} else if(retval == 2) {
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continue;
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}
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|
if(fabs(evs[0]) < fabs(evs[1]))
|
|
SWAP(double, evs[0], evs[1]);
|
|
if(fabs(evs[1]) < fabs(evs[2]))
|
|
SWAP(double, evs[1], evs[2]);
|
|
if(fabs(evs[0]) < fabs(evs[1]))
|
|
SWAP(double, evs[0], evs[1]);
|
|
|
|
double x = log(fabs(evs[0]));
|
|
double y = -log(fabs(evs[2]));
|
|
|
|
if(y/x > max_slope && (x > 0.1 || y > 0.1)) {
|
|
max_slope_index = distinct_invariants[i] - invariants;
|
|
max_slope = y/x;
|
|
}
|
|
|
|
#ifdef OUTPUT_POINTS
|
|
gmp_printf("%Qd %Qd %f %f %f\n", distinct_invariants[i]->tr, distinct_invariants[i]->trinv, x, y, y/x);
|
|
#endif
|
|
}
|
|
|
|
#ifdef OUTPUT_SUMMARY
|
|
// fprintf(stdout, "%.5f %.5f %.5f %f %s\n", mpq_get_d(t), mpq_get_d(m), s, max_slope, print_word(&group->elements[max_slope_index], buf));
|
|
fprintf(stdout, "%.5f %.5f %.5f %f %s\n", mpq_get_d(t), mpq_get_d(m), s, max_slope, print_word(&group->elements[max_slope_index], buf));
|
|
#endif
|
|
}
|
|
}
|
|
|
|
DEBUG("Clean up\n");
|
|
for(int i = 0; i < nmax; i++) {
|
|
mpq_clear(invariants[i].tr);
|
|
mpq_clear(invariants[i].trinv);
|
|
}
|
|
free(invariants);
|
|
free(distinct_invariants);
|
|
for(int i = 0; i < nmax; i++)
|
|
mat_clear(matrices[i]);
|
|
free(matrices);
|
|
coxeter_clear(group);
|
|
mpq_clears(m, t, tmp, NULL);
|
|
mps_context_free(solver);
|
|
}
|