triangle_reflection_complex/singular_values.c

#include "triangle.h"
#include "linalg.h"
#include "mat.h"

#include <gsl/gsl_poly.h>

//#define MAX_ELEMENTS 2800000
//#define MAX_ELEMENTS 720000
#define MAX_ELEMENTS 14000
//#define DRAW_PICTURE 1

void continued_fraction_approximation(mpq_t out, double in, int level)
{
	mpq_t tmp;

	if(in < 0) {
		mpq_init(tmp);
		mpq_set_ui(tmp, 0, 1);
		continued_fraction_approximation(out, -in, level);
		mpq_sub(out, tmp, out);
		mpq_clear(tmp);
		return;
	}

	if(level == 0) {
		mpq_set_si(out, (signed long int)round(in), 1); // floor(in)
	} else {
		continued_fraction_approximation(out, 1/(in - floor(in)), level - 1);
		mpq_init(tmp);
		mpq_set_ui(tmp, 1, 1);
		mpq_div(out, tmp, out); // out -> 1/out
		mpq_set_si(tmp, (signed long int)in, 1); // floor(in)
		mpq_add(out, out, tmp);
		mpq_clear(tmp);
	}
}

void quartic(mpq_t out, mpq_t in, int a, int b, int c, int d, int e)
{
	mpq_t tmp;
	mpq_init(tmp);

	mpq_set_si(out, a, 1);
	mpq_mul(out, out, in);
	mpq_set_si(tmp, b, 1);
	mpq_add(out, out, tmp);
	mpq_mul(out, out, in);
	mpq_set_si(tmp, c, 1);
	mpq_add(out, out, tmp);
	mpq_mul(out, out, in);
	mpq_set_si(tmp, d, 1);
	mpq_add(out, out, tmp);
	mpq_mul(out, out, in);
	mpq_set_si(tmp, e, 1);
	mpq_add(out, out, tmp);

	mpq_clear(tmp);
}

void initialize_triangle_generators(mat_workspace *ws, mat *gen, mpq_t s, mpq_t t)
{
	mpq_set_ui(*mat_ref(gen[0], 0, 0), 0, 1);
	mpq_set_ui(*mat_ref(gen[0], 0, 1), 0, 1);
	mpq_set_ui(*mat_ref(gen[0], 0, 2), 1, 1);
	mpq_set_ui(*mat_ref(gen[0], 1, 0), 1, 1);
	mpq_set_ui(*mat_ref(gen[0], 1, 1), 0, 1);
	mpq_set_ui(*mat_ref(gen[0], 1, 2), 0, 1);
	mpq_set_ui(*mat_ref(gen[0], 2, 0), 0, 1);
	mpq_set_ui(*mat_ref(gen[0], 2, 1), 1, 1);
	mpq_set_ui(*mat_ref(gen[0], 2, 2), 0, 1);

	mpq_set_ui(*mat_ref(gen[1], 0, 0), 1, 1);
	mpq_set_ui(*mat_ref(gen[1], 1, 0), 0, 1);
	mpq_set_ui(*mat_ref(gen[1], 2, 0), 0, 1);
	quartic(*mat_ref(gen[1], 0, 1), t, 0, 0, 1, -1, 2);
	quartic(*mat_ref(gen[1], 1, 1), t, 0, 0, -1, 2, -2);
	quartic(*mat_ref(gen[1], 2, 1), t, 0, 0, 1, -3, 3);
	quartic(*mat_ref(gen[1], 0, 2), t, 0, 0, 1, 0, 3);
	quartic(*mat_ref(gen[1], 1, 2), t, 0, 0, -1, 1, -1);
	quartic(*mat_ref(gen[1], 2, 2), t, 0, 0, 1, -2, 1);

	mat_pseudoinverse(ws, gen[3], gen[0]); // p^{-1}
	mat_pseudoinverse(ws, gen[4], gen[1]); // q^{-1}
	mat_multiply(ws, gen[2], gen[4], gen[3]); // r = q^{-1}p^{-1}
	mat_pseudoinverse(ws, gen[5], gen[2]);

//	mat_print(gen[0]);
//	mat_print(gen[1]);
//	mat_print(gen[2]);
//	mat_print(gen[3]);
//	mat_print(gen[4]);
//	mat_print(gen[5]);
}

char *print_word(groupelement_t *g, char *str)
{
  int i = g->length - 1;

  str[g->length] = 0;
  while(g->parent) {
    str[i--] = 'a' + g->letter;
    g = g->parent;
  }

  return str;
}

void enumerate(groupelement_t *group, mat *matrices, mpq_t s, mpq_t t)
{
	mat_workspace *ws;
	mat tmp;
	mat gen[6];
	char buf[100], buf2[100], buf3[100];

	// allocate stuff
	ws = mat_workspace_init(3);
	for(int i = 0; i < 6; i++)
		mat_init(gen[i], 3);
	mat_init(tmp, 3);

	initialize_triangle_generators(ws, gen, s, t);

	mat_identity(matrices[0]);
	for(int i = 1; i < MAX_ELEMENTS; i++) {
		if(group[i].length % 2 != 0)
			continue;
		if(!group[i].inverse)
			continue;

		int parent = group[i].parent->id;
		int grandparent = group[i].parent->parent->id;
		int letter;

		if(group[parent].letter == 1 && group[i].letter == 2)
			letter = 0; // p = bc
		else if(group[parent].letter == 2 && group[i].letter == 0)
			letter = 1; // q = ca
		else if(group[parent].letter == 0 && group[i].letter == 1)
			letter = 2; // r = ab
		if(group[parent].letter == 2 && group[i].letter == 1)
			letter = 3; // p^{-1} = cb
		else if(group[parent].letter == 0 && group[i].letter == 2)
			letter = 4; // q^{-1} = ac
		else if(group[parent].letter == 1 && group[i].letter == 0)
			letter = 5; // r^{-1} = ba

		mat_multiply(ws, matrices[i], matrices[grandparent], gen[letter]);
	}

	// free stuff
	for(int i = 0; i < 6; i++)
		mat_clear(gen[i]);
	mat_clear(tmp);
	mat_workspace_clear(ws);
}

void output_invariants(groupelement_t *group, mat *matrices, mpq_t s, mpq_t t)
{
	mpq_t tr, trinv;
	char buf[100];

	mpq_inits(tr, trinv, NULL);

	for(int i = 0; i < MAX_ELEMENTS; i++) {
		if(group[i].length % 2 != 0 || !group[i].inverse)
			continue;

		mat_trace(tr, matrices[i]);
		mat_trace(trinv, matrices[group[i].inverse->id]);

		gmp_printf("%d %d %s %Qd %Qd %f %f\n", i, group[i].length, print_word(&group[i], buf), tr, trinv, log(mpq_get_d(tr)), log(mpq_get_d(trinv)));
	}

	mpq_clears(tr, trinv, NULL);
}

double max_slope(groupelement_t *group, mat *matrices, mpq_t s, mpq_t t, int *index)
{
	double max = 0;
	double slope;

	mpq_t tr, trinv;
	char buf[100];

	mpq_inits(tr, trinv, NULL);

	for(int i = 0; i < MAX_ELEMENTS; i++) {
		if(group[i].length % 2 != 0 || !group[i].inverse)
			continue;

		mat_trace(tr, matrices[i]);
		mat_trace(trinv, matrices[group[i].inverse->id]);

		slope = log(mpq_get_d(trinv))/log(mpq_get_d(tr));
		if(slope > max)
		{
			*index = i;
			max = slope;
		}
	}

	mpq_clears(tr, trinv, NULL);

	return max;
}

int main(int argc, char *argv[])
{
	mpq_t s, t, tmp;
	mpz_t accuracy;
	double t_;
	mat *matrices;
	groupelement_t *group;
	int index;

	mpq_inits(s, t, tmp, NULL);
	mpz_init(accuracy);
	group = malloc(MAX_ELEMENTS*sizeof(groupelement_t));
	matrices = malloc(MAX_ELEMENTS*sizeof(mat));
	for(int i = 0; i < MAX_ELEMENTS; i++)
		mat_init(matrices[i], 3);

	// mpq_set_str(t, argv[1], 10);
	mpz_set_ui(accuracy, 100);
	for(int i = 0; ; i++) {
		mpq_set(t, tmp);
		continued_fraction_approximation(tmp, atof(argv[1]), i);
		if(mpz_cmp(mpq_numref(tmp),accuracy) > 0 && mpz_cmp(mpq_denref(tmp),accuracy) > 0)
			break;
	}
	mpq_canonicalize(t);
	gmp_fprintf(stdout, "\"t = %Qd = %.3f\"\n", mpq_get_d(t), t);

	if(argc > 2 && strcmp(argv[2],"p") == 0) {
		gmp_fprintf(stdout, "%Qd\n", t);
		return 0;
	}

	generate_triangle_group(group, MAX_ELEMENTS, 3, 3, 4);

//	for(int i = 0; i < 10; i++) {
//		mpq_set_ui(t,100+i,100);
//		mpq_canonicalize(t);

		enumerate(group, matrices, s, t);
		//printf("%f %f\n", mpq_get_d(t), max_slope(group, matrices, s, t, &index));
		output_invariants(group, matrices, s, t);
//	}

	for(int i = 0; i < MAX_ELEMENTS; i++)
		mat_clear(matrices[i]);
	free(matrices);
	free(group);
	mpq_clears(s, t, tmp, NULL);
	mpz_clear(accuracy);
}