triangle_reflection_complex/singular_values.c

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

#include <gsl/gsl_poly.h>
#include <mps/mps.h>

//#define MAX_ELEMENTS 2800000
//#define MAX_ELEMENTS 720000
#define MAX_ELEMENTS 1000
//#define DRAW_PICTURE 1

#define SWAP(t,x,y) do { t _tmp = (x); (x) = (y); (y) = _tmp; } while (0);

int solve_characteristic_polynomial(mps_context *solv, mpq_t tr, mpq_t trinv, double *eigenvalues)
{
	mpq_t coeff1, coeff2, zero;
	cplx_t *roots;
	double radii[3];
	double *radii_p[3];
	mps_monomial_poly *poly;
	mps_boolean errors;
	int result = 0;

	mpq_inits(coeff1, coeff2, zero, NULL);
	mpq_set(coeff1, trinv);
	mpq_sub(coeff2, zero, tr);

	poly = mps_monomial_poly_new(solv, 3);
	mps_monomial_poly_set_coefficient_int(solv, poly, 0, -1, 0);
	mps_monomial_poly_set_coefficient_q(solv, poly, 1, coeff1, zero);
	mps_monomial_poly_set_coefficient_q(solv, poly, 2, coeff2, zero);
	mps_monomial_poly_set_coefficient_int(solv, poly, 3, 1, 0);

	mps_context_set_input_poly(solv, (mps_polynomial*)poly);
	mps_mpsolve(solv);

	roots = cplx_valloc(3);
	for(int i = 0; i < 3; i++)
		radii_p[i] = &(radii[i]);
	mps_context_get_roots_d(solv, &roots, radii_p);
	errors = mps_context_has_errors(solv);

	if(errors) {
		result = 1;
	} else {
		for(int i = 0; i < 3; i++) {
			eigenvalues[i] = cplx_Re(roots[i]);
			if(fabs(cplx_Im(roots[i])) > radii[i]) // non-real root
				result = 2;
		}
	}

	cplx_vfree(roots);
	mpq_clears(coeff1, coeff2, zero, NULL);

	return result;
}

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 q)
{
	mat r1,r2,r3;
	mpq_t rho1, rho2, rho3;
	mpq_t b1,c1,a2,c2,a3,b3;
	mpq_t sinv;

	mpq_inits(sinv,rho1,rho2,rho3,b1,c1,a2,c2,a3,b3,NULL);
	mat_init(r1, 3);
	mat_init(r2, 3);
	mat_init(r3, 3);

	mpq_set_ui(sinv, 1, 1);
	mpq_div(sinv, sinv, s);

	quartic(rho1, s, 0, 0, 1, -1, 1);
	quartic(rho2, s, 0, 0, 1, -1, 1);
	quartic(rho3, s, 0, 0, 1,  0, 1);

	mpq_mul(c1, rho2, q);
	mpq_mul(a2, rho3, q);
	mpq_mul(b3, rho1, q);
	mpq_set_ui(b1, 1, 1);
	mpq_set_ui(c2, 1, 1);
	mpq_set_ui(a3, 1, 1);
	mpq_div(b1, b1, q);
	mpq_div(c2, c2, q);
	mpq_div(a3, a3, q);

	// actually, we want minus everything
	mat_zero(r1);
	mat_zero(r2);
	mat_zero(r3);
	mpq_set_si(*mat_ref(r1, 0, 0), -1, 1);
	mpq_set_si(*mat_ref(r1, 1, 1), 1, 1);
	mpq_set_si(*mat_ref(r1, 2, 2), 1, 1);
	mpq_set_si(*mat_ref(r2, 0, 0), 1, 1);
	mpq_set_si(*mat_ref(r2, 1, 1), -1, 1);
	mpq_set_si(*mat_ref(r2, 2, 2), 1, 1);
	mpq_set_si(*mat_ref(r3, 0, 0), 1, 1);
	mpq_set_si(*mat_ref(r3, 1, 1), 1, 1);
	mpq_set_si(*mat_ref(r3, 2, 2), -1, 1);

	mpq_set(*mat_ref(r1, 1, 0), b1);
	mpq_set(*mat_ref(r1, 2, 0), c1);
	mpq_set(*mat_ref(r2, 0, 1), a2);
	mpq_set(*mat_ref(r2, 2, 1), c2);
	mpq_set(*mat_ref(r3, 0, 2), a3);
	mpq_set(*mat_ref(r3, 1, 2), b3);

	mat_zero(gen[0]);
	mat_zero(gen[1]);
	mat_zero(gen[2]);

	mpq_set_ui(*mat_ref(gen[0], 0, 0), 1, 1);
	mat_set(gen[0], 1, 1, sinv);
	mat_set(gen[0], 2, 2, s);

	mat_set(gen[1], 0, 0, s);
	mpq_set_ui(*mat_ref(gen[1], 1, 1), 1, 1);
	mat_set(gen[1], 2, 2, sinv);

	mat_set(gen[2], 0, 0, sinv);
	mat_set(gen[2], 1, 1, s);
	mpq_set_ui(*mat_ref(gen[2], 2, 2), 1, 1);

	mat_multiply(ws, gen[0], r2, gen[0]);
	mat_multiply(ws, gen[0], gen[0], r3);
	mat_multiply(ws, gen[1], r3, gen[1]);
	mat_multiply(ws, gen[1], gen[1], r1);
	mat_multiply(ws, gen[2], r1, gen[2]);
	mat_multiply(ws, gen[2], gen[2], r2);

	mat_pseudoinverse(ws, gen[3], gen[0]);
	mat_pseudoinverse(ws, gen[4], gen[1]);
	mat_pseudoinverse(ws, gen[5], gen[2]);

	/*
	mat_print(r1);
	mat_print(r2);
	mat_print(r3);
	mat_print(gen[0]);
	mat_print(gen[1]);
	mat_print(gen[2]);
	mat_print(gen[3]);
	mat_print(gen[4]);
	mat_print(gen[5]);
	*/

	mpq_clears(sinv,rho1,rho2,rho3,b1,c1,a2,c2,a3,b3,NULL);
	mat_clear(r1);
	mat_clear(r2);
	mat_clear(r3);
}

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 q, mps_context *solver)
{
	mpq_t tr, trinv;
	char buf[100];
	double evs[3];
	int retval;

	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]);

		retval = solve_characteristic_polynomial(solver, tr, trinv, evs);
		if(retval == 1) {
			fprintf(stderr, "Error! Could not solve polynomial.\n");
			continue;
		} else if(retval == 2) {
			continue;
		}

		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]);

		gmp_printf("%d %d %s %Qd %Qd %f %f\n", i, group[i].length, print_word(&group[i], buf), tr, trinv, log(evs[0]), -log(evs[2]));
	}

	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, q, t, tmp;
	double sapprox, tapprox, qapprox, tqfactor;
	mat *matrices;
	groupelement_t *group;
	int index;
	mps_context *solver;

	mpq_inits(s, q, t, tmp, NULL);
	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);

	solver = mps_context_new();
	mps_context_set_output_prec(solver, 20); // relative precision
	mps_context_set_output_goal(solver, MPS_OUTPUT_GOAL_APPROXIMATE);

	int acc = 100;

	sapprox = atof(argv[1]);
	tapprox = atof(argv[2]);
	tqfactor = pow((sapprox*sapprox-sapprox+1)*(sapprox*sapprox-sapprox+1)*(sapprox*sapprox+1), 1/6.0);
	qapprox = tapprox/tqfactor;

	for(int i = 0; ; i++) {
		continued_fraction_approximation(tmp, sapprox, i);
//		gmp_fprintf(stdout, "%Qd\n", tmp);
		if(fabs(mpq_get_d(t)-sapprox) < 1e-10 || (mpz_cmpabs_ui(mpq_numref(tmp),acc) > 0 && mpz_cmpabs_ui(mpq_denref(tmp),acc) > 0))
			break;
		mpq_set(s, tmp);
	}
	mpq_canonicalize(s);

	for(int i = 0; ; i++) {
		continued_fraction_approximation(tmp, qapprox, i);
//		gmp_fprintf(stdout, "%Qd\n", tmp);
		if(fabs(mpq_get_d(t)-qapprox) < 1e-10 || (mpz_cmpabs_ui(mpq_numref(tmp),acc) > 0 && mpz_cmpabs_ui(mpq_denref(tmp),acc) > 0))
			break;
		mpq_set(q, tmp);
	}
	mpq_canonicalize(q);

	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);

	gmp_fprintf(stdout, "\"s = %Qd = %.3f, q = %Qd, t = %.3f\"\n", s, mpq_get_d(s), q, mpq_get_d(q)*tqfactor);

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

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