florian/triangle_reflection_complex
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triangle_reflection_complex/singular_values.c

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C
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#include "coxeter.h"
#include "linalg.h"
#include "mat.h"
#include <gsl/gsl_poly.h>
#include <mps/mps.h>
#define SWAP(t,x,y) do { t _tmp = (x); (x) = (y); (y) = _tmp; } while (0);
//#define DEBUG(msg, ...) fprintf(stderr, msg, ##__VA_ARGS__)
#define DEBUG(msg, ...)
#define OUTPUT_POINTS
//#define OUTPUT_POINTS
struct result {
mpq_t tr;
mpq_t trinv;
};
static int compare_result(const void *a_, const void *b_)
{
int d = 0;
struct result **a = (struct result **)a_;
struct result **b = (struct result **)b_;
d = mpq_cmp((*a)->tr,(*b)->tr);
if(d == 0)
d = mpq_cmp((*a)->trinv, (*b)->trinv);
return d;
}
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(group_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 < group->size; i++) {
if(group->elements[i].length % 2 != 0)
continue;
if(!group->elements[i].inverse)
continue;
int parent = group->elements[i].parent->id;
int grandparent = group->elements[i].parent->parent->id;
int letter;
if(group->elements[parent].letter == 1 && group->elements[i].letter == 2)
letter = 0; // p = bc
else if(group->elements[parent].letter == 2 && group->elements[i].letter == 0)
letter = 1; // q = ca
else if(group->elements[parent].letter == 0 && group->elements[i].letter == 1)
letter = 2; // r = ab
if(group->elements[parent].letter == 2 && group->elements[i].letter == 1)
letter = 3; // p^{-1} = cb
else if(group->elements[parent].letter == 0 && group->elements[i].letter == 2)
letter = 4; // q^{-1} = ac
else if(group->elements[parent].letter == 1 && group->elements[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(group_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 < group->size; i++) {
if(group->elements[i].length % 2 != 0 || !group->elements[i].inverse)
continue;
mat_trace(tr, matrices[i]);
mat_trace(trinv, matrices[group->elements[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->elements[i].length, print_word(&group->elements[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;
group_t *group;
int index;
mps_context *solver;
int acc = 100;
int n, nuniq, nmax;
int retval;
double evs[3];
double max_slope;
char buf[100];
char buf2[100];
struct result *invariants;
struct result **distinct_invariants;
nmax = atoi(argv[1]);
DEBUG("Allocate\n");
mpq_inits(s, q, t, tmp, NULL);
matrices = malloc(nmax*sizeof(mat));
for(int i = 0; i < nmax; i++)
mat_init(matrices[i], 3);
invariants = malloc(nmax*sizeof(struct result));
distinct_invariants = malloc(nmax*sizeof(struct result));
for(int i = 0; i < nmax; i++) {
mpq_init(invariants[i].tr);
mpq_init(invariants[i].trinv);
distinct_invariants[i] = &invariants[i];
}
solver = mps_context_new();
mps_context_set_output_prec(solver, 20); // relative precision
mps_context_set_output_goal(solver, MPS_OUTPUT_GOAL_APPROXIMATE);
DEBUG("Approximate parameters\n");
// get approximate s and q values
sapprox = atof(argv[2]);
tapprox = atof(argv[3]);
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);
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);
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);
#ifdef OUTPUT_POINTS
gmp_fprintf(stdout, "\"s = %Qd = %.3f, q = %Qd, t = %.3f\"\n", s, mpq_get_d(s), q, mpq_get_d(q)*tqfactor);
#endif
// group
DEBUG("Generate group\n");
group = coxeter_init_triangle(4, 3, 3, nmax);
DEBUG("Compute matrices\n");
enumerate(group, matrices, s, q);
n = 0;
DEBUG("Compute traces\n");
for(int i = 0; i < nmax; i++) {
if(group->elements[i].length % 2 != 0 || !group->elements[i].inverse)
continue;
mat_trace(invariants[i].tr, matrices[i]);
mat_trace(invariants[i].trinv, matrices[group->elements[i].inverse->id]);
distinct_invariants[n++] = &invariants[i];
}
DEBUG("Get unique traces\n");
qsort(distinct_invariants, n, sizeof(struct result*), compare_result);
nuniq = 0;
for(int i = 0; i < n; i++) {
if(i == 0 || compare_result(&distinct_invariants[i], &distinct_invariants[nuniq-1]) != 0)
distinct_invariants[nuniq++] = distinct_invariants[i];
}
max_slope = 0;
int max_slope_index;
DEBUG("Solve characteristic polynomials\n");
for(int i = 0; i < nuniq; i++) {
retval = solve_characteristic_polynomial(solver, distinct_invariants[i]->tr, distinct_invariants[i]->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]);
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 %d\n", distinct_invariants[i]->tr, distinct_invariants[i]->trinv, x, y, y/x);
#endif
}
#ifdef OUTPUT_SUMMARY
fprintf(stdout, "%.3f %.3f %f %s\n", mpq_get_d(s), mpq_get_d(q)*tqfactor, max_slope, print_word(&group->elements[max_slope_index], buf));
#endif
// output_invariants(group, matrices, s, q, solver);
// for(int i = 0; i < 10; i++) {
// mpq_set_ui(t,100+i,100);
// mpq_canonicalize(t);
//printf("%f %f\n", mpq_get_d(t), max_slope(group, matrices, s, t, &index));
// }
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(s, q, t, tmp, NULL);
mps_context_free(solver);
}
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