florian/triangle_reflection_complex
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

draw character variety picture

Browse Source
This commit is contained in:
Florian Stecker
2021-03-31 22:45:25 -05:00
parent 7656408f57
commit b5bd9d398f
2 changed files with 148 additions and 143 deletions
Download Patch File Download Diff File Expand all files Collapse all files

10
Makefile
View File

@@ -7,14 +7,20 @@ SPECIAL_OPTIONS=-O3 -flto -funroll-loops -Winline
OPTIONS=-m64 -march=native -mtune=native -std=gnu99 -D_GNU_SOURCE $(SPECIAL_OPTIONS) OPTIONS=-m64 -march=native -mtune=native -std=gnu99 -D_GNU_SOURCE $(SPECIAL_OPTIONS)
all: singular_values all: singular_values special_element
singular_values: singular_values.o coxeter.o linalg.o mat.o singular_values: singular_values.o coxeter.o linalg.o mat.o
gcc $(OPTIONS) -o singular_values coxeter.o linalg.o singular_values.o mat.o -lm -lgsl -lcblas -lgmp -lmps -lpthread gcc $(OPTIONS) -o singular_values coxeter.o linalg.o singular_values.o mat.o -lm -lgsl -lcblas -lgmp -lmps -lpthread
special_element: special_element.o coxeter.o linalg.o mat.o
gcc $(OPTIONS) -o special_element coxeter.o linalg.o special_element.o mat.o -lm -lgsl -lcblas -lgmp -lmps -lpthread
singular_values.o: singular_values.c $(HEADERS) singular_values.o: singular_values.c $(HEADERS)
gcc $(OPTIONS) -c singular_values.c gcc $(OPTIONS) -c singular_values.c
special_element.o: special_element.c $(HEADERS)
gcc $(OPTIONS) -c special_element.c
linalg.o: linalg.c $(HEADERS) linalg.o: linalg.c $(HEADERS)
gcc $(OPTIONS) -c linalg.c gcc $(OPTIONS) -c linalg.c
@@ -25,4 +31,4 @@ mat.o: mat.c $(HEADERS)
gcc $(OPTIONS) -c mat.c gcc $(OPTIONS) -c mat.c
clean: clean:
rm -f singular_values coxeter.o linalg.o singular_values.o mat.o rm -f singular_values special_element coxeter.o linalg.o singular_values.o mat.o special_element.o

281
singular_values.c
View File

@@ -6,11 +6,26 @@
#include <mps/mps.h> #include <mps/mps.h>
#define SWAP(t,x,y) do { t _tmp = (x); (x) = (y); (y) = _tmp; } while (0); #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, ...) do { print_time(); fprintf(stderr, msg, ##__VA_ARGS__); } while (0);
#define DEBUG(msg, ...) //#define DEBUG(msg, ...)
#define OUTPUT_POINTS
//#define OUTPUT_POINTS //#define OUTPUT_POINTS
#define OUTPUT_SUMMARY
struct timespec starttime;
void print_time()
{
double diff;
struct timespec current;
clock_gettime(CLOCK_REALTIME, &current);
diff = (current.tv_sec - starttime.tv_sec) + (current.tv_nsec - starttime.tv_nsec)*1e-9;
fprintf(stderr, "[%.3f] ", diff);
}
struct result { struct result {
mpq_t tr; mpq_t tr;
@@ -124,99 +139,75 @@ void quartic(mpq_t out, mpq_t in, int a, int b, int c, int d, int e)
mpq_clear(tmp); mpq_clear(tmp);
} }
void initialize_triangle_generators(mat_workspace *ws, mat *gen, mpq_t s, mpq_t q) // this version is only for the (4,4,4) group
void initialize_triangle_generators(mat_workspace *ws, mat *gen, mpq_t m, mpq_t t)
{ {
mat r1,r2,r3; mpq_t s,sinv,q,x,y;
mpq_t rho1, rho2, rho3; mpq_t zero, one, two;
mpq_t b1,c1,a2,c2,a3,b3; mpq_t tmp;
mpq_t sinv;
mpq_inits(sinv,rho1,rho2,rho3,b1,c1,a2,c2,a3,b3,NULL); mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
mat_init(r1, 3); mpq_set_ui(zero, 0, 1);
mat_init(r2, 3); mpq_set_ui(one, 1, 1);
mat_init(r3, 3); mpq_set_ui(two, 2, 1);
mpq_set_ui(sinv, 1, 1); // s = (1-m^2)/2m
mpq_div(sinv, sinv, s); mpq_mul(s, m, m);
mpq_sub(s, one, s);
mpq_div(s, s, m);
mpq_div(s, s, two);
mpq_div(sinv, one, s);
quartic(rho1, s, 0, 0, 1, -1, 1); // q = (1+m^2)/(1-m^2) = 2/(1-m^2) - 1
quartic(rho2, s, 0, 0, 1, -1, 1); mpq_mul(q, m, m);
quartic(rho3, s, 0, 0, 1, 0, 1); mpq_sub(q, one, q);
mpq_div(q, two, q);
mpq_sub(q, q, one);
mpq_mul(c1, rho2, q); // x = -tq, y = -q/t
mpq_mul(a2, rho3, q); mpq_mul(x, q, t);
mpq_mul(b3, rho1, q); mpq_sub(x, zero, x);
mpq_set_ui(b1, 1, 1); mpq_div(y, q, t);
mpq_set_ui(c2, 1, 1); mpq_sub(y, zero, y);
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 // q^2 = xy = 1 + 1/s^2
mat_zero(r1); // [ -s s*y 0]
mat_zero(r2); // [ -s*x s*x*y - 1/s 0]
mat_zero(r3); // [ -s*y s*y^2 - x 1]
mpq_set_si(*mat_ref(r1, 0, 0), -1, 1); LOOP(i,3) {
mpq_set_si(*mat_ref(r1, 1, 1), 1, 1); mat_zero(gen[i]);
mpq_set_si(*mat_ref(r1, 2, 2), 1, 1); mpq_sub(tmp, zero, s);
mpq_set_si(*mat_ref(r2, 0, 0), 1, 1); mat_set(gen[i%3], i%3, i%3, tmp);
mpq_set_si(*mat_ref(r2, 1, 1), -1, 1); mpq_mul(tmp, s, y);
mpq_set_si(*mat_ref(r2, 2, 2), 1, 1); mat_set(gen[i%3], i%3, (i+1)%3, tmp);
mpq_set_si(*mat_ref(r3, 0, 0), 1, 1); mpq_mul(tmp, s, x);
mpq_set_si(*mat_ref(r3, 1, 1), 1, 1); mpq_sub(tmp, zero, tmp);
mpq_set_si(*mat_ref(r3, 2, 2), -1, 1); mat_set(gen[i%3], (i+1)%3, i%3, tmp);
mpq_mul(tmp, s, x);
mpq_mul(tmp, tmp, y);
mpq_sub(tmp, tmp, sinv);
mat_set(gen[i%3], (i+1)%3, (i+1)%3, tmp);
mpq_mul(tmp, s, y);
mpq_sub(tmp, zero, tmp);
mat_set(gen[i%3], (i+2)%3, i%3, tmp);
mpq_mul(tmp, s, y);
mpq_mul(tmp, tmp, y);
mpq_sub(tmp, tmp, x);
mat_set(gen[i%3], (i+2)%3, (i+1)%3, tmp);
mat_set(gen[i%3], (i+2)%3, (i+2)%3, one);
}
mpq_set(*mat_ref(r1, 1, 0), b1); LOOP(i,3) mat_pseudoinverse(ws, gen[i+3], gen[i]);
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]);
// debug output
/* /*
mat_print(r1); gmp_printf("m = %Qd, s = %Qd, t = %Qd, q = %Qd, x = %Qd, y = %Qd\n", m, s, t, q, x, y);
mat_print(r2);
mat_print(r3);
mat_print(gen[0]); mat_print(gen[0]);
mat_print(gen[1]); mat_print(gen[1]);
mat_print(gen[2]); 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); mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
mat_clear(r1);
mat_clear(r2);
mat_clear(r3);
} }
char *print_word(groupelement_t *g, char *str) char *print_word(groupelement_t *g, char *str)
@@ -232,7 +223,7 @@ char *print_word(groupelement_t *g, char *str)
return str; return str;
} }
void enumerate(group_t *group, mat *matrices, mpq_t s, mpq_t t) void enumerate(group_t *group, mat *matrices, mpq_t m, mpq_t t)
{ {
mat_workspace *ws; mat_workspace *ws;
mat tmp; mat tmp;
@@ -245,7 +236,7 @@ void enumerate(group_t *group, mat *matrices, mpq_t s, mpq_t t)
mat_init(gen[i], 3); mat_init(gen[i], 3);
mat_init(tmp, 3); mat_init(tmp, 3);
initialize_triangle_generators(ws, gen, s, t); initialize_triangle_generators(ws, gen, m, t);
mat_identity(matrices[0]); mat_identity(matrices[0]);
for(int i = 1; i < group->size; i++) { for(int i = 1; i < group->size; i++) {
@@ -352,8 +343,8 @@ double max_slope(groupelement_t *group, mat *matrices, mpq_t s, mpq_t t, int *in
int main(int argc, char *argv[]) int main(int argc, char *argv[])
{ {
mpq_t s, q, t, tmp; mpq_t m, t, tmp;
double sapprox, tapprox, qapprox, tqfactor; double s;
mat *matrices; mat *matrices;
group_t *group; group_t *group;
int index; int index;
@@ -369,11 +360,13 @@ int main(int argc, char *argv[])
struct result *invariants; struct result *invariants;
struct result **distinct_invariants; struct result **distinct_invariants;
clock_gettime(CLOCK_REALTIME, &starttime);
nmax = atoi(argv[1]); nmax = atoi(argv[1]);
DEBUG("Allocate\n"); DEBUG("Allocate\n");
mpq_inits(s, q, t, tmp, NULL); mpq_inits(m, t, tmp, NULL);
matrices = malloc(nmax*sizeof(mat)); matrices = malloc(nmax*sizeof(mat));
for(int i = 0; i < nmax; i++) for(int i = 0; i < nmax; i++)
mat_init(matrices[i], 3); mat_init(matrices[i], 3);
@@ -389,6 +382,7 @@ int main(int argc, char *argv[])
mps_context_set_output_prec(solver, 20); // relative precision mps_context_set_output_prec(solver, 20); // relative precision
mps_context_set_output_goal(solver, MPS_OUTPUT_GOAL_APPROXIMATE); mps_context_set_output_goal(solver, MPS_OUTPUT_GOAL_APPROXIMATE);
/*
DEBUG("Approximate parameters\n"); DEBUG("Approximate parameters\n");
// get approximate s and q values // get approximate s and q values
@@ -420,81 +414,86 @@ int main(int argc, char *argv[])
#ifdef OUTPUT_POINTS #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); gmp_fprintf(stdout, "\"s = %Qd = %.3f, q = %Qd, t = %.3f\"\n", s, mpq_get_d(s), q, mpq_get_d(q)*tqfactor);
#endif #endif
*/
// group // group
DEBUG("Generate group\n"); DEBUG("Generate group\n");
group = coxeter_init_triangle(4, 3, 3, nmax); group = coxeter_init_triangle(4, 4, 4, nmax);
DEBUG("Compute matrices\n"); fprintf(stderr, "max word length = %d\n", group->elements[nmax-1].length);
enumerate(group, matrices, s, q);
n = 0; for(int ttick = 45; ttick <= 65; ttick++) {
DEBUG("Compute traces\n"); for(int mtick = 45; mtick < 65; mtick++) {
for(int i = 0; i < nmax; i++) { mpq_set_ui(t, ttick, 100);
if(group->elements[i].length % 2 != 0 || !group->elements[i].inverse) mpq_set_ui(m, mtick, 100); // 414/1000 ~ sqrt(2)-1 <-> s=1
continue; s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m));
mat_trace(invariants[i].tr, matrices[i]); DEBUG("Compute matrices\n");
mat_trace(invariants[i].trinv, matrices[group->elements[i].inverse->id]); enumerate(group, matrices, m, t);
distinct_invariants[n++] = &invariants[i]; 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;
DEBUG("Get unique traces\n"); mat_trace(invariants[i].tr, matrices[i]);
mat_trace(invariants[i].trinv, matrices[group->elements[i].inverse->id]);
qsort(distinct_invariants, n, sizeof(struct result*), compare_result); distinct_invariants[n++] = &invariants[i];
nuniq = 0; // gmp_printf("%Qd %Qd %d %s\n", invariants[i].tr, invariants[i].trinv, i, print_word(&group->elements[i], buf));
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; DEBUG("Get unique traces\n");
int max_slope_index;
DEBUG("Solve characteristic polynomials\n"); qsort(distinct_invariants, n, sizeof(struct result*), compare_result);
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])) nuniq = 0;
SWAP(double, evs[0], evs[1]); for(int i = 0; i < n; i++) {
if(fabs(evs[1]) < fabs(evs[2])) if(i == 0 || compare_result(&distinct_invariants[i], &distinct_invariants[nuniq-1]) != 0)
SWAP(double, evs[1], evs[2]); distinct_invariants[nuniq++] = distinct_invariants[i];
if(fabs(evs[0]) < fabs(evs[1])) }
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]); 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 x = log(fabs(evs[0]));
double y = -log(fabs(evs[2])); double y = -log(fabs(evs[2]));
if(y/x > max_slope && (x > 0.1 || y > 0.1)) { if(y/x > max_slope && (x > 0.1 || y > 0.1)) {
max_slope_index = distinct_invariants[i] - invariants; max_slope_index = distinct_invariants[i] - invariants;
max_slope = y/x; max_slope = y/x;
} }
#ifdef OUTPUT_POINTS #ifdef OUTPUT_POINTS
gmp_printf("%Qd %Qd %f %f %f %d\n", distinct_invariants[i]->tr, distinct_invariants[i]->trinv, x, y, y/x); gmp_printf("%Qd %Qd %f %f %f\n", distinct_invariants[i]->tr, distinct_invariants[i]->trinv, x, y, y/x);
#endif #endif
} }
#ifdef OUTPUT_SUMMARY #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)); // 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 #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"); DEBUG("Clean up\n");
for(int i = 0; i < nmax; i++) { for(int i = 0; i < nmax; i++) {
@@ -507,6 +506,6 @@ int main(int argc, char *argv[])
mat_clear(matrices[i]); mat_clear(matrices[i]);
free(matrices); free(matrices);
coxeter_clear(group); coxeter_clear(group);
mpq_clears(s, q, t, tmp, NULL); mpq_clears(m, t, tmp, NULL);
mps_context_free(solver); mps_context_free(solver);
} }