draw character variety picture

Makefile

@@ -7,14 +7,20 @@ SPECIAL_OPTIONS=-O3 -flto -funroll-loops -Winline
 
 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
 	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)
 	gcc $(OPTIONS) -c singular_values.c
 
+special_element.o: special_element.c $(HEADERS)
+	gcc $(OPTIONS) -c special_element.c
+
 linalg.o: linalg.c $(HEADERS)
 	gcc $(OPTIONS) -c linalg.c
 
@@ -25,4 +31,4 @@ mat.o: mat.c $(HEADERS)
 	gcc $(OPTIONS) -c mat.c
 
 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

singular_values.c

@@ -6,11 +6,26 @@
 #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, ...) do { print_time(); fprintf(stderr, msg, ##__VA_ARGS__); } while (0);
-#define DEBUG(msg, ...)
+//#define DEBUG(msg, ...)
 
-#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 {
 	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);
 }
 
-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[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);
+	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)
@@ -232,7 +223,7 @@ char *print_word(groupelement_t *g, char *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 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(tmp, 3);
 
-	initialize_triangle_generators(ws, gen, s, t);
+	initialize_triangle_generators(ws, gen, m, t);
 
 	mat_identity(matrices[0]);
 	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[])
 {
-	mpq_t s, q, t, tmp;
+	mpq_t m, t, tmp;
-	double sapprox, tapprox, qapprox, tqfactor;
+	double s;
 	mat *matrices;
 	group_t *group;
 	int index;
@@ -369,11 +360,13 @@ int main(int argc, char *argv[])
 	struct result *invariants;
 	struct result **distinct_invariants;
 
+    clock_gettime(CLOCK_REALTIME, &starttime);
+
 	nmax = atoi(argv[1]);
 
 	DEBUG("Allocate\n");
 
-	mpq_inits(s, q, t, tmp, NULL);
+	mpq_inits(m, t, tmp, NULL);
 	matrices = malloc(nmax*sizeof(mat));
 	for(int i = 0; i < nmax; i++)
 		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_goal(solver, MPS_OUTPUT_GOAL_APPROXIMATE);
 
+	/*
 	DEBUG("Approximate parameters\n");
 
 	// get approximate s and q values
@@ -420,13 +414,22 @@ int main(int argc, char *argv[])
 #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);
+	group = coxeter_init_triangle(4, 4, 4, nmax);
+
+	fprintf(stderr, "max word length = %d\n", group->elements[nmax-1].length);
+
+	for(int ttick = 45; ttick <= 65; ttick++) {
+		for(int mtick = 45; mtick < 65; mtick++) {
+			mpq_set_ui(t, ttick, 100);
+			mpq_set_ui(m, mtick, 100); // 414/1000 ~ sqrt(2)-1 <-> s=1
+			s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m));
 
 			DEBUG("Compute matrices\n");
-	enumerate(group, matrices, s, q);
+			enumerate(group, matrices, m, t);
 
 			n = 0;
 			DEBUG("Compute traces\n");
@@ -438,6 +441,8 @@ int main(int argc, char *argv[])
 				mat_trace(invariants[i].trinv, matrices[group->elements[i].inverse->id]);
 
 				distinct_invariants[n++] = &invariants[i];
+
+//		gmp_printf("%Qd %Qd %d %s\n", invariants[i].tr, invariants[i].trinv, i, print_word(&group->elements[i], buf));
 			}
 
 			DEBUG("Get unique traces\n");
@@ -479,22 +484,16 @@ int main(int argc, char *argv[])
 				}
 
 #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
 			}
 
 #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
-
+		}
-//	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++) {
@@ -507,6 +506,6 @@ int main(int argc, char *argv[])
 		mat_clear(matrices[i]);
 	free(matrices);
 	coxeter_clear(group);
-	mpq_clears(s, q, t, tmp, NULL);
+	mpq_clears(m, t, tmp, NULL);
 	mps_context_free(solver);
 }