missing files from previous commit + clean up of special_element program + billiard picture script

billiard_picture.sh

@@ -0,0 +1,20 @@
+#!/bin/bash
+
+trap 'exit 130' INT
+
+wordlength=30
+sdenom=100
+sstart=1
+send=100
+qdenom=100
+qstart=1
+qend=100  # 1/sqrt(2) = 0.7071...
+
+words="$(./billiard_words $wordlength | awk '{print $1}')"
+
+for s in $(seq $sstart $send); do
+	for q in $(seq $qstart $qend); do
+		echo -n "$s/$sdenom $q/$qdenom "
+		MAXIMUM=only ./special_element $s/$sdenom $q/$qdenom $words
+	done
+done

billiard_words.hs

@@ -9,14 +9,23 @@ main = do
 
 listWordsUpToLength :: Int -> IO ()
 listWordsUpToLength n = do
-  putStrLn $ unlines [printf "%s %d/%d %f" w (x `div` gcd x y) (y `div` gcd x y) (fromIntegral x / fromIntegral y :: Double) | ((p,q),w) <- wordlist (n`div`2) (n`div`2), length w <= n, let x = 2*q + p, let y = 2*p + q]
+  putStrLn $ unlines [printf "%s %d/%d %f"
---  putStrLn $ unlines [printf "%d/%d\t%d/%d\t%.7f\t%d\t%s" p q (x `div` gcd x y) (y `div` gcd x y) (sqrt 3 / (1 + 2*fromIntegral q / fromIntegral p) :: Double) (length w) w | ((p,q),w) <- wordlist (n`div`2) (n`div`2), length w <= n, let x = 2*q + p, let y = 2*p + q]
+                             w
-
+                             (x `div` gcd x y)
-
+                             (y `div` gcd x y)
-  --  putStrLn $ unlines [printf "%d/%d\t%.5f\t%.5f\t%d\t%s" p q (fromIntegral p / fromIntegral q :: Double) (sqrt 3 / (1 + 2*fromIntegral q / fromIntegral p) :: Double) (length w) w | ((p,q),w) <- wordlist (n`div`2) (n`div`2), length w <= n]
+                             (fromIntegral x / fromIntegral y :: Double) |
+                      ((p,q),w) <- wordlist (n `div` 2) (n `div` 2),
+                      length w <= n,
+                      let x = 2*q + p,
+                      let y = 2*p + q]
 
 wordlist :: Int -> Int -> [((Int,Int),String)]
-wordlist pmax qmax = nub $ sortBy (comparing sl) [((p `div` gcd p q, q `div` gcd p q), slopeWord "bca" p q) | p <- [0..pmax], q <- [0..qmax], p /= 0 || q /= 0]
+wordlist pmax qmax = nub $
+                     sortBy (comparing sl)
+                     [((p `div` gcd p q, q `div` gcd p q), slopeWord "bca" p q) |
+                      p <- [0..pmax],
+                      q <- [0..qmax],
+                      q /= 0]   -- use p /= 0 || q /= 0 for more symmetric output
     where
       sl ((p,q),_) = fromIntegral p / fromIntegral q
 

enumerate_triangle_group.c

@@ -0,0 +1,254 @@
+#include "enumerate_triangle_group.h"
+#include "linalg.h"
+
+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);
+}
+
+// p1,p2,p3 are only allowed to be 2,3,4,6
+void initialize_triangle_generators(mat_workspace *ws, mat *gen, int p1, int p2, int p3, 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);
+
+	// coefficient 2 is the value for p=infinity, not sure if that would even work
+	quartic(rho1, s, 0, 0, 1, p1 == 2 ? -2 : p1 == 3 ? -1 : p1 == 4 ? 0 : p1 == 6 ? 1 : 2, 1);
+	quartic(rho2, s, 0, 0, 1, p2 == 2 ? -2 : p2 == 3 ? -1 : p2 == 4 ? 0 : p2 == 6 ? 1 : 2, 1);
+	quartic(rho3, s, 0, 0, 1, p3 == 2 ? -2 : p3 == 3 ? -1 : p3 == 4 ? 0 : p3 == 6 ? 1 : 2, 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, int p1, int p2, int p3, 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, p1, p2, p3, 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);
+}

enumerate_triangle_group.h

@@ -0,0 +1,17 @@
+#ifndef ENUMERATE_TRIANGLE_GROUP_H
+#define ENUMERATE_TRIANGLE_GROUP_H
+
+#include "mat.h"
+#include "coxeter.h"
+
+#include <mps/mps.h>
+
+int solve_characteristic_polynomial(mps_context *solv, mpq_t tr, mpq_t trinv, double *eigenvalues);
+void continued_fraction_approximation(mpq_t out, double in, int level);
+void quartic(mpq_t out, mpq_t in, int a, int b, int c, int d, int e);
+// p1,p2,p3 are only allowed to be 2,3,4,6
+void initialize_triangle_generators(mat_workspace *ws, mat *gen, int p1, int p2, int p3, mpq_t s, mpq_t q);
+char *print_word(groupelement_t *g, char *str);
+void enumerate(group_t *group, mat *matrices, int p1, int p2, int p3, mpq_t s, mpq_t t);
+
+#endif

special_element.c

@@ -5,11 +5,6 @@
 
 #define SWAP(t,x,y) do { t _tmp = (x); (x) = (y); (y) = _tmp; } while (0);
 #define DEBUG(msg, ...)
-#define DENOMINATOR 300
-#define WIDTH 135
-#define STARTX 121
-#define HEIGHT 300
-#define IDX(i,j) (((i)-1)*HEIGHT + ((j)-1))
 
 double mpq_log(mpq_t m_op)
 {
@@ -45,10 +40,25 @@ int main(int argc, char *argv[])
 	int retval;
 	double evs[3];
 	char buf[100];
-	double *max_slope;
+	double max_slope = 0;
-	int *max_slope_index;
+	int max_slope_index = 0;
+	double min_slope = INFINITY;
+	int min_slope_index = 0;
+	char *env;
+	int mode;
 
-	DEBUG("Allocate\n");
+	if(argc < 2) {
+		fprintf(stderr,
+		        "Usage: %s <s> <q> <word1> <word2> ...\n"
+		        "Computes jordan slopes of a list of group elements for a fixed representation.\n"
+		        "s,q: representation in the Hitchin component, given as rational numbers, e.g. 2/7\n"
+		        "word1, word2, ...: elements in the triangle rotation group, given as reflection group words\n"
+		        "output: word - jordan slope pairs\n"
+		        "+max slope index, max slope value, max slope word, min slope index, min slope value, min slope word\n"
+		        "controlled by environment variable MAXIMUM=no/yes/only, default yes\n",
+		        argv[0]);
+		exit(0);
+	}
 
 	mpq_inits(m, t, s, q, tmp, tmp2, tr, trinv, NULL);
 	ws = mat_workspace_init(3);
@@ -56,10 +66,6 @@ int main(int argc, char *argv[])
 		mat_init(gen[i], 3);
 	mat_init(element, 3);
 	mat_init(inverse, 3);
-	max_slope = malloc(sizeof(double)*WIDTH*HEIGHT);
-	max_slope_index = malloc(sizeof(int)*WIDTH*HEIGHT);
-	memset(max_slope_index, 0, sizeof(int)*WIDTH*HEIGHT);
-	memset(max_slope, 0, sizeof(int)*WIDTH*HEIGHT);
 
 	solver = mps_context_new();
 	mps_context_set_output_prec(solver, 20); // relative precision
@@ -68,21 +74,23 @@ int main(int argc, char *argv[])
 	mpq_set_str(s, argv[1], 10);
 	mpq_set_str(q, argv[2], 10);
 
-//	for(int i = STARTX; i <= WIDTH; i++) {
+	env = getenv("MAXIMUM");
-//		for(int j = 1; j <= HEIGHT; j++) {
+	if(!env || strcmp(env, "yes") == 0) {
-			for(int w = 3; w < argc; w++) {
+		mode = 1; // yes
-//				mpq_set_ui(t, j, DENOMINATOR);
+	} else if(strcmp(env, "no") == 0) {
-//				mpq_set_ui(m, i, DENOMINATOR); // 414/1000 ~ sqrt(2)-1 <-> s=1
+		mode = 0; // no
-//				s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m));
+	} else if(strcmp(env, "only") == 0) {
+		mode = 2; // only
+	}
 
-				DEBUG("Compute matrix\n");
+	for(int w = 0; w < argc - 3; w++) {
-				initialize_triangle_generators(ws, gen, 6, 4, 3, s, q);
+		initialize_triangle_generators(ws, gen, 4, 4, 4, s, q);
 
 		mat_identity(element);
 		mat_identity(inverse);
-				for(int k = 0; k < strlen(argv[w]); k+=2) {
+		for(int k = 0; k < strlen(argv[w+3]); k+=2) {
-					letter1 = argv[w][k] - 'a';
+			letter1 = argv[w+3][k] - 'a';
-					letter2 = argv[w][k+1] - 'a';
+			letter2 = argv[w+3][k+1] - 'a';
 
 			if(letter1 == 1 && letter2 == 2)
 				letter = 0; // p = bc
@@ -101,12 +109,9 @@ int main(int argc, char *argv[])
 			mat_multiply(ws, inverse, gen[(letter+3)%6], inverse);
 		}
 
-				DEBUG("Compute traces\n");
-
 		mat_trace(tr, element);
 		mat_trace(trinv, inverse);
 
-				DEBUG("Solve characteristic polynomials\n");
 		retval = solve_characteristic_polynomial(solver, tr, trinv, evs);
 		if(retval == 1) {
 			fprintf(stderr, "Error! Could not solve polynomial.\n");
@@ -134,26 +139,26 @@ int main(int argc, char *argv[])
 		if(slope < 1)
 			slope = 1/slope;
 
-
+		if(slope > max_slope) {
-//				if(slope > max_slope[IDX(i,j)]) {
+			max_slope = slope;
-//					max_slope[IDX(i,j)] = slope;
+			max_slope_index = w;
-//					max_slope_index[IDX(i,j)] = w;
-//				}
-
-//	gmp_printf("%Qd %Qd %f %f %f\n", tr, trinv, x, y, y/x);
-//				gmp_printf("%s %.5f %.5f %Qd %Qd %.9f %.9f %.9f\n", argv[w], mpq_get_d(s), mpq_get_d(q),
-//				           tr, trinv,
-//				           x, y, slope);
-				gmp_printf("%s %.9f\n", argv[w], slope);
-
 		}
 
-//			printf("%.5f %.5f %d %.9f\n", (double)i/DENOMINATOR, (double)j/DENOMINATOR, max_slope_index[IDX(i,j)], max_slope[IDX(i,j)]);
+		if(slope < min_slope) {
-			fflush(stdout);
+			min_slope = slope;
-//		}
+			min_slope_index = w;
-//	}
+		}
+
+		if(mode != 2)
+			gmp_printf("%s %.9f\n", argv[w+3], slope);
+	}
+
+	if(mode != 0)
+		printf("%d %.9f %s %d %.9f %s\n",
+		       max_slope_index, max_slope, argv[max_slope_index+3],
+		       min_slope_index, min_slope, argv[min_slope_index+3]);
+	fflush(stdout);
 
-	DEBUG("Clean up\n");
 	mpq_clears(m, t, s, q, tmp, tmp2, tr, trinv, NULL);
 	mat_workspace_clear(ws);
 	for(int i = 0; i < 6; i++)
@@ -161,6 +166,4 @@ int main(int argc, char *argv[])
 	mat_clear(element);
 	mat_clear(inverse);
 	mps_context_free(solver);
-	free(max_slope);
-	free(max_slope_index);
 }