diff --git a/.gitignore b/.gitignore
index ebf22f7..1f23877 100644
--- a/.gitignore
+++ b/.gitignore
@@ -2,6 +2,7 @@
 triangle_group/singular_values
 .#*
 singular_values
+singular_values_mpi
 output/
 special_element
 max_slope_picture/generate
diff --git a/Makefile b/Makefile
index 0ff4f43..a88cd22 100644
--- a/Makefile
+++ b/Makefile
@@ -8,7 +8,7 @@ SPECIAL_OPTIONS=-O3 -pg -funroll-loops -fno-inline
 
 OPTIONS=-I../mps/include -L../mps/lib -pthread -m64 -std=gnu99 -D_GNU_SOURCE $(SPECIAL_OPTIONS)
 
-all: singular_values special_element convert billiard_words
+all: singular_values special_element singular_values_mpi convert billiard_words
 
 convert: convert.hs
 	ghc --make -dynamic convert.hs
@@ -17,13 +17,19 @@ billiard_words: billiard_words.hs
 	ghc --make -dynamic billiard_words.hs
 
 singular_values: singular_values.o coxeter.o mat.o
-	mpicc $(OPTIONS) -o singular_values coxeter.o singular_values.o mat.o -lm -lgmp -lmps
+	gcc $(OPTIONS) -o singular_values coxeter.o singular_values.o mat.o -lm -lgmp -lmps
+
+singular_values_mpi: singular_values_mpi.o coxeter.o mat.o
+	mpicc $(OPTIONS) -o singular_values_mpi coxeter.o singular_values_mpi.o mat.o -lm -lgmp -lmps
 
 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 -lgmp -lmps -lgsl -lcblas
 
 singular_values.o: singular_values.c $(HEADERS)
-	mpicc $(OPTIONS) -c singular_values.c
+	gcc $(OPTIONS) -c singular_values.c
+
+singular_values_mpi.o: singular_values_mpi.c $(HEADERS)
+	mpicc $(OPTIONS) -c singular_values_mpi.c
 
 special_element.o: special_element.c $(HEADERS)
 	gcc $(OPTIONS) -c special_element.c
@@ -38,4 +44,4 @@ mat.o: mat.c $(HEADERS)
 	gcc $(OPTIONS) -c mat.c
 
 clean:
-	rm -f singular_values special_element coxeter.o linalg.o singular_values.o mat.o special_element.o convert.hi convert.o convert billiard_words.hi billiard_words.o billiard_words
+	rm -f singular_values special_element singular_values_mpi coxeter.o linalg.o singular_values.o singular_values_mpi.o mat.o special_element.o convert.hi convert.o convert billiard_words.hi billiard_words.o billiard_words
diff --git a/cdf.plt b/cdf.plt
new file mode 100644
index 0000000..f6b68a2
--- /dev/null
+++ b/cdf.plt
@@ -0,0 +1,57 @@
+#if(!exists("logt")) logt = log(1.80)
+if(!exists("n")) n = 263
+if(!exists("logt")) logt = log(1)
+if(!exists("logs")) logs = log(1)
+
+#logt = 0.01*n
+logt = log(1000000000)
+
+file = sprintf("< ./singular_values 713698 %f %f", exp(logs), exp(logt))
+#file = sprintf("< ./singular_values 1621 %f %f", exp(logs), exp(logt))
+#outfile = sprintf("cdf/cdf_hires_%05d.png", n)
+outfile = sprintf("cdf/cdf_hires_limit.png")
+
+set log x
+set zeroaxis
+set samples 1000
+set size square
+set xrange [0.5:2]
+set yrange [0:500000]
+#set yrange [0:1000]
+set trange [0:30]
+set grid
+set parametric
+
+set terminal pngcairo enhanced size 1024, 1024
+set output outfile
+print sprintf("n = %d, t = %.2f", n, exp(logt))
+
+# plot file using 2:3 w p pt 7 ps 0.5 lc 1 t title
+
+#tr(a,b) = exp((2*a+b)/3) + exp((b-a)/3) + exp(-(a+2*b)/3)
+#trinv(a,b) = exp(-(2*a+b)/3) + exp((a-b)/3) + exp((a+2*b)/3)
+
+tr(a,b) = exp(a) + exp(b-a) + exp(-b)
+trinv(a,b) = exp(-a) + exp(a-b) + exp(b)
+
+#plot file using 6:7 w p pt 7 ps 0.5 lc 1 t columnheader,
+#	 log(tr(t,t*2)),log(trinv(t,2*t)) w l lw 2 t "", \
+#	 log(tr(t,t/2)),log(trinv(t,t/2)) w l lw 2 t ""
+
+plot file using 8:3 w steps lw 2 lc 1 t sprintf("t = %.2f", exp(logt))
+
+#plot for[i=-10:10] log(tr(t,t*exp(log(2)*i/10.0))),log(trinv(t,t*exp(log(2)*i/10.0))) w l lw 2 t ""
+
+#plot for[i=-10:10] t,log(tr(t,t*exp(log(2)*i/10.0)))-t w l lw 2 t ""
+
+##plot for[i=20:20] t,log(tr(1/t,exp(2*log(2)*i/20.0-log(2)))) w l lw 2 t ""
+
+#n=n+1
+#if(n < 1000) reread
+
+# pause mouse keypress
+# if(MOUSE_KEY == 60) logt=logt-0.02
+# if(MOUSE_KEY == 62) logt=logt+0.02
+# if(MOUSE_KEY == 44) logs=logs-0.02
+# if(MOUSE_KEY == 46) logs=logs+0.02
+# if(MOUSE_KEY != 113) reread
diff --git a/singular_values.c b/singular_values.c
index 0f9e1e4..973fe6c 100644
--- a/singular_values.c
+++ b/singular_values.c
@@ -1,113 +1,26 @@
 #include "coxeter.h"
-//#include "linalg.h"
+#include "linalg.h"
 #include "mat.h"
 
-//#include <gsl/gsl_poly.h>
+#include <gsl/gsl_poly.h>
 #include <mps/mps.h>
-#include <mpi.h>
-#include <sys/stat.h>
-#include <sys/mman.h>
-#include <fcntl.h>
-#include <errno.h>
-#include <string.h>
-#include <unistd.h>
 
-#define MIN(x,y) ((x)<(y)?(x):(y))
 #define SWAP(t,x,y) do { t _tmp = (x); (x) = (y); (y) = _tmp; } while (0);
-#define DEBUG(msg, ...) do { print_time(); fprintf(stderr, msg, ##__VA_ARGS__); } while (0);
-//#define DEBUG(msg, ...)
+//#define DEBUG(msg, ...) fprintf(stderr, msg, ##__VA_ARGS__)
+#define DEBUG(msg, ...)
 
-#define TDIV 10
-#define TFROM 1
-#define TTO 9
-#define MDIV 10
-#define MFROM 1
-#define MTO 9
-#define JOBNR(t,m) (((t)-TFROM) + ((m)-MFROM)*(TTO-TFROM+1))
-#define NJOBS ((TTO-TFROM+1)*(MTO-MFROM+1))
-#define FLUSH_INTERVAL 100
-
-enum message_tag {
-	JOB_ORDER,
-	JOB_RESULT,
-	JOB_SHUTDOWN,
-};
-
-struct job {
-	int tparam, mparam;
-	int done;
-	double max_slope;
-	double time;
-};
+#define OUTPUT_POINTS
+//#define OUTPUT_POINTS
 
 struct result {
+	int id;
+	int count;
 	mpq_t tr;
 	mpq_t trinv;
+	double x;
+	double y;
 };
 
-struct global_data {
-	int n;
-	group_t *group;
-	mat* matrices;
-	struct result *invariants;
-	struct result **distinct_invariants;
-	mps_context *solver;
-};
-
-
-struct timespec starttime;
-char processor_name[MPI_MAX_PROCESSOR_NAME];
-int world_rank;
-int world_size;
-MPI_Datatype job_datatype;
-
-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, "[%04d %.3f] ", world_rank, diff);
-}
-
-static struct global_data allocate_global_data(int n)
-{
-	struct global_data result;
-	result.n = n;
-	result.matrices = malloc(n*sizeof(mat));
-	for(int i = 0; i < n; i++)
-		mat_init(result.matrices[i], 3);
-	result.invariants = malloc(n*sizeof(struct result));
-	result.distinct_invariants = malloc(n*sizeof(struct result*));
-	for(int i = 0; i < n; i++) {
-		mpq_init(result.invariants[i].tr);
-		mpq_init(result.invariants[i].trinv);
-		result.distinct_invariants[i] = &result.invariants[i];
-	}
-	result.solver = mps_context_new();
-	mps_context_set_output_prec(result.solver, 20); // relative precision
-	mps_context_set_output_goal(result.solver, MPS_OUTPUT_GOAL_APPROXIMATE);
-
-	return result;
-}
-
-void free_global_data(struct global_data dat)
-{
-	for(int i = 0; i < dat.n; i++)
-		mat_clear(dat.matrices[i]);
-	free(dat.matrices);
-	for(int i = 0; i < dat.n; i++) {
-		mpq_clear(dat.invariants[i].tr);
-		mpq_clear(dat.invariants[i].trinv);
-	}
-	free(dat.invariants);
-	free(dat.distinct_invariants);
-	mps_context_free(dat.solver);
-}
-
 static int compare_result(const void *a_, const void *b_)
 {
 	int d = 0;
@@ -116,12 +29,45 @@ static int compare_result(const void *a_, const void *b_)
 	struct result **b = (struct result **)b_;
 
 	d = mpq_cmp((*a)->tr,(*b)->tr);
-	if(d == 0)
+	if(d == 0) {
 		d = mpq_cmp((*a)->trinv, (*b)->trinv);
+	}
 
 	return d;
 }
 
+static int compare_result_with_id(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);
+		if(d == 0) {
+			d = (*b)->id - (*a)->id;
+		}
+	}
+
+	return d;
+}
+
+static int compare_result_by_slope(const void *a_, const void *b_)
+{
+	int d = 0;
+
+	struct result **a = (struct result **)a_;
+	struct result **b = (struct result **)b_;
+
+	double slopea = (*a)->x / (*a)->y;
+	double slopeb = (*b)->x / (*b)->y;
+
+	return slopea > slopeb ? -1 : slopea < slopeb ? 1 : 0;
+}
+
+
 int solve_characteristic_polynomial(mps_context *solv, mpq_t tr, mpq_t trinv, double *eigenvalues)
 {
 	mpq_t coeff1, coeff2, zero;
@@ -215,75 +161,99 @@ void quartic(mpq_t out, mpq_t in, int a, int b, int c, int d, int e)
 	mpq_clear(tmp);
 }
 
-// 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)
+void initialize_triangle_generators(mat_workspace *ws, mat *gen, mpq_t s, mpq_t q)
 {
-	mpq_t s,sinv,q,x,y;
-	mpq_t zero, one, two;
-	mpq_t tmp;
+	mat r1,r2,r3;
+	mpq_t rho1, rho2, rho3;
+	mpq_t b1,c1,a2,c2,a3,b3;
+	mpq_t sinv;
 
-	mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
-	mpq_set_ui(zero, 0, 1);
-	mpq_set_ui(one, 1, 1);
-	mpq_set_ui(two, 2, 1);
+	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);
 
-	// s = (1-m^2)/2m
-	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);
+	mpq_set_ui(sinv, 1, 1);
+	mpq_div(sinv, sinv, s);
 
-	// q = (1+m^2)/(1-m^2) = 2/(1-m^2) - 1
-	mpq_mul(q, m, m);
-	mpq_sub(q, one, q);
-	mpq_div(q, two, q);
-	mpq_sub(q, q, one);
+	quartic(rho1, s, 0, 0, 1, -1, 1);
+	quartic(rho2, s, 0, 0, 1, -1, 1);
+	quartic(rho3, s, 0, 0, 1,  0, 1);
 
-	// x = -tq, y = -q/t
-	mpq_mul(x, q, t);
-	mpq_sub(x, zero, x);
-	mpq_div(y, q, t);
-	mpq_sub(y, zero, y);
+	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);
 
-	// q^2 = xy = 1 + 1/s^2
-	// [         -s         s*y           0]
-	// [       -s*x s*x*y - 1/s           0]
-	// [       -s*y   s*y^2 - x           1]
-	LOOP(i,3) {
-		mat_zero(gen[i]);
-		mpq_sub(tmp, zero, s);
-		mat_set(gen[i%3], i%3, i%3, tmp);
-		mpq_mul(tmp, s, y);
-		mat_set(gen[i%3], i%3, (i+1)%3, tmp);
-		mpq_mul(tmp, s, x);
-		mpq_sub(tmp, zero, tmp);
-		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);
-	}
+	// 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);
 
-	LOOP(i,3) mat_pseudoinverse(ws, gen[i+3], gen[i]);
+	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]);
 
-	// debug output
 	/*
-	gmp_printf("m = %Qd, s = %Qd, t = %Qd, q = %Qd, x = %Qd, y = %Qd\n", m, s, t, q, x, y);
+	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_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
+	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)
@@ -299,7 +269,7 @@ char *print_word(groupelement_t *g, char *str)
   return str;
 }
 
-void enumerate(group_t *group, mat *matrices, mpq_t m, mpq_t t)
+void enumerate(group_t *group, mat *matrices, mpq_t s, mpq_t t)
 {
 	mat_workspace *ws;
 	mat tmp;
@@ -312,7 +282,7 @@ void enumerate(group_t *group, mat *matrices, mpq_t m, mpq_t t)
 		mat_init(gen[i], 3);
 	mat_init(tmp, 3);
 
-	initialize_triangle_generators(ws, gen, m, t);
+	initialize_triangle_generators(ws, gen, s, t);
 
 	mat_identity(matrices[0]);
 	for(int i = 1; i < group->size; i++) {
@@ -348,55 +318,194 @@ void enumerate(group_t *group, mat *matrices, mpq_t m, mpq_t t)
 	mat_workspace_clear(ws);
 }
 
-
-double compute_max_slope(struct global_data dat, mpq_t t, mpq_t m)
+void output_invariants(group_t *group, mat *matrices, mpq_t s, mpq_t q, mps_context *solver)
 {
-//	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));
+	mpq_t tr, trinv;
+	char buf[100];
+	double evs[3];
+	int retval;
 
-	int n = 0;
-	int nmax = dat.n;
-	int nuniq;
-	double max_slope;
+	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];
 
-	group_t *group = dat.group;
-	mat *matrices = dat.matrices;
-	struct result *invariants = dat.invariants;
-	struct result **distinct_invariants = dat.distinct_invariants;
-	mps_context *solver = dat.solver;
+	struct result *invariants;
+	struct result **distinct_invariants;
 
-//	DEBUG("Compute matrices\n");
-	enumerate(group, matrices, m, t);
+	if(argc < 4) {
+		fprintf(stderr, "Usage: %s <N> <s> <t>\n", argv[0]);
+		exit(1);
+	}
+
+	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);
 
-//	DEBUG("Compute traces\n");
 	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;
 
+		invariants[i].id = i;
 		mat_trace(invariants[i].tr, matrices[i]);
-				mat_trace(invariants[i].trinv, matrices[group->elements[i].inverse->id]);
+		mat_trace(invariants[i].trinv, matrices[group->elements[i].inverse->id]);
 
-				distinct_invariants[n++] = &invariants[i];
+		distinct_invariants[n++] = &invariants[i];
 	}
 
-//	DEBUG("Get unique traces\n");
-
-	qsort(distinct_invariants, n, sizeof(struct result*), compare_result);
+	DEBUG("Get unique traces\n");
 
 	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];
+		if(i == 0 || compare_result(&distinct_invariants[i], &distinct_invariants[nuniq-1]) != 0) {
+			distinct_invariants[nuniq] = distinct_invariants[i];
+			distinct_invariants[nuniq]->count = 1;
+			nuniq++;
+		} else {
+			distinct_invariants[nuniq-1]->count++;
+			int oldlength = group->elements[distinct_invariants[nuniq-1]->id].length;
+			int newlength = group->elements[distinct_invariants[i]->id].length;
+			if(newlength < oldlength)
+				distinct_invariants[nuniq-1]->id = distinct_invariants[i]->id;
+		}
+
+		gmp_printf("%d %d %s\n", i, nuniq-1, print_word(&group->elements[i], buf));
 	}
 
 	max_slope = 0;
 	int max_slope_index;
 
-//	DEBUG("Solve characteristic polynomials\n");
+	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) {
@@ -416,227 +525,75 @@ double compute_max_slope(struct global_data dat, mpq_t t, mpq_t m)
 		double x = log(fabs(evs[0]));
 		double y = -log(fabs(evs[2]));
 
+		distinct_invariants[i]->x = x;
+		distinct_invariants[i]->y = y;
+
 		if(y/x > max_slope && (x > 0.1 || y > 0.1)) {
 			max_slope_index = distinct_invariants[i] - invariants;
 			max_slope = y/x;
 		}
-
-//				gmp_printf("%Qd %Qd %f %f %f\n", distinct_invariants[i]->tr, distinct_invariants[i]->trinv, x, y, y/x);
 	}
 
-	return max_slope;
-}
+	qsort(distinct_invariants, nuniq, sizeof(struct result*), compare_result_by_slope);
 
-void write_results_and_end(struct job *jobs, const char *outfile)
-{
-	DEBUG("writing output and shutting down\n");
+//	printf("- 0 0 - - - - 0.5\n");
+	int cumulative = 0;
+	double slope;
+	for(int i = 0; i < nuniq; i++) {
+		slope = distinct_invariants[i]->y/distinct_invariants[i]->x;
 
-	FILE *f = fopen(outfile, "w");
-	for(int i = 0; i < NJOBS; i++)
-		fprintf(f, "%d/%d %d/%d %.10f %.10f %.10f %.3f\n",
-		        jobs[i].tparam, TDIV, jobs[i].mparam, MDIV,
-		        (double)jobs[i].tparam/TDIV, (double)jobs[i].mparam/MDIV, jobs[i].max_slope,
-		        jobs[i].time);
-	fclose(f);
-
-	for(int i = 1; i < world_size; i++)
-		MPI_Send(NULL, 0, job_datatype, i, JOB_SHUTDOWN, MPI_COMM_WORLD);
-
-}
-
-void run_master_process(int nmax, const char *restart, const char *outfile)
-{
-	int total_jobs = NJOBS;
-	int completed = 0;
-	int queue_jobs = MIN(total_jobs, 2*world_size);
-	struct job current_job;
-	MPI_Status status;
-	FILE *f;
-	int continuing = 1;
-	int restartf;
-	struct job *alljobs;
-	struct job *current;
-
-	restartf = open(restart, O_RDWR);
-	if(restartf == -1 && errno == ENOENT) {
-		restartf = open(restart, O_RDWR | O_CREAT, 0666);
-		continuing = 0;
-	}
-	if(restartf == -1) {
-		DEBUG("error opening restart file: %s\n", strerror(errno));
-		exit(1);
-	}
-	ftruncate(restartf, total_jobs*sizeof(struct job));
-	alljobs = (struct job*) mmap(0, total_jobs*sizeof(struct job), PROT_READ | PROT_WRITE, MAP_SHARED, restartf, 0);
-	if(alljobs == MAP_FAILED) {
-		DEBUG("error mapping restart file: %s\n", strerror(errno));
-		exit(1);
-	}
-
-	if(continuing) {
-		for(int i = 0; i < total_jobs; i++)
-			if(alljobs[i].done)
-				completed++;
-	} else {
-		for(int tparam = TFROM; tparam <= TTO; tparam++) {
-			for(int mparam = MFROM; mparam <= MTO; mparam++) {
-				alljobs[JOBNR(tparam,mparam)].tparam = tparam;
-				alljobs[JOBNR(tparam,mparam)].mparam = mparam;
-				alljobs[JOBNR(tparam,mparam)].done = 0;
-			}
+		mpq_set_si(tmp, 1, 1);
+		if(mpq_cmp(distinct_invariants[i]->tr, tmp) == 0 && mpq_cmp(distinct_invariants[i]->trinv, tmp) == 0) {
+			continue;
 		}
-	}
-
-	fsync(restartf);
-
-	if(continuing) {
-		DEBUG("continuing from restart file, %d/%d jobs completed, %d nodes\n", completed, total_jobs, world_size);
-	} else {
-		DEBUG("starting from scratch, %d jobs, %d nodes\n", total_jobs, world_size);
-	}
-
-	if(completed >= total_jobs)
-	{
-		write_results_and_end(alljobs, outfile);
-		goto cleanup;
-	}
-
-	// assign initial jobs
-	current = alljobs-1;
-	for(int i = 0; i < 2*world_size; i++) {
-		do {
-			current++;
-		} while(current < alljobs + total_jobs && current->done);
-		if(current >= alljobs + total_jobs) // all jobs are assigned
-			break;
-		MPI_Send(current, 1, job_datatype, 1 + i%(world_size-1), JOB_ORDER, MPI_COMM_WORLD);
-	}
-
-	while(1) {
-		MPI_Probe(MPI_ANY_SOURCE, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
-		if(status.MPI_TAG == JOB_RESULT) {
-			MPI_Recv(&current_job, 1, job_datatype, MPI_ANY_SOURCE, JOB_RESULT, MPI_COMM_WORLD, &status);
-			completed++;
-
-			DEBUG("job (%d,%d) completed by instance %d in %f seconds, result = %.3f, %d/%d done\n",
-			      current_job.tparam, current_job.mparam,
-			      status.MPI_SOURCE, current_job.time, current_job.max_slope, completed, total_jobs);
-
-			int nr = JOBNR(current_job.tparam, current_job.mparam);
-			memcpy(&alljobs[nr], &current_job, sizeof(struct job));
-			alljobs[nr].done = 1;
-
-			if(completed % FLUSH_INTERVAL == 0)
-				fsync(restartf);
-
-			// find the next unassigned job
-			do {
-				current++;
-			} while(current < alljobs + total_jobs && current->done);
-
-			if(current < alljobs + total_jobs) {
-				MPI_Send(current, 1, job_datatype, status.MPI_SOURCE, JOB_ORDER, MPI_COMM_WORLD);
-			}
-
-			if(completed >= total_jobs) {
-				write_results_and_end(alljobs, outfile);
-				goto cleanup;
-			}
+		mpq_set_si(tmp, 0, 1);
+		if(mpq_cmp(distinct_invariants[i]->tr, tmp) == 0 && mpq_cmp(distinct_invariants[i]->trinv, tmp) == 0) {
+			continue;
 		}
+		mpq_set_si(tmp, -1, 1);
+		if(mpq_cmp(distinct_invariants[i]->tr, tmp) == 0 && mpq_cmp(distinct_invariants[i]->trinv, tmp) == 0) {
+			continue;
+		}
+		mpq_set_si(tmp, 3, 1);
+		if(mpq_cmp(distinct_invariants[i]->tr, tmp) == 0 && mpq_cmp(distinct_invariants[i]->trinv, tmp) == 0) {
+			continue;
+		}
+
+		cumulative += distinct_invariants[i]->count;
+		gmp_printf("%d %d %d %f %f %f %f %f %s\n",
+		           distinct_invariants[i]->id, distinct_invariants[i]->count, cumulative,
+		           distinct_invariants[i]->tr, distinct_invariants[i]->trinv,
+		           log(fabs(mpq_get_d(distinct_invariants[i]->tr))), log(fabs(mpq_get_d(distinct_invariants[i]->trinv))),
+		           distinct_invariants[i]->x, distinct_invariants[i]->y, slope,
+		           print_word(&group->elements[distinct_invariants[i]->id], buf)
+			);
 	}
+//	printf("- 0 %d - - - - 2.0\n", cumulative);
 
-cleanup:
+#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
 
-	munmap(alljobs, total_jobs*sizeof(struct job));
-	close(restartf);
-}
-
-int main(int argc, char *argv[])
-{
-	int name_len;
-
-	MPI_Status status;
-
-	mpq_t m, t;
-	double s;
-	struct job current_job;
-	int nmax;
-	double max_slope;
-	struct global_data dat;
-	double jobtime;
-
-    clock_gettime(CLOCK_REALTIME, &starttime);
-
-    if(argc < 4) {
-	    fprintf(stderr, "Usage: mpirun -n <nr> --hostfile <hostfile> %s <number of elements> <restartfile> <outfile>\n", argv[0]);
-	    return 0;
-    }
-	nmax = atoi(argv[1]);
-
-	MPI_Init(NULL, NULL);
-	MPI_Comm_size(MPI_COMM_WORLD, &world_size);
-	MPI_Comm_rank(MPI_COMM_WORLD, &world_rank);
-	MPI_Get_processor_name(processor_name, &name_len);
-
-//	DEBUG("instance %d/%d started on %s\n", world_rank, world_size, processor_name);
-
-	int blocklengths[2] = {3, 2};
-	MPI_Datatype types[2] = {MPI_INT, MPI_DOUBLE};
-	MPI_Aint displacements[2] = {(size_t)&((struct job*)0)->tparam, (size_t)&((struct job*)0)->max_slope};
-	MPI_Type_create_struct(2, blocklengths, displacements, types, &job_datatype);
-	MPI_Type_commit(&job_datatype);
-
-	if(world_rank == 0) { // master processor
-		run_master_process(nmax, argv[2], argv[3]);
-		MPI_Finalize();
-		return 0;
-	}
-
-//	DEBUG("Allocate & generate group\n");
-	mpq_inits(m, t, NULL);
-	dat = allocate_global_data(nmax);
-	dat.group = coxeter_init_triangle(4, 4, 4, nmax);
-
-//	fprintf(stderr, "max word length = %d\n", dat.group->elements[nmax-1].length);
-
-	while(1) {
-		MPI_Probe(0, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
-//		MPI_Recv(&current_job, 1, job_datatype, 0, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
-		if(status.MPI_TAG == JOB_SHUTDOWN) {
-//			DEBUG("instance %d shutting down\n", world_rank);
-			break;
-		}
-		else if(status.MPI_TAG == JOB_ORDER) {
-			MPI_Recv(&current_job, 1, job_datatype, 0, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
-			DEBUG("instance %d starting order (%d,%d)\n", world_rank, current_job.tparam, current_job.mparam);
-
-			jobtime = -MPI_Wtime();
-
-			// do the actual work
-			mpq_set_ui(t, current_job.tparam, TDIV);
-			mpq_set_ui(m, current_job.mparam, MDIV);
-			s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m));
-
-			max_slope = compute_max_slope(dat, t, m);
-
-			jobtime += MPI_Wtime();
-
-//			fprintf(stdout, "%.5f %.5f %.5f %f\n",
-//			        mpq_get_d(t), mpq_get_d(m), s, max_slope);
-			current_job.max_slope = max_slope;
-			current_job.time = jobtime;
-
-			DEBUG("instance %d finished order (%d,%d) in %f seconds\n", world_rank, current_job.tparam, current_job.mparam, jobtime);
-
-			MPI_Send(&current_job, 1, job_datatype, 0, JOB_RESULT, MPI_COMM_WORLD);
-		}
-	}
-
-//	DEBUG("Clean up\n");
-	coxeter_clear(dat.group);
-	free_global_data(dat);
-	mpq_clears(m, t, NULL);
-
-	MPI_Type_free(&job_datatype);
-	MPI_Finalize();
+//	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);
 }
diff --git a/singular_values.plt b/singular_values.plt
index f85674c..ef88c1e 100644
--- a/singular_values.plt
+++ b/singular_values.plt
@@ -1,5 +1,5 @@
-if(!exists("logt")) logt = log(1.78)
-if(!exists("logs")) logs = log(0.5)
+if(!exists("logt")) logt = log(1)
+if(!exists("logs")) logs = log(1)
 
 #file = sprintf("< ./singular_values 713698 %f %f", exp(logs), exp(logt))
 file = sprintf("< ./singular_values 1621 %f %f", exp(logs), exp(logt))
@@ -7,28 +7,13 @@ file = sprintf("< ./singular_values 1621 %f %f", exp(logs), exp(logt))
 set zeroaxis
 set samples 1000
 set size square
-set xrange [0:30]
-set yrange [0:30]
-set trange [0:30]
+set xrange [0:3]
+set yrange [0:3]
+set trange [0:5]
 set grid
 set parametric
 
-# plot file using 2:3 w p pt 7 ps 0.5 lc 1 t title
-
-#tr(a,b) = exp((2*a+b)/3) + exp((b-a)/3) + exp(-(a+2*b)/3)
-#trinv(a,b) = exp(-(2*a+b)/3) + exp((a-b)/3) + exp((a+2*b)/3)
-
-tr(a,b) = exp(a) + exp(b-a) + exp(-b)
-trinv(a,b) = exp(-a) + exp(a-b) + exp(b)
-
-#plot file using 6:7 w p pt 7 ps 0.5 lc 1 t columnheader,
-#	 log(tr(t,t*2)),log(trinv(t,2*t)) w l lw 2 t "", \
-#	 log(tr(t,t/2)),log(trinv(t,t/2)) w l lw 2 t ""
-
-plot file using 3:4 w p pt 7 ps 0.5 lc 1 t columnheader, \
-	 t,2*t w l lw 2 t "", \
-	 t,t/2 w l lw 2 t ""
-
+plot file using ($8/$9):($6/$7) w p pt 7 ps 0.3 lc 1 t sprintf("t = %.2f", exp(logt)), t, t
 
 #plot for[i=-10:10] log(tr(t,t*exp(log(2)*i/10.0))),log(trinv(t,t*exp(log(2)*i/10.0))) w l lw 2 t ""
 
@@ -36,6 +21,9 @@ plot file using 3:4 w p pt 7 ps 0.5 lc 1 t columnheader, \
 
 ##plot for[i=20:20] t,log(tr(1/t,exp(2*log(2)*i/20.0-log(2)))) w l lw 2 t ""
 
+#n=n+1
+#if(n < 1000) reread
+
 pause mouse keypress
 if(MOUSE_KEY == 60) logt=logt-0.02
 if(MOUSE_KEY == 62) logt=logt+0.02
diff --git a/singular_values_mpi.c b/singular_values_mpi.c
new file mode 100644
index 0000000..0f9e1e4
--- /dev/null
+++ b/singular_values_mpi.c
@@ -0,0 +1,642 @@
+#include "coxeter.h"
+//#include "linalg.h"
+#include "mat.h"
+
+//#include <gsl/gsl_poly.h>
+#include <mps/mps.h>
+#include <mpi.h>
+#include <sys/stat.h>
+#include <sys/mman.h>
+#include <fcntl.h>
+#include <errno.h>
+#include <string.h>
+#include <unistd.h>
+
+#define MIN(x,y) ((x)<(y)?(x):(y))
+#define SWAP(t,x,y) do { t _tmp = (x); (x) = (y); (y) = _tmp; } while (0);
+#define DEBUG(msg, ...) do { print_time(); fprintf(stderr, msg, ##__VA_ARGS__); } while (0);
+//#define DEBUG(msg, ...)
+
+#define TDIV 10
+#define TFROM 1
+#define TTO 9
+#define MDIV 10
+#define MFROM 1
+#define MTO 9
+#define JOBNR(t,m) (((t)-TFROM) + ((m)-MFROM)*(TTO-TFROM+1))
+#define NJOBS ((TTO-TFROM+1)*(MTO-MFROM+1))
+#define FLUSH_INTERVAL 100
+
+enum message_tag {
+	JOB_ORDER,
+	JOB_RESULT,
+	JOB_SHUTDOWN,
+};
+
+struct job {
+	int tparam, mparam;
+	int done;
+	double max_slope;
+	double time;
+};
+
+struct result {
+	mpq_t tr;
+	mpq_t trinv;
+};
+
+struct global_data {
+	int n;
+	group_t *group;
+	mat* matrices;
+	struct result *invariants;
+	struct result **distinct_invariants;
+	mps_context *solver;
+};
+
+
+struct timespec starttime;
+char processor_name[MPI_MAX_PROCESSOR_NAME];
+int world_rank;
+int world_size;
+MPI_Datatype job_datatype;
+
+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, "[%04d %.3f] ", world_rank, diff);
+}
+
+static struct global_data allocate_global_data(int n)
+{
+	struct global_data result;
+	result.n = n;
+	result.matrices = malloc(n*sizeof(mat));
+	for(int i = 0; i < n; i++)
+		mat_init(result.matrices[i], 3);
+	result.invariants = malloc(n*sizeof(struct result));
+	result.distinct_invariants = malloc(n*sizeof(struct result*));
+	for(int i = 0; i < n; i++) {
+		mpq_init(result.invariants[i].tr);
+		mpq_init(result.invariants[i].trinv);
+		result.distinct_invariants[i] = &result.invariants[i];
+	}
+	result.solver = mps_context_new();
+	mps_context_set_output_prec(result.solver, 20); // relative precision
+	mps_context_set_output_goal(result.solver, MPS_OUTPUT_GOAL_APPROXIMATE);
+
+	return result;
+}
+
+void free_global_data(struct global_data dat)
+{
+	for(int i = 0; i < dat.n; i++)
+		mat_clear(dat.matrices[i]);
+	free(dat.matrices);
+	for(int i = 0; i < dat.n; i++) {
+		mpq_clear(dat.invariants[i].tr);
+		mpq_clear(dat.invariants[i].trinv);
+	}
+	free(dat.invariants);
+	free(dat.distinct_invariants);
+	mps_context_free(dat.solver);
+}
+
+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);
+}
+
+// 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)
+{
+	mpq_t s,sinv,q,x,y;
+	mpq_t zero, one, two;
+	mpq_t tmp;
+
+	mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
+	mpq_set_ui(zero, 0, 1);
+	mpq_set_ui(one, 1, 1);
+	mpq_set_ui(two, 2, 1);
+
+	// s = (1-m^2)/2m
+	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);
+
+	// q = (1+m^2)/(1-m^2) = 2/(1-m^2) - 1
+	mpq_mul(q, m, m);
+	mpq_sub(q, one, q);
+	mpq_div(q, two, q);
+	mpq_sub(q, q, one);
+
+	// x = -tq, y = -q/t
+	mpq_mul(x, q, t);
+	mpq_sub(x, zero, x);
+	mpq_div(y, q, t);
+	mpq_sub(y, zero, y);
+
+	// q^2 = xy = 1 + 1/s^2
+	// [         -s         s*y           0]
+	// [       -s*x s*x*y - 1/s           0]
+	// [       -s*y   s*y^2 - x           1]
+	LOOP(i,3) {
+		mat_zero(gen[i]);
+		mpq_sub(tmp, zero, s);
+		mat_set(gen[i%3], i%3, i%3, tmp);
+		mpq_mul(tmp, s, y);
+		mat_set(gen[i%3], i%3, (i+1)%3, tmp);
+		mpq_mul(tmp, s, x);
+		mpq_sub(tmp, zero, tmp);
+		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);
+	}
+
+	LOOP(i,3) mat_pseudoinverse(ws, gen[i+3], gen[i]);
+
+	// debug output
+	/*
+	gmp_printf("m = %Qd, s = %Qd, t = %Qd, q = %Qd, x = %Qd, y = %Qd\n", m, s, t, q, x, y);
+	mat_print(gen[0]);
+	mat_print(gen[1]);
+	mat_print(gen[2]);
+	*/
+
+	mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
+}
+
+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 m, 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, m, 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);
+}
+
+
+double compute_max_slope(struct global_data dat, mpq_t t, mpq_t m)
+{
+//	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));
+
+	int n = 0;
+	int nmax = dat.n;
+	int nuniq;
+	double max_slope;
+	int retval;
+	double evs[3];
+
+	group_t *group = dat.group;
+	mat *matrices = dat.matrices;
+	struct result *invariants = dat.invariants;
+	struct result **distinct_invariants = dat.distinct_invariants;
+	mps_context *solver = dat.solver;
+
+//	DEBUG("Compute matrices\n");
+	enumerate(group, matrices, m, t);
+
+//	DEBUG("Compute traces\n");
+	n = 0;
+	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;
+		}
+
+//				gmp_printf("%Qd %Qd %f %f %f\n", distinct_invariants[i]->tr, distinct_invariants[i]->trinv, x, y, y/x);
+	}
+
+	return max_slope;
+}
+
+void write_results_and_end(struct job *jobs, const char *outfile)
+{
+	DEBUG("writing output and shutting down\n");
+
+	FILE *f = fopen(outfile, "w");
+	for(int i = 0; i < NJOBS; i++)
+		fprintf(f, "%d/%d %d/%d %.10f %.10f %.10f %.3f\n",
+		        jobs[i].tparam, TDIV, jobs[i].mparam, MDIV,
+		        (double)jobs[i].tparam/TDIV, (double)jobs[i].mparam/MDIV, jobs[i].max_slope,
+		        jobs[i].time);
+	fclose(f);
+
+	for(int i = 1; i < world_size; i++)
+		MPI_Send(NULL, 0, job_datatype, i, JOB_SHUTDOWN, MPI_COMM_WORLD);
+
+}
+
+void run_master_process(int nmax, const char *restart, const char *outfile)
+{
+	int total_jobs = NJOBS;
+	int completed = 0;
+	int queue_jobs = MIN(total_jobs, 2*world_size);
+	struct job current_job;
+	MPI_Status status;
+	FILE *f;
+	int continuing = 1;
+	int restartf;
+	struct job *alljobs;
+	struct job *current;
+
+	restartf = open(restart, O_RDWR);
+	if(restartf == -1 && errno == ENOENT) {
+		restartf = open(restart, O_RDWR | O_CREAT, 0666);
+		continuing = 0;
+	}
+	if(restartf == -1) {
+		DEBUG("error opening restart file: %s\n", strerror(errno));
+		exit(1);
+	}
+	ftruncate(restartf, total_jobs*sizeof(struct job));
+	alljobs = (struct job*) mmap(0, total_jobs*sizeof(struct job), PROT_READ | PROT_WRITE, MAP_SHARED, restartf, 0);
+	if(alljobs == MAP_FAILED) {
+		DEBUG("error mapping restart file: %s\n", strerror(errno));
+		exit(1);
+	}
+
+	if(continuing) {
+		for(int i = 0; i < total_jobs; i++)
+			if(alljobs[i].done)
+				completed++;
+	} else {
+		for(int tparam = TFROM; tparam <= TTO; tparam++) {
+			for(int mparam = MFROM; mparam <= MTO; mparam++) {
+				alljobs[JOBNR(tparam,mparam)].tparam = tparam;
+				alljobs[JOBNR(tparam,mparam)].mparam = mparam;
+				alljobs[JOBNR(tparam,mparam)].done = 0;
+			}
+		}
+	}
+
+	fsync(restartf);
+
+	if(continuing) {
+		DEBUG("continuing from restart file, %d/%d jobs completed, %d nodes\n", completed, total_jobs, world_size);
+	} else {
+		DEBUG("starting from scratch, %d jobs, %d nodes\n", total_jobs, world_size);
+	}
+
+	if(completed >= total_jobs)
+	{
+		write_results_and_end(alljobs, outfile);
+		goto cleanup;
+	}
+
+	// assign initial jobs
+	current = alljobs-1;
+	for(int i = 0; i < 2*world_size; i++) {
+		do {
+			current++;
+		} while(current < alljobs + total_jobs && current->done);
+		if(current >= alljobs + total_jobs) // all jobs are assigned
+			break;
+		MPI_Send(current, 1, job_datatype, 1 + i%(world_size-1), JOB_ORDER, MPI_COMM_WORLD);
+	}
+
+	while(1) {
+		MPI_Probe(MPI_ANY_SOURCE, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
+		if(status.MPI_TAG == JOB_RESULT) {
+			MPI_Recv(&current_job, 1, job_datatype, MPI_ANY_SOURCE, JOB_RESULT, MPI_COMM_WORLD, &status);
+			completed++;
+
+			DEBUG("job (%d,%d) completed by instance %d in %f seconds, result = %.3f, %d/%d done\n",
+			      current_job.tparam, current_job.mparam,
+			      status.MPI_SOURCE, current_job.time, current_job.max_slope, completed, total_jobs);
+
+			int nr = JOBNR(current_job.tparam, current_job.mparam);
+			memcpy(&alljobs[nr], &current_job, sizeof(struct job));
+			alljobs[nr].done = 1;
+
+			if(completed % FLUSH_INTERVAL == 0)
+				fsync(restartf);
+
+			// find the next unassigned job
+			do {
+				current++;
+			} while(current < alljobs + total_jobs && current->done);
+
+			if(current < alljobs + total_jobs) {
+				MPI_Send(current, 1, job_datatype, status.MPI_SOURCE, JOB_ORDER, MPI_COMM_WORLD);
+			}
+
+			if(completed >= total_jobs) {
+				write_results_and_end(alljobs, outfile);
+				goto cleanup;
+			}
+		}
+	}
+
+cleanup:
+
+	munmap(alljobs, total_jobs*sizeof(struct job));
+	close(restartf);
+}
+
+int main(int argc, char *argv[])
+{
+	int name_len;
+
+	MPI_Status status;
+
+	mpq_t m, t;
+	double s;
+	struct job current_job;
+	int nmax;
+	double max_slope;
+	struct global_data dat;
+	double jobtime;
+
+    clock_gettime(CLOCK_REALTIME, &starttime);
+
+    if(argc < 4) {
+	    fprintf(stderr, "Usage: mpirun -n <nr> --hostfile <hostfile> %s <number of elements> <restartfile> <outfile>\n", argv[0]);
+	    return 0;
+    }
+	nmax = atoi(argv[1]);
+
+	MPI_Init(NULL, NULL);
+	MPI_Comm_size(MPI_COMM_WORLD, &world_size);
+	MPI_Comm_rank(MPI_COMM_WORLD, &world_rank);
+	MPI_Get_processor_name(processor_name, &name_len);
+
+//	DEBUG("instance %d/%d started on %s\n", world_rank, world_size, processor_name);
+
+	int blocklengths[2] = {3, 2};
+	MPI_Datatype types[2] = {MPI_INT, MPI_DOUBLE};
+	MPI_Aint displacements[2] = {(size_t)&((struct job*)0)->tparam, (size_t)&((struct job*)0)->max_slope};
+	MPI_Type_create_struct(2, blocklengths, displacements, types, &job_datatype);
+	MPI_Type_commit(&job_datatype);
+
+	if(world_rank == 0) { // master processor
+		run_master_process(nmax, argv[2], argv[3]);
+		MPI_Finalize();
+		return 0;
+	}
+
+//	DEBUG("Allocate & generate group\n");
+	mpq_inits(m, t, NULL);
+	dat = allocate_global_data(nmax);
+	dat.group = coxeter_init_triangle(4, 4, 4, nmax);
+
+//	fprintf(stderr, "max word length = %d\n", dat.group->elements[nmax-1].length);
+
+	while(1) {
+		MPI_Probe(0, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
+//		MPI_Recv(&current_job, 1, job_datatype, 0, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
+		if(status.MPI_TAG == JOB_SHUTDOWN) {
+//			DEBUG("instance %d shutting down\n", world_rank);
+			break;
+		}
+		else if(status.MPI_TAG == JOB_ORDER) {
+			MPI_Recv(&current_job, 1, job_datatype, 0, MPI_ANY_TAG, MPI_COMM_WORLD, &status);
+			DEBUG("instance %d starting order (%d,%d)\n", world_rank, current_job.tparam, current_job.mparam);
+
+			jobtime = -MPI_Wtime();
+
+			// do the actual work
+			mpq_set_ui(t, current_job.tparam, TDIV);
+			mpq_set_ui(m, current_job.mparam, MDIV);
+			s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m));
+
+			max_slope = compute_max_slope(dat, t, m);
+
+			jobtime += MPI_Wtime();
+
+//			fprintf(stdout, "%.5f %.5f %.5f %f\n",
+//			        mpq_get_d(t), mpq_get_d(m), s, max_slope);
+			current_job.max_slope = max_slope;
+			current_job.time = jobtime;
+
+			DEBUG("instance %d finished order (%d,%d) in %f seconds\n", world_rank, current_job.tparam, current_job.mparam, jobtime);
+
+			MPI_Send(&current_job, 1, job_datatype, 0, JOB_RESULT, MPI_COMM_WORLD);
+		}
+	}
+
+//	DEBUG("Clean up\n");
+	coxeter_clear(dat.group);
+	free_global_data(dat);
+	mpq_clears(m, t, NULL);
+
+	MPI_Type_free(&job_datatype);
+	MPI_Finalize();
+}