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

This commit is contained in:
Florian Stecker 2022-01-30 19:09:56 -06:00
parent 7f0cb08787
commit 2b384447ca
5 changed files with 394 additions and 91 deletions

20
billiard_picture.sh Executable file
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@ -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

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@ -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 "%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]
-- 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]
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]
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

254
enumerate_triangle_group.c Normal file
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@ -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);
}

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@ -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

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@ -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;
int *max_slope_index;
double max_slope = 0;
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++) {
// for(int j = 1; j <= HEIGHT; j++) {
for(int w = 3; w < argc; w++) {
// mpq_set_ui(t, j, DENOMINATOR);
// mpq_set_ui(m, i, DENOMINATOR); // 414/1000 ~ sqrt(2)-1 <-> s=1
// s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m));
env = getenv("MAXIMUM");
if(!env || strcmp(env, "yes") == 0) {
mode = 1; // yes
} else if(strcmp(env, "no") == 0) {
mode = 0; // no
} else if(strcmp(env, "only") == 0) {
mode = 2; // only
}
DEBUG("Compute matrix\n");
initialize_triangle_generators(ws, gen, 6, 4, 3, s, q);
for(int w = 0; w < argc - 3; w++) {
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) {
letter1 = argv[w][k] - 'a';
letter2 = argv[w][k+1] - 'a';
for(int k = 0; k < strlen(argv[w+3]); k+=2) {
letter1 = argv[w+3][k] - '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[IDX(i,j)]) {
// max_slope[IDX(i,j)] = slope;
// 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);
if(slope > max_slope) {
max_slope = slope;
max_slope_index = w;
}
// printf("%.5f %.5f %d %.9f\n", (double)i/DENOMINATOR, (double)j/DENOMINATOR, max_slope_index[IDX(i,j)], max_slope[IDX(i,j)]);
fflush(stdout);
// }
// }
if(slope < min_slope) {
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);
}