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combine singular_values and special_elements and test Charlie's prediction

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This commit is contained in:
Florian Stecker 2021-10-23 16:04:12 -05:00
parent ef1b48e86e
commit 7f0cb08787
5 changed files with 80 additions and 546 deletions
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15
Makefile
View File

@ -1,4 +1,4 @@
HEADERS=linalg.h mat.h coxeter.h HEADERS=linalg.h mat.h coxeter.h enumerate_triangle_group.h
#SPECIAL_OPTIONS=-O0 -g -D_DEBUG #SPECIAL_OPTIONS=-O0 -g -D_DEBUG
SPECIAL_OPTIONS=-O3 -pg -funroll-loops -fno-inline SPECIAL_OPTIONS=-O3 -pg -funroll-loops -fno-inline
@ -16,14 +16,14 @@ convert: convert.hs
billiard_words: billiard_words.hs billiard_words: billiard_words.hs
ghc --make -dynamic billiard_words.hs ghc --make -dynamic billiard_words.hs
singular_values: singular_values.o coxeter.o mat.o singular_values: singular_values.o coxeter.o mat.o enumerate_triangle_group.o
gcc $(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 enumerate_triangle_group.o -lm -lgmp -lmps
singular_values_mpi: singular_values_mpi.o coxeter.o mat.o 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 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 special_element: special_element.o coxeter.o linalg.o mat.o enumerate_triangle_group.o
gcc $(OPTIONS) -o special_element coxeter.o linalg.o special_element.o mat.o -lm -lgmp -lmps -lgsl -lcblas gcc $(OPTIONS) -o special_element coxeter.o linalg.o special_element.o mat.o enumerate_triangle_group.o -lm -lgmp -lmps -lgsl -lcblas
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
@ -34,6 +34,9 @@ singular_values_mpi.o: singular_values_mpi.c $(HEADERS)
special_element.o: special_element.c $(HEADERS) special_element.o: special_element.c $(HEADERS)
gcc $(OPTIONS) -c special_element.c gcc $(OPTIONS) -c special_element.c
enumerate_triangle_group.o: enumerate_triangle_group.c $(HEADERS)
gcc $(OPTIONS) -c enumerate_triangle_group.c
linalg.o: linalg.c $(HEADERS) linalg.o: linalg.c $(HEADERS)
gcc $(OPTIONS) -c linalg.c gcc $(OPTIONS) -c linalg.c
@ -44,4 +47,4 @@ mat.o: mat.c $(HEADERS)
gcc $(OPTIONS) -c mat.c gcc $(OPTIONS) -c mat.c
clean: clean:
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 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 enumerate_triangle_group.o

10
billiard_words.hs
View File

@ -1,18 +1,22 @@
import Data.List import Data.List
import Data.Ord import Data.Ord
import Text.Printf import Text.Printf
import System.Environment
main = listWordsUpToLength 200 main = do
argv <- getArgs
listWordsUpToLength $ read $ argv !! 0
listWordsUpToLength :: Int -> IO () listWordsUpToLength :: Int -> IO ()
listWordsUpToLength n = do listWordsUpToLength n = do
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 "%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 "%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]
wordlist :: Int -> Int -> [((Int,Int),String)] 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..200], q <- [0..200], 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], p /= 0 || q /= 0]
where where
sl ((p,q),_) = fromIntegral p / fromIntegral q sl ((p,q),_) = fromIntegral p / fromIntegral q

6
coxeter.c
View File

@ -9,9 +9,9 @@
group_t *coxeter_init_triangle(int p, int q, int r, int nmax) group_t *coxeter_init_triangle(int p, int q, int r, int nmax)
{ {
int coxeter_matrix[9] = { int coxeter_matrix[9] = {
1, p, r, 1, r, q,
p, 1, q, r, 1, p,
r, q, 1}; q, p, 1};
return coxeter_init(3, coxeter_matrix, nmax); return coxeter_init(3, coxeter_matrix, nmax);
} }

359
singular_values.c
View File

@ -1,17 +1,12 @@
#include "coxeter.h" #include "coxeter.h"
#include "linalg.h" #include "linalg.h"
#include "mat.h" #include "mat.h"
#include "enumerate_triangle_group.h"
#include <gsl/gsl_poly.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, ...) fprintf(stderr, msg, ##__VA_ARGS__)
#define DEBUG(msg, ...) #define DEBUG(msg, ...)
#define OUTPUT_POINTS
//#define OUTPUT_POINTS
struct result { struct result {
int id; int id;
int count; int count;
@ -67,326 +62,6 @@ static int compare_result_by_slope(const void *a_, const void *b_)
return slopea > slopeb ? -1 : slopea < slopeb ? 1 : 0; 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;
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);
}
void initialize_triangle_generators(mat_workspace *ws, mat *gen, 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);
quartic(rho1, s, 0, 0, 1, -1, 1);
quartic(rho2, s, 0, 0, 1, -1, 1);
quartic(rho3, s, 0, 0, 1, 0, 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, 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, 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);
}
void output_invariants(group_t *group, mat *matrices, mpq_t s, mpq_t q, mps_context *solver)
{
mpq_t tr, trinv;
char buf[100];
double evs[3];
int retval;
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[]) int main(int argc, char *argv[])
{ {
mpq_t s, q, t, tmp; mpq_t s, q, t, tmp;
@ -407,7 +82,7 @@ int main(int argc, char *argv[])
struct result **distinct_invariants; struct result **distinct_invariants;
if(argc < 4) { if(argc < 4) {
fprintf(stderr, "Usage: %s <N> <s> <t>\n", argv[0]); fprintf(stderr, "Usage: %s <N> <s> <q>\n", argv[0]);
exit(1); exit(1);
} }
@ -435,9 +110,10 @@ int main(int argc, char *argv[])
// get approximate s and q values // get approximate s and q values
sapprox = atof(argv[2]); sapprox = atof(argv[2]);
tapprox = atof(argv[3]); qapprox = atof(argv[3]);
tqfactor = pow((sapprox*sapprox-sapprox+1)*(sapprox*sapprox-sapprox+1)*(sapprox*sapprox+1), 1/6.0); // tapprox = atof(argv[3]);
qapprox = tapprox/tqfactor; // tqfactor = pow((sapprox*sapprox-sapprox+1)*(sapprox*sapprox-sapprox+1)*(sapprox*sapprox+1), 1/6.0);
// qapprox = tapprox/tqfactor;
for(int i = 0; ; i++) { for(int i = 0; ; i++) {
continued_fraction_approximation(tmp, sapprox, i); continued_fraction_approximation(tmp, sapprox, i);
@ -459,16 +135,14 @@ int main(int argc, char *argv[])
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); 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 // group
// order of the triangle reflection generators: a, b, c
// order of the rotation orders: bc, ac, ab
DEBUG("Generate group\n"); DEBUG("Generate group\n");
group = coxeter_init_triangle(4, 3, 3, nmax); group = coxeter_init_triangle(4, 3, 3, nmax);
DEBUG("Compute matrices\n"); DEBUG("Compute matrices\n");
enumerate(group, matrices, s, q); enumerate(group, matrices, 4, 3, 3, s, q);
n = 0; n = 0;
DEBUG("Compute traces\n"); DEBUG("Compute traces\n");
@ -538,6 +212,8 @@ int main(int argc, char *argv[])
qsort(distinct_invariants, nuniq, sizeof(struct result*), compare_result_by_slope); qsort(distinct_invariants, nuniq, sizeof(struct result*), compare_result_by_slope);
gmp_fprintf(stdout, "\"s = %Qd = %.3f, q = %Qd, t = %.3f\"\n", s, mpq_get_d(s), q, mpq_get_d(q)*tqfactor);
// printf("- 0 0 - - - - 0.5\n"); // printf("- 0 0 - - - - 0.5\n");
int cumulative = 0; int cumulative = 0;
double slope; double slope;
@ -572,19 +248,6 @@ int main(int argc, char *argv[])
} }
// printf("- 0 %d - - - - 2.0\n", cumulative); // printf("- 0 %d - - - - 2.0\n", cumulative);
#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
// 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++) {
mpq_clear(invariants[i].tr); mpq_clear(invariants[i].tr);

236
special_element.c
View File

@ -1,10 +1,7 @@
#include "coxeter.h" #include "coxeter.h"
#include "linalg.h" #include "linalg.h"
#include "mat.h" #include "mat.h"
#include "enumerate_triangle_group.h"
#include <gsl/gsl_poly.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, ...) #define DEBUG(msg, ...)
@ -14,180 +11,30 @@
#define HEIGHT 300 #define HEIGHT 300
#define IDX(i,j) (((i)-1)*HEIGHT + ((j)-1)) #define IDX(i,j) (((i)-1)*HEIGHT + ((j)-1))
int solve_characteristic_polynomial(mps_context *solv, mpq_t tr, mpq_t trinv, double *eigenvalues) double mpq_log(mpq_t m_op)
{ {
mpq_t coeff1, coeff2, zero; static double logB = log(ULONG_MAX);
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); // Undefined logs (should probably return NAN in second case?)
mpq_set(coeff1, trinv); if (mpz_get_ui(mpq_numref(m_op)) == 0 || mpz_sgn(mpq_numref(m_op)) < 0)
mpq_sub(coeff2, zero, tr); return -INFINITY;
poly = mps_monomial_poly_new(solv, 3); // Log of numerator
mps_monomial_poly_set_coefficient_int(solv, poly, 0, -1, 0); double lognum = log(mpq_numref(m_op)->_mp_d[abs(mpq_numref(m_op)->_mp_size) - 1]);
mps_monomial_poly_set_coefficient_q(solv, poly, 1, coeff1, zero); lognum += (abs(mpq_numref(m_op)->_mp_size)-1) * logB;
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); // Subtract log of denominator, if it exists
mps_mpsolve(solv); if (abs(mpq_denref(m_op)->_mp_size) > 0)
{
roots = cplx_valloc(3); lognum -= log(mpq_denref(m_op)->_mp_d[abs(mpq_denref(m_op)->_mp_size)-1]);
for(int i = 0; i < 3; i++) lognum -= (abs(mpq_denref(m_op)->_mp_size)-1) * logB;
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;
}
} }
return lognum;
cplx_vfree(roots);
mpq_clears(coeff1, coeff2, zero, NULL);
return result;
}
// 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]);
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);
} }
int main(int argc, char *argv[]) int main(int argc, char *argv[])
{ {
mpq_t m, t, tmp; mpq_t m, t, s, q, tmp, tmp2;
double s;
mat_workspace *ws; mat_workspace *ws;
mat gen[6]; mat gen[6];
mps_context *solver; mps_context *solver;
@ -203,7 +50,7 @@ int main(int argc, char *argv[])
DEBUG("Allocate\n"); DEBUG("Allocate\n");
mpq_inits(m, t, tmp, tr, trinv, NULL); mpq_inits(m, t, s, q, tmp, tmp2, tr, trinv, NULL);
ws = mat_workspace_init(3); ws = mat_workspace_init(3);
for(int i = 0; i < 6; i++) for(int i = 0; i < 6; i++)
mat_init(gen[i], 3); mat_init(gen[i], 3);
@ -218,15 +65,18 @@ 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);
for(int i = STARTX; i <= WIDTH; i++) { mpq_set_str(s, argv[1], 10);
for(int j = 1; j <= HEIGHT; j++) { mpq_set_str(q, argv[2], 10);
for(int w = 1; w < argc; w++) {
mpq_set_ui(t, j, DENOMINATOR); // for(int i = STARTX; i <= WIDTH; i++) {
mpq_set_ui(m, i, DENOMINATOR); // 414/1000 ~ sqrt(2)-1 <-> s=1 // for(int j = 1; j <= HEIGHT; j++) {
s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m)); 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));
DEBUG("Compute matrix\n"); DEBUG("Compute matrix\n");
initialize_triangle_generators(ws, gen, m, t); initialize_triangle_generators(ws, gen, 6, 4, 3, s, q);
mat_identity(element); mat_identity(element);
mat_identity(inverse); mat_identity(inverse);
@ -272,25 +122,39 @@ int main(int argc, char *argv[])
x = log(fabs(evs[0])); x = log(fabs(evs[0]));
y = -log(fabs(evs[2])); y = -log(fabs(evs[2]));
slope = y/x > 1 ? y/x : x/y;
if(slope > max_slope[IDX(i,j)]) { if(x > DBL_MAX || y > DBL_MAX) {
max_slope[IDX(i,j)] = slope; mpq_abs(tmp, tr);
max_slope_index[IDX(i,j)] = w; mpq_abs(tmp2, trinv);
slope = mpq_log(tmp)/mpq_log(tmp2);
} else {
slope = y/x;
} }
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("%Qd %Qd %f %f %f\n", tr, trinv, x, y, y/x);
// gmp_printf("%.5f %.5f %.7f %.9f\n", mpq_get_d(t), mpq_get_d(m), s, slope); // 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)]); // 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); fflush(stdout);
} // }
} // }
DEBUG("Clean up\n"); DEBUG("Clean up\n");
mpq_clears(m, t, tmp, tr, trinv, NULL); mpq_clears(m, t, s, q, tmp, tmp2, tr, trinv, NULL);
mat_workspace_clear(ws); mat_workspace_clear(ws);
for(int i = 0; i < 6; i++) for(int i = 0; i < 6; i++)
mat_clear(gen[i]); mat_clear(gen[i]);