florian/triangle_reflection_complex
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

compute complex traces

Browse Source
This commit is contained in:
Florian Stecker 2022-06-14 14:22:22 +02:00
parent 15681c308b
commit 244784794d
11 changed files with 446 additions and 59 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

18
.gitignore vendored
View File

@ -1,17 +1,7 @@
*.o
triangle_group/singular_values
.#*
singular_values
singular_values_mpi
singular_values_barbot
output/
special_element
max_slope_picture/generate
convert
billiard_words
*.pnm
*.png
*.hi
.\#*
\#*\#
gmon.out
restart
core
output/
complex_anosov

141
complex_anosov.c
View File

@ -5,9 +5,148 @@
#include "qext.h"
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define LOOP(i,n) for(int i = 0; i < (n); i++)
/*
Elements up to length 0: 1
Elements up to length 1: 4
Elements up to length 2: 10
Elements up to length 3: 22
Elements up to length 4: 46
Elements up to length 5: 91
Elements up to length 6: 175
Elements up to length 7: 334
Elements up to length 8: 634
Elements up to length 9: 1198
Elements up to length 10: 2260
Elements up to length 11: 4261
Elements up to length 12: 8029
Elements up to length 13: 15124
Elements up to length 14: 28486
Elements up to length 15: 53650
Elements up to length 16: 101038
Elements up to length 17: 190279
Elements up to length 18: 358339
Elements up to length 19: 674830
Elements up to length 20: 1270846
Elements up to length 21: 2393266
Elements up to length 22: 4507012
Elements up to length 23: 8487625
*/
static double gaussian_sqrt5_real(NUMBER x)
{
double result = 0.0;
mpq_t tmp;
mpq_init(tmp);
// a_0 + sqrt(5)a_1 + 4a_2 + 2sqrt(5)a_3
mpq_set_si(tmp, 4, 1);
mpq_mul(tmp, tmp, x->a[2]);
mpq_add(tmp, tmp, x->a[0]);
result = mpq_get_d(tmp);
mpq_set_si(tmp, 2, 1);
mpq_mul(tmp, tmp, x->a[3]);
mpq_add(tmp, tmp, x->a[1]);
result += mpq_get_d(tmp)*sqrt(5);
mpq_clear(tmp);
return result;
}
static double gaussian_sqrt5_imag(NUMBER x)
{
double result = 0.0;
mpq_t tmp;
mpq_init(tmp);
// a_1 + 2sqrt(5)a_2 + 14a_3
mpq_set_si(tmp, 14, 1);
mpq_mul(tmp, tmp, x->a[3]);
mpq_add(tmp, tmp, x->a[1]);
result = mpq_get_d(tmp);
mpq_set_si(tmp, 2, 1);
mpq_mul(tmp, tmp, x->a[2]);
result += mpq_get_d(tmp)*sqrt(5);
mpq_clear(tmp);
return result;
}
int main(int argc, char *argv[])
{
printf("initial main function.\n");
char buf[100];
int n = atoi(argv[1]);
mpq_t qreal, qimag;
mpq_inits(qreal, qimag, NULL);
mpq_set_si(qreal, 50, 10);
mpq_set_si(qimag, 1, 10);
mpq_canonicalize(qreal);
mpq_canonicalize(qimag);
/*
int length = 0;
LOOP(i, n) {
if(group->elements[i].length > length) {
printf("Elements up to length %d: %d\n", length, i);
length = group->elements[i].length;
}
}
return 0;
*/
/*
LOOP(i, n) {
groupelement_t *cur = &group->elements[i];
groupelement_t *other;
cur->conjugacy_class = cur; // start with itself and reduce if possible
LOOP(j, 3) {
if(cur->left[j] && cur->left[j]->right[j]) {
other = cur->left[j]->right[j];
if(other->id < cur->id)
cur->conjugacy_class = other->conjugacy_class;
}
if(cur->right[j] && cur->right[j]->left[j]) {
other = cur->right[j]->left[j];
if(other->id < cur->id)
cur->conjugacy_class = other->conjugacy_class;
}
}
}
*/
group_t *group = coxeter_init_triangle(5, 5, 5, n);
mat gen[3];
LOOP(i, 3) mat_init(gen[i], 3, QT_GAUSS_SQRT5);
generators_triangle_reflection_group_555_complex(gen, 2, 2, 2, qreal, qimag);
struct tracedata *traces;
int nuniq = enumerate_coxeter_group_traces(group, gen, &traces);
LOOP(i, nuniq) {
printf("%d %f %f %f %f\n",
traces[i].id,
gaussian_sqrt5_real(traces[i].tr),
gaussian_sqrt5_imag(traces[i].tr),
gaussian_sqrt5_real(traces[i].trinv),
gaussian_sqrt5_imag(traces[i].trinv));
}
enumerate_tracedata_clear(traces, nuniq);
LOOP(i, 3) mat_clear(gen[i]);
coxeter_clear(group);
mpq_clears(qreal, qimag, NULL);
return 0;
}

16
coxeter.c
View File

@ -5,6 +5,8 @@
#include "coxeter.h"
#define LOOP(i,n) for(int i = 0; i < (n); i++)
#define MIN(x,y) ((x) > (y) ? (y) : (x))
#define MAX(x,y) ((x) > (y) ? (x) : (y))
group_t *coxeter_init_triangle(int p, int q, int r, int nmax)
{
@ -185,3 +187,17 @@ void coxeter_clear(group_t *g)
free(g->lists);
free(g);
}
int coxeter_snprint(char *str, int buflen, groupelement_t *g)
{
int n = MIN(g->length, buflen-1); // number of characters without null byte
str[n] = 0;
int i = n-1;
while(g->parent) {
str[i--] = 'a' + g->letter;
g = g->parent;
}
return n;
}

2
coxeter.h
View File

@ -13,6 +13,7 @@ typedef struct _groupelement {
struct _groupelement *inverse;
struct _groupelement **left;
struct _groupelement **right;
struct _groupelement *conjugacy_class;
} groupelement_t;
typedef struct _group {
@ -26,5 +27,6 @@ typedef struct _group {
group_t *coxeter_init(int rank, int *coxeter_matrix, int nmax);
group_t *coxeter_init_triangle(int p, int q, int r, int nmax);
void coxeter_clear(group_t *g);
int coxeter_snprint(char *str, int buflen, groupelement_t *g);
#endif

98
enumerate.c
View File

@ -1,5 +1,7 @@
#include "enumerate.h"
#include <stdlib.h>
#define LOOP(i,n) for(int i = 0; i < (n); i++)
void enumerate_coxeter_group(group_t *group, mat *gen, mat *matrices)
@ -22,3 +24,99 @@ void enumerate_coxeter_group(group_t *group, mat *gen, mat *matrices)
mat_workspace_clear(ws);
}
static int compare_tracedata(const void *a_, const void *b_)
{
int d = 0;
struct tracedata **a = (struct tracedata **)a_;
struct tracedata **b = (struct tracedata **)b_;
d = CMP((*a)->tr,(*b)->tr);
if(d == 0) {
d = CMP((*a)->trinv, (*b)->trinv);
}
return d;
}
static int compare_tracedata_id(const void *a, const void *b)
{
int ida = (*(struct tracedata **)a)->id;
int idb = (*(struct tracedata **)b)->id;
return ida > idb ? 1 : ida < idb ? -1 : 0;
}
int enumerate_coxeter_group_traces(group_t *group, mat *gen, struct tracedata **traces_out)
{
TYPE t = GETTYPE(gen[0]->x[0]);
int n = group->size;
mat *matrices = malloc(n*sizeof(mat));
struct tracedata *traces = malloc(n*sizeof(struct tracedata));
struct tracedata **distinct_traces = malloc(n*sizeof(struct tracedata*));
LOOP(i, n) mat_init(matrices[i], 3, t);
enumerate_coxeter_group(group, gen, matrices);
LOOP(i, n) {
INIT(traces[i].tr, t);
INIT(traces[i].trinv, t);
}
LOOP(i, n) {
if(!group->elements[i].inverse)
continue;
mat_trace(traces[i].tr, matrices[i]);
mat_trace(traces[i].trinv, matrices[group->elements[i].inverse->id]);
traces[i].id = i;
distinct_traces[i] = &traces[i];
}
qsort(distinct_traces, n, sizeof(struct tracedata*), compare_tracedata);
int nuniq = 0;
LOOP(i, n) {
if(i == 0 || compare_tracedata(&distinct_traces[i], &distinct_traces[nuniq-1]) != 0) {
distinct_traces[nuniq] = distinct_traces[i];
nuniq++;
} else {
int oldlength = group->elements[distinct_traces[nuniq-1]->id].length;
int newlength = group->elements[distinct_traces[i]->id].length;
if(newlength < oldlength)
distinct_traces[nuniq-1]->id = distinct_traces[i]->id;
}
}
qsort(distinct_traces, nuniq, sizeof(struct tracedata*), compare_tracedata_id);
struct tracedata *unique_traces = malloc(nuniq*sizeof(struct tracedata));
LOOP(i, nuniq) {
INIT(unique_traces[i].tr, t);
INIT(unique_traces[i].trinv, t);
SET(unique_traces[i].tr, distinct_traces[i]->tr);
SET(unique_traces[i].trinv, distinct_traces[i]->trinv);
unique_traces[i].id = distinct_traces[i]->id;
}
LOOP(i, n) mat_clear(matrices[i]);
LOOP(i, n) {
CLEAR(traces[i].tr);
CLEAR(traces[i].trinv);
}
free(matrices);
free(traces);
free(distinct_traces);
*traces_out = unique_traces;
return nuniq;
}
void enumerate_tracedata_clear(struct tracedata *traces, int n)
{
LOOP(i, n) {
CLEAR(traces[i].tr);
CLEAR(traces[i].trinv);
}
free(traces);
}

9
enumerate.h
View File

@ -4,6 +4,15 @@
#include "mat.h"
#include "coxeter.h"
struct tracedata {
int id;
NUMBER tr;
NUMBER trinv;
};
void enumerate_coxeter_group(group_t *group, mat *gen, mat *matrices);
int enumerate_coxeter_group_traces(group_t *group, mat *gen, struct tracedata **traces_out);
void enumerate_tracedata_clear(struct tracedata *traces, int n);
#endif

103
generators.c
View File

@ -1,33 +1,17 @@
#include "generators.h"
#include "mat.h"
#define LOOP(i,n) for(int i = 0; i < (n); i++)
void generators_triangle_reflection_generic(mat *gen, NUMBER rho1, NUMBER rho2, NUMBER rho3, mpq_t q)
void generators_triangle_reflection_generic(mat *gen, NUMBER rho1, NUMBER rho2, NUMBER rho3, NUMBER q)
{
NUMBER b1,c1,a2,c2,a3,b3;
mpq_t qinv;
NUMBER tmp;
NUMBER qinv;
TYPE t = GETTYPE(rho1);
INIT(b1,t);INIT(c1,t);INIT(a2,t);INIT(c2,t);INIT(a3,t);INIT(b3,t);
// qinv = q^{-1}
mpq_init(qinv);
mpq_inv(qinv, q);
// c1 = rho2 q, a2 = rho3 q, b3 = rho1 q, b1 = c2 = a3 = 1/q
SET_ZERO(c1);
SET_Q(c1, q);
SET_Q(a2, q);
SET_Q(b3, q);
MUL(c1, c1, rho2);
MUL(a2, a2, rho3);
MUL(b3, b3, rho1);
SET_INT(b1, 1);
SET_INT(c2, 1);
SET_INT(a3, 1);
SET_Q(b1, qinv);
SET_Q(c2, qinv);
SET_Q(a3, qinv);
INIT(tmp, t);
INIT(qinv, t);
INV(qinv, q);
LOOP(i, 3) {
mat_init(gen[i], 3, t);
@ -37,18 +21,21 @@ void generators_triangle_reflection_generic(mat *gen, NUMBER rho1, NUMBER rho2,
}
}
NEG(*mat_ref(gen[0], 1, 0), b1);
NEG(*mat_ref(gen[0], 2, 0), c1);
NEG(*mat_ref(gen[1], 0, 1), a2);
NEG(*mat_ref(gen[1], 2, 1), c2);
NEG(*mat_ref(gen[2], 0, 2), a3);
NEG(*mat_ref(gen[2], 1, 2), b3);
MUL(tmp, q, rho1);
NEG(*mat_ref(gen[2], 1, 2), tmp);
MUL(tmp, q, rho2);
NEG(*mat_ref(gen[0], 2, 0), tmp);
MUL(tmp, q, rho3);
NEG(*mat_ref(gen[1], 0, 1), tmp);
CLEAR(b1);CLEAR(c1);CLEAR(a2);CLEAR(c2);CLEAR(a3);CLEAR(b3);
mpq_clear(qinv);
NEG(*mat_ref(gen[1], 2, 1), qinv);
NEG(*mat_ref(gen[2], 0, 2), qinv);
NEG(*mat_ref(gen[0], 1, 0), qinv);
CLEAR(tmp);
CLEAR(qinv);
}
int generators_triangle_reflection_group(mat *gen, int p1, int p2, int p3, int q1, int q2, int q3, mpq_t t)
{
int p[3] = {p1, p2, p3};
@ -103,9 +90,59 @@ int generators_triangle_reflection_group(mat *gen, int p1, int p2, int p3, int q
}
}
generators_triangle_reflection_generic(gen, rho[0], rho[1], rho[2], t);
NUMBER param;
INIT(param, type);
SET_Q(param, t);
generators_triangle_reflection_generic(gen, rho[0], rho[1], rho[2], param);
LOOP(i, 3) CLEAR(rho[i]);
CLEAR(param);
return 1;
}
int generators_triangle_reflection_group_555_complex(mat *gen, int q1, int q2, int q3, mpq_t treal, mpq_t timag)
{
int q[3] = {q1, q2, q3};
NUMBER rho[3];
LOOP(i, 3) INIT(rho[i], QT_GAUSS_SQRT5);
// compute rho1, rho2, rho3
LOOP(i, 3) {
if(q[i] == 1) {
// 4cos(pi/5)^2 = 3/2 + 1/2*sqrt(5)
mpq_set_si(rho[i]->a[0], 3, 2);
mpq_set_si(rho[i]->a[1], 7, 12);
mpq_set_si(rho[i]->a[2], 0, 1);
mpq_set_si(rho[i]->a[3], -1, 24);
} else if(q[i] == 2) {
// 4cos(pi/5)^2 = 3/2 - 1/2*sqrt(5)
mpq_set_si(rho[i]->a[0], 3, 2);
mpq_set_si(rho[i]->a[1], -7, 12);
mpq_set_si(rho[i]->a[2], 0, 1);
mpq_set_si(rho[i]->a[3], 1, 24);
} else {
return 0;
}
}
NUMBER param;
INIT(param, QT_GAUSS_SQRT5);
mpq_set(param->a[0], treal);
mpq_set_si(param->a[1], -1, 6);
mpq_mul(param->a[1], param->a[1], timag);
mpq_set_si(param->a[2], 0, 1);
mpq_set_si(param->a[3], 1, 12);
mpq_mul(param->a[3], param->a[3], timag);
generators_triangle_reflection_generic(gen, rho[0], rho[1], rho[2], param);
LOOP(i, 3) CLEAR(rho[i]);
CLEAR(param);
return 1;
}

6
generators.h
View File

@ -3,7 +3,11 @@
#include "mat.h"
void generators_triangle_reflection_generic(mat *gen, NUMBER rho1, NUMBER rho2, NUMBER rho3, mpq_t q);
void generators_triangle_reflection_generic(mat *gen, NUMBER rho1, NUMBER rho2, NUMBER rho3, NUMBER q);
int generators_triangle_reflection_group(mat *gen, int p1, int p2, int p3, int q1, int q2, int q3, mpq_t q);
#ifdef QEXT
int generators_triangle_reflection_group_555_complex(mat *gen, int q1, int q2, int q3, mpq_t real, mpq_t imag);
#endif
#endif

4
mat.h
View File

@ -24,7 +24,8 @@
#define SUB qext_sub
#define NEG qext_neg
#define MUL qext_mul
// #define DIV qext_div
#define DIV qext_div
#define INV qext_inv
#define CMP qext_cmp
#define PRINT qext_print
#define SNPRINT qext_snprint
@ -46,6 +47,7 @@
#define NEG mpq_neg
#define MUL mpq_mul
#define DIV mpq_div
#define INV mpq_inv
#define PRINT(x) gmp_printf("%Qd", x)
#define SNPRINT(out, size, x) gmp_snprintf(out, size, "%Qd", x)
#define TYPE int

102
qext.c
View File

@ -1,5 +1,7 @@
#include "qext.h"
#include <assert.h>
int qext_rank;
mpq_t *qext_coefficient;
@ -9,13 +11,20 @@ mpq_t *qext_coefficient;
#define LOOP(i,n) for(int i = 0; i < (n); i++)
#define RANK(x) ((x)->type->rank)
const int qext_type_trivial_coeffs[] = {-1, 1};
struct qext_type qext_type_trivial_v = {1, qext_type_trivial_coeffs, NULL};
struct qext_type *QT_TRIVIAL = &qext_type_trivial_v;
struct qext_type *QT_TRIVIAL = &(struct qext_type) {
.rank = 1,
.integer_coeffs = (const int[]) {-1, 1}
};
const int qext_type_sqrt5_coeffs[] = {-5, 0, 1};
struct qext_type qext_type_sqrt5_v = {2, qext_type_sqrt5_coeffs, NULL};
struct qext_type *QT_SQRT5 = &qext_type_sqrt5_v;
struct qext_type *QT_SQRT5 = &(struct qext_type) {
.rank = 2,
.integer_coeffs = (const int[]) {-5, 0, 1}
};
struct qext_type *QT_GAUSS_SQRT5 = &(struct qext_type) {
.rank = 4,
.integer_coeffs = (const int[]) {36, 0, -8, 0, 1}
};
static void qext_init_type(struct qext_type *type)
{
@ -161,7 +170,86 @@ void qext_mul(qext_number out, qext_number x, qext_number y)
mpq_clear(tmp);
}
void qext_inv(qext_number out, qext_number in)
{
qext_number one;
qext_init(one, in->type);
qext_set_int(one, 1);
qext_div(out, one, in);
qext_clear(one);
}
void qext_div(qext_number out, qext_number x, qext_number y)
{
// todo: implement this by solving a linear system
int rk = RANK(x);
struct qext_type *type = x->type;
mpq_t tmp;
mpq_t result[rk];
mpq_t matrix[rk*rk];
mpq_init(tmp);
LOOP(i, rk) mpq_init(result[i]);
LOOP(i, rk*rk) mpq_init(matrix[i]);
// initialize a matrix which represents multiplication by y
// that means the i-th column is y*a^i
LOOP(i, rk)
mpq_set(matrix[i*rk], y->a[i]);
LOOP(j, rk-1) { // columns
mpq_mul(matrix[j+1], matrix[(rk-1)*rk+j], type->coeffs[0]);
LOOP(i, rk-1) { // rows
mpq_mul(matrix[(i+1)*rk+(j+1)], matrix[(rk-1)*rk+j], type->coeffs[i+1]);
mpq_add(matrix[(i+1)*rk+(j+1)], matrix[(i+1)*rk+(j+1)], matrix[i*rk+j]);
}
}
// initialize result as x
LOOP(i, rk) mpq_set(result[i], x->a[i]);
// use Gaussian elimination to solve the system matrix * out = result
LOOP(j, rk)
{
// find nonzero entry
int row = j;
while(row < rk && mpq_sgn(matrix[row*rk+j]) == 0) row++;
// if the entire column is zero, the matrix was not invertible
// this could happen if the polynomial is not irreducible / we're not in a field
assert(row != rk);
// permute rows
if(row != j) {
mpq_swap(result[row], result[j]);
LOOP(k, rk) {
mpq_swap(matrix[row*rk+k], matrix[j*rk+k]);
}
}
// normalize
LOOP(k, rk)
if(k != j)
mpq_div(matrix[j*rk+k], matrix[j*rk+k], matrix[j*rk+j]);
mpq_div(result[j], result[j], matrix[j*rk+j]);
mpq_set_ui(matrix[j*rk+j], 1, 1);
// subtract
LOOP(i, rk) {
if(i == j)
continue;
mpq_mul(tmp, matrix[i*rk+j], result[j]);
mpq_sub(result[i], result[i], tmp);
for(int k = j+1; k < rk; k++) {
mpq_mul(tmp, matrix[i*rk+j], matrix[j*rk+k]);
mpq_sub(matrix[i*rk+k], matrix[i*rk+k], tmp);
}
mpq_set_ui(matrix[i*rk+j], 0, 1); // it isn't strictly necessary to do this as we won't use this entry again, but helps debugging
}
}
LOOP(i, rk) mpq_set(out->a[i], result[i]);
mpq_clear(tmp);
LOOP(i, rk) mpq_clear(result[i]);
LOOP(i, rk*rk) mpq_clear(matrix[i]);
}

6
qext.h
View File

@ -5,8 +5,8 @@
struct qext_type {
int rank;
const int *integer_coeffs;
mpq_t *coeffs;
const int *integer_coeffs; // actually the rank+1 coefficients of the polynomial
mpq_t *coeffs; // components of a^rank
};
struct qext_number_internal {
@ -18,6 +18,7 @@ typedef struct qext_number_internal qext_number[1];
extern struct qext_type *QT_TRIVIAL;
extern struct qext_type *QT_SQRT5;
extern struct qext_type *QT_GAUSS_SQRT5;
struct qext_type *qext_newtype(int rank, const int *coeffs);
void qext_init(qext_number x, struct qext_type *type);
@ -29,6 +30,7 @@ void qext_add(qext_number result, qext_number x, qext_number y);
void qext_sub(qext_number result, qext_number x, qext_number y);
void qext_neg(qext_number result, qext_number x);
void qext_mul(qext_number out, qext_number x, qext_number y);
void qext_inv(qext_number out, qext_number in);
void qext_div(qext_number out, qext_number x, qext_number y);
int qext_cmp(qext_number x, qext_number y);
void qext_print(qext_number x);