draw character variety picture
This commit is contained in:
Florian Stecker
parent
7656408f57
commit
b5bd9d398f
10
Makefile
10
Makefile
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@ -7,14 +7,20 @@ SPECIAL_OPTIONS=-O3 -flto -funroll-loops -Winline
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OPTIONS=-m64 -march=native -mtune=native -std=gnu99 -D_GNU_SOURCE $(SPECIAL_OPTIONS)
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all: singular_values
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all: singular_values special_element
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singular_values: singular_values.o coxeter.o linalg.o mat.o
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gcc $(OPTIONS) -o singular_values coxeter.o linalg.o singular_values.o mat.o -lm -lgsl -lcblas -lgmp -lmps -lpthread
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special_element: special_element.o coxeter.o linalg.o mat.o
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gcc $(OPTIONS) -o special_element coxeter.o linalg.o special_element.o mat.o -lm -lgsl -lcblas -lgmp -lmps -lpthread
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singular_values.o: singular_values.c $(HEADERS)
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gcc $(OPTIONS) -c singular_values.c
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special_element.o: special_element.c $(HEADERS)
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gcc $(OPTIONS) -c special_element.c
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linalg.o: linalg.c $(HEADERS)
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gcc $(OPTIONS) -c linalg.c
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@ -25,4 +31,4 @@ mat.o: mat.c $(HEADERS)
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gcc $(OPTIONS) -c mat.c
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clean:
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rm -f singular_values coxeter.o linalg.o singular_values.o mat.o
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rm -f singular_values special_element coxeter.o linalg.o singular_values.o mat.o special_element.o
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@ -6,11 +6,26 @@
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#include <mps/mps.h>
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#define SWAP(t,x,y) do { t _tmp = (x); (x) = (y); (y) = _tmp; } while (0);
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//#define DEBUG(msg, ...) fprintf(stderr, msg, ##__VA_ARGS__)
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#define DEBUG(msg, ...)
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#define DEBUG(msg, ...) do { print_time(); fprintf(stderr, msg, ##__VA_ARGS__); } while (0);
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//#define DEBUG(msg, ...)
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#define OUTPUT_POINTS
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//#define OUTPUT_POINTS
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#define OUTPUT_SUMMARY
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struct timespec starttime;
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void print_time()
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{
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double diff;
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struct timespec current;
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clock_gettime(CLOCK_REALTIME, ¤t);
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diff = (current.tv_sec - starttime.tv_sec) + (current.tv_nsec - starttime.tv_nsec)*1e-9;
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fprintf(stderr, "[%.3f] ", diff);
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}
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struct result {
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mpq_t tr;
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@ -124,99 +139,75 @@ void quartic(mpq_t out, mpq_t in, int a, int b, int c, int d, int e)
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mpq_clear(tmp);
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}
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void initialize_triangle_generators(mat_workspace *ws, mat *gen, mpq_t s, mpq_t q)
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// this version is only for the (4,4,4) group
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void initialize_triangle_generators(mat_workspace *ws, mat *gen, mpq_t m, mpq_t t)
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{
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mat r1,r2,r3;
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mpq_t rho1, rho2, rho3;
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mpq_t b1,c1,a2,c2,a3,b3;
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mpq_t sinv;
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mpq_t s,sinv,q,x,y;
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mpq_t zero, one, two;
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mpq_t tmp;
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mpq_inits(sinv,rho1,rho2,rho3,b1,c1,a2,c2,a3,b3,NULL);
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mat_init(r1, 3);
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mat_init(r2, 3);
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mat_init(r3, 3);
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mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
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mpq_set_ui(zero, 0, 1);
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mpq_set_ui(one, 1, 1);
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mpq_set_ui(two, 2, 1);
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mpq_set_ui(sinv, 1, 1);
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mpq_div(sinv, sinv, s);
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// s = (1-m^2)/2m
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mpq_mul(s, m, m);
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mpq_sub(s, one, s);
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mpq_div(s, s, m);
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mpq_div(s, s, two);
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mpq_div(sinv, one, s);
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quartic(rho1, s, 0, 0, 1, -1, 1);
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quartic(rho2, s, 0, 0, 1, -1, 1);
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quartic(rho3, s, 0, 0, 1, 0, 1);
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// q = (1+m^2)/(1-m^2) = 2/(1-m^2) - 1
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mpq_mul(q, m, m);
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mpq_sub(q, one, q);
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mpq_div(q, two, q);
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mpq_sub(q, q, one);
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mpq_mul(c1, rho2, q);
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mpq_mul(a2, rho3, q);
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mpq_mul(b3, rho1, q);
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mpq_set_ui(b1, 1, 1);
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mpq_set_ui(c2, 1, 1);
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mpq_set_ui(a3, 1, 1);
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mpq_div(b1, b1, q);
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mpq_div(c2, c2, q);
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mpq_div(a3, a3, q);
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// x = -tq, y = -q/t
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mpq_mul(x, q, t);
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mpq_sub(x, zero, x);
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mpq_div(y, q, t);
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mpq_sub(y, zero, y);
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// actually, we want minus everything
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mat_zero(r1);
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mat_zero(r2);
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mat_zero(r3);
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mpq_set_si(*mat_ref(r1, 0, 0), -1, 1);
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mpq_set_si(*mat_ref(r1, 1, 1), 1, 1);
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mpq_set_si(*mat_ref(r1, 2, 2), 1, 1);
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mpq_set_si(*mat_ref(r2, 0, 0), 1, 1);
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mpq_set_si(*mat_ref(r2, 1, 1), -1, 1);
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mpq_set_si(*mat_ref(r2, 2, 2), 1, 1);
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mpq_set_si(*mat_ref(r3, 0, 0), 1, 1);
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mpq_set_si(*mat_ref(r3, 1, 1), 1, 1);
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mpq_set_si(*mat_ref(r3, 2, 2), -1, 1);
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// q^2 = xy = 1 + 1/s^2
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// [ -s s*y 0]
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// [ -s*x s*x*y - 1/s 0]
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// [ -s*y s*y^2 - x 1]
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LOOP(i,3) {
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mat_zero(gen[i]);
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mpq_sub(tmp, zero, s);
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mat_set(gen[i%3], i%3, i%3, tmp);
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mpq_mul(tmp, s, y);
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mat_set(gen[i%3], i%3, (i+1)%3, tmp);
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mpq_mul(tmp, s, x);
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mpq_sub(tmp, zero, tmp);
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mat_set(gen[i%3], (i+1)%3, i%3, tmp);
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mpq_mul(tmp, s, x);
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mpq_mul(tmp, tmp, y);
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mpq_sub(tmp, tmp, sinv);
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mat_set(gen[i%3], (i+1)%3, (i+1)%3, tmp);
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mpq_mul(tmp, s, y);
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mpq_sub(tmp, zero, tmp);
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mat_set(gen[i%3], (i+2)%3, i%3, tmp);
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mpq_mul(tmp, s, y);
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mpq_mul(tmp, tmp, y);
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mpq_sub(tmp, tmp, x);
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mat_set(gen[i%3], (i+2)%3, (i+1)%3, tmp);
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mat_set(gen[i%3], (i+2)%3, (i+2)%3, one);
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}
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mpq_set(*mat_ref(r1, 1, 0), b1);
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mpq_set(*mat_ref(r1, 2, 0), c1);
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mpq_set(*mat_ref(r2, 0, 1), a2);
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mpq_set(*mat_ref(r2, 2, 1), c2);
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mpq_set(*mat_ref(r3, 0, 2), a3);
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mpq_set(*mat_ref(r3, 1, 2), b3);
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mat_zero(gen[0]);
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mat_zero(gen[1]);
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mat_zero(gen[2]);
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mpq_set_ui(*mat_ref(gen[0], 0, 0), 1, 1);
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mat_set(gen[0], 1, 1, sinv);
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mat_set(gen[0], 2, 2, s);
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mat_set(gen[1], 0, 0, s);
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mpq_set_ui(*mat_ref(gen[1], 1, 1), 1, 1);
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mat_set(gen[1], 2, 2, sinv);
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mat_set(gen[2], 0, 0, sinv);
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mat_set(gen[2], 1, 1, s);
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mpq_set_ui(*mat_ref(gen[2], 2, 2), 1, 1);
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mat_multiply(ws, gen[0], r2, gen[0]);
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mat_multiply(ws, gen[0], gen[0], r3);
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mat_multiply(ws, gen[1], r3, gen[1]);
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mat_multiply(ws, gen[1], gen[1], r1);
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mat_multiply(ws, gen[2], r1, gen[2]);
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mat_multiply(ws, gen[2], gen[2], r2);
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mat_pseudoinverse(ws, gen[3], gen[0]);
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mat_pseudoinverse(ws, gen[4], gen[1]);
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mat_pseudoinverse(ws, gen[5], gen[2]);
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LOOP(i,3) mat_pseudoinverse(ws, gen[i+3], gen[i]);
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// debug output
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/*
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mat_print(r1);
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mat_print(r2);
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mat_print(r3);
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gmp_printf("m = %Qd, s = %Qd, t = %Qd, q = %Qd, x = %Qd, y = %Qd\n", m, s, t, q, x, y);
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mat_print(gen[0]);
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mat_print(gen[1]);
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mat_print(gen[2]);
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mat_print(gen[3]);
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mat_print(gen[4]);
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mat_print(gen[5]);
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*/
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mpq_clears(sinv,rho1,rho2,rho3,b1,c1,a2,c2,a3,b3,NULL);
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mat_clear(r1);
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mat_clear(r2);
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mat_clear(r3);
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mpq_inits(s,sinv,q,x,y,zero,one,two,tmp,NULL);
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}
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char *print_word(groupelement_t *g, char *str)
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@ -232,7 +223,7 @@ char *print_word(groupelement_t *g, char *str)
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return str;
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}
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void enumerate(group_t *group, mat *matrices, mpq_t s, mpq_t t)
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void enumerate(group_t *group, mat *matrices, mpq_t m, mpq_t t)
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{
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mat_workspace *ws;
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mat tmp;
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@ -245,7 +236,7 @@ void enumerate(group_t *group, mat *matrices, mpq_t s, mpq_t t)
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mat_init(gen[i], 3);
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mat_init(tmp, 3);
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initialize_triangle_generators(ws, gen, s, t);
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initialize_triangle_generators(ws, gen, m, t);
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mat_identity(matrices[0]);
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for(int i = 1; i < group->size; i++) {
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@ -352,8 +343,8 @@ double max_slope(groupelement_t *group, mat *matrices, mpq_t s, mpq_t t, int *in
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int main(int argc, char *argv[])
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{
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mpq_t s, q, t, tmp;
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double sapprox, tapprox, qapprox, tqfactor;
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mpq_t m, t, tmp;
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double s;
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mat *matrices;
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group_t *group;
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int index;
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@ -369,11 +360,13 @@ int main(int argc, char *argv[])
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struct result *invariants;
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struct result **distinct_invariants;
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clock_gettime(CLOCK_REALTIME, &starttime);
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nmax = atoi(argv[1]);
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DEBUG("Allocate\n");
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mpq_inits(s, q, t, tmp, NULL);
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mpq_inits(m, t, tmp, NULL);
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matrices = malloc(nmax*sizeof(mat));
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for(int i = 0; i < nmax; i++)
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mat_init(matrices[i], 3);
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@ -389,6 +382,7 @@ int main(int argc, char *argv[])
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mps_context_set_output_prec(solver, 20); // relative precision
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mps_context_set_output_goal(solver, MPS_OUTPUT_GOAL_APPROXIMATE);
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/*
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DEBUG("Approximate parameters\n");
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// get approximate s and q values
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@ -420,81 +414,86 @@ int main(int argc, char *argv[])
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#ifdef OUTPUT_POINTS
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gmp_fprintf(stdout, "\"s = %Qd = %.3f, q = %Qd, t = %.3f\"\n", s, mpq_get_d(s), q, mpq_get_d(q)*tqfactor);
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#endif
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*/
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// group
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DEBUG("Generate group\n");
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group = coxeter_init_triangle(4, 3, 3, nmax);
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group = coxeter_init_triangle(4, 4, 4, nmax);
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DEBUG("Compute matrices\n");
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enumerate(group, matrices, s, q);
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fprintf(stderr, "max word length = %d\n", group->elements[nmax-1].length);
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n = 0;
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DEBUG("Compute traces\n");
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for(int i = 0; i < nmax; i++) {
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if(group->elements[i].length % 2 != 0 || !group->elements[i].inverse)
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continue;
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for(int ttick = 45; ttick <= 65; ttick++) {
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for(int mtick = 45; mtick < 65; mtick++) {
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mpq_set_ui(t, ttick, 100);
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mpq_set_ui(m, mtick, 100); // 414/1000 ~ sqrt(2)-1 <-> s=1
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s = (1-mpq_get_d(m)*mpq_get_d(m))/(2*mpq_get_d(m));
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mat_trace(invariants[i].tr, matrices[i]);
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mat_trace(invariants[i].trinv, matrices[group->elements[i].inverse->id]);
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DEBUG("Compute matrices\n");
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enumerate(group, matrices, m, t);
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distinct_invariants[n++] = &invariants[i];
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}
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n = 0;
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DEBUG("Compute traces\n");
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for(int i = 0; i < nmax; i++) {
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if(group->elements[i].length % 2 != 0 || !group->elements[i].inverse)
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continue;
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DEBUG("Get unique traces\n");
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mat_trace(invariants[i].tr, matrices[i]);
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mat_trace(invariants[i].trinv, matrices[group->elements[i].inverse->id]);
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qsort(distinct_invariants, n, sizeof(struct result*), compare_result);
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distinct_invariants[n++] = &invariants[i];
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nuniq = 0;
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for(int i = 0; i < n; i++) {
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if(i == 0 || compare_result(&distinct_invariants[i], &distinct_invariants[nuniq-1]) != 0)
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distinct_invariants[nuniq++] = distinct_invariants[i];
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}
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// gmp_printf("%Qd %Qd %d %s\n", invariants[i].tr, invariants[i].trinv, i, print_word(&group->elements[i], buf));
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}
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max_slope = 0;
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int max_slope_index;
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DEBUG("Get unique traces\n");
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DEBUG("Solve characteristic polynomials\n");
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for(int i = 0; i < nuniq; i++) {
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retval = solve_characteristic_polynomial(solver, distinct_invariants[i]->tr, distinct_invariants[i]->trinv, evs);
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if(retval == 1) {
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fprintf(stderr, "Error! Could not solve polynomial.\n");
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continue;
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} else if(retval == 2) {
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continue;
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}
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qsort(distinct_invariants, n, sizeof(struct result*), compare_result);
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if(fabs(evs[0]) < fabs(evs[1]))
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SWAP(double, evs[0], evs[1]);
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if(fabs(evs[1]) < fabs(evs[2]))
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SWAP(double, evs[1], evs[2]);
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if(fabs(evs[0]) < fabs(evs[1]))
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nuniq = 0;
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for(int i = 0; i < n; i++) {
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if(i == 0 || compare_result(&distinct_invariants[i], &distinct_invariants[nuniq-1]) != 0)
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distinct_invariants[nuniq++] = distinct_invariants[i];
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}
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max_slope = 0;
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int max_slope_index;
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DEBUG("Solve characteristic polynomials\n");
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for(int i = 0; i < nuniq; i++) {
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retval = solve_characteristic_polynomial(solver, distinct_invariants[i]->tr, distinct_invariants[i]->trinv, evs);
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if(retval == 1) {
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fprintf(stderr, "Error! Could not solve polynomial.\n");
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continue;
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} else if(retval == 2) {
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continue;
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}
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if(fabs(evs[0]) < fabs(evs[1]))
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SWAP(double, evs[0], evs[1]);
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if(fabs(evs[1]) < fabs(evs[2]))
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SWAP(double, evs[1], evs[2]);
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if(fabs(evs[0]) < fabs(evs[1]))
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SWAP(double, evs[0], evs[1]);
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double x = log(fabs(evs[0]));
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double y = -log(fabs(evs[2]));
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double x = log(fabs(evs[0]));
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double y = -log(fabs(evs[2]));
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if(y/x > max_slope && (x > 0.1 || y > 0.1)) {
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max_slope_index = distinct_invariants[i] - invariants;
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max_slope = y/x;
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}
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if(y/x > max_slope && (x > 0.1 || y > 0.1)) {
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max_slope_index = distinct_invariants[i] - invariants;
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max_slope = y/x;
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}
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#ifdef OUTPUT_POINTS
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gmp_printf("%Qd %Qd %f %f %f %d\n", distinct_invariants[i]->tr, distinct_invariants[i]->trinv, x, y, y/x);
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gmp_printf("%Qd %Qd %f %f %f\n", distinct_invariants[i]->tr, distinct_invariants[i]->trinv, x, y, y/x);
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#endif
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}
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}
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#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));
|
||||
// fprintf(stdout, "%.5f %.5f %.5f %f %s\n", mpq_get_d(t), mpq_get_d(m), s, max_slope, print_word(&group->elements[max_slope_index], buf));
|
||||
fprintf(stdout, "%.5f %.5f %.5f %f %s\n", mpq_get_d(t), mpq_get_d(m), s, 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");
|
||||
for(int i = 0; i < nmax; i++) {
|
||||
|
|
@ -507,6 +506,6 @@ int main(int argc, char *argv[])
|
|||
mat_clear(matrices[i]);
|
||||
free(matrices);
|
||||
coxeter_clear(group);
|
||||
mpq_clears(s, q, t, tmp, NULL);
|
||||
mpq_clears(m, t, tmp, NULL);
|
||||
mps_context_free(solver);
|
||||
}
|
||||
|
|
|
|||
Loading…
Reference in New Issue