long-reid-thistlethwaite groups with rationals
This commit is contained in:
Florian Stecker
parent
c566a8741e
commit
b7c7d7245c
@ -9,7 +9,33 @@
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#define MAX_ELEMENTS 14000
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//#define DRAW_PICTURE 1
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void mpq_quartic(mpq_t out, mpq_t in, int a, int b, int c, int d, int e)
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void continued_fraction_approximation(mpq_t out, double in, int level)
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{
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mpq_t tmp;
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if(in < 0) {
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mpq_init(tmp);
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mpq_set_ui(tmp, 0, 1);
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continued_fraction_approximation(out, -in, level);
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mpq_sub(out, tmp, out);
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mpq_clear(tmp);
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return;
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}
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if(level == 0) {
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mpq_set_si(out, (signed long int)round(in), 1); // floor(in)
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} else {
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continued_fraction_approximation(out, 1/(in - floor(in)), level - 1);
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mpq_init(tmp);
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mpq_set_ui(tmp, 1, 1);
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mpq_div(out, tmp, out); // out -> 1/out
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mpq_set_si(tmp, (signed long int)in, 1); // floor(in)
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mpq_add(out, out, tmp);
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mpq_clear(tmp);
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}
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}
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void quartic(mpq_t out, mpq_t in, int a, int b, int c, int d, int e)
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{
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mpq_t tmp;
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mpq_init(tmp);
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@ -46,12 +72,12 @@ void initialize_triangle_generators(mat_workspace *ws, mat *gen, mpq_t s, mpq_t
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mpq_set_ui(*mat_ref(gen[1], 0, 0), 1, 1);
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mpq_set_ui(*mat_ref(gen[1], 1, 0), 0, 1);
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mpq_set_ui(*mat_ref(gen[1], 2, 0), 0, 1);
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mpq_quartic(*mat_ref(gen[1], 0, 1), t, 0, 0, 1, -1, 2);
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mpq_quartic(*mat_ref(gen[1], 1, 1), t, 0, 0, -1, 2, -2);
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mpq_quartic(*mat_ref(gen[1], 2, 1), t, 0, 0, 1, -3, 3);
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mpq_quartic(*mat_ref(gen[1], 0, 2), t, 0, 0, 1, 0, 3);
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mpq_quartic(*mat_ref(gen[1], 1, 2), t, 0, 0, -1, 1, -1);
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mpq_quartic(*mat_ref(gen[1], 2, 2), t, 0, 0, 1, -2, 1);
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quartic(*mat_ref(gen[1], 0, 1), t, 0, 0, 1, -1, 2);
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quartic(*mat_ref(gen[1], 1, 1), t, 0, 0, -1, 2, -2);
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quartic(*mat_ref(gen[1], 2, 1), t, 0, 0, 1, -3, 3);
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quartic(*mat_ref(gen[1], 0, 2), t, 0, 0, 1, 0, 3);
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quartic(*mat_ref(gen[1], 1, 2), t, 0, 0, -1, 1, -1);
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quartic(*mat_ref(gen[1], 2, 2), t, 0, 0, 1, -2, 1);
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mat_pseudoinverse(ws, gen[3], gen[0]); // p^{-1}
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mat_pseudoinverse(ws, gen[4], gen[1]); // q^{-1}
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@ -79,95 +105,152 @@ char *print_word(groupelement_t *g, char *str)
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return str;
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}
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void enumerate(groupelement_t *group, mat *matrices, mpq_t s, mpq_t t)
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{
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mat_workspace *ws;
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mat tmp;
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mat gen[6];
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char buf[100], buf2[100], buf3[100];
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// allocate stuff
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ws = mat_workspace_init(3);
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for(int i = 0; i < 6; i++)
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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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mat_identity(matrices[0]);
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for(int i = 1; i < MAX_ELEMENTS; i++) {
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if(group[i].length % 2 != 0)
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continue;
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if(!group[i].inverse)
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continue;
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int parent = group[i].parent->id;
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int grandparent = group[i].parent->parent->id;
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int letter;
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if(group[parent].letter == 1 && group[i].letter == 2)
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letter = 0; // p = bc
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else if(group[parent].letter == 2 && group[i].letter == 0)
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letter = 1; // q = ca
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else if(group[parent].letter == 0 && group[i].letter == 1)
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letter = 2; // r = ab
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if(group[parent].letter == 2 && group[i].letter == 1)
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letter = 3; // p^{-1} = cb
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else if(group[parent].letter == 0 && group[i].letter == 2)
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letter = 4; // q^{-1} = ac
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else if(group[parent].letter == 1 && group[i].letter == 0)
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letter = 5; // r^{-1} = ba
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mat_multiply(ws, matrices[i], matrices[grandparent], gen[letter]);
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}
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// free stuff
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for(int i = 0; i < 6; i++)
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mat_clear(gen[i]);
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mat_clear(tmp);
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mat_workspace_clear(ws);
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}
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void output_invariants(groupelement_t *group, mat *matrices, mpq_t s, mpq_t t)
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{
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mpq_t tr, trinv;
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char buf[100];
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mpq_inits(tr, trinv, NULL);
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for(int i = 0; i < MAX_ELEMENTS; i++) {
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if(group[i].length % 2 != 0 || !group[i].inverse)
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continue;
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mat_trace(tr, matrices[i]);
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mat_trace(trinv, matrices[group[i].inverse->id]);
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gmp_printf("%d %d %s %Qd %Qd %f %f\n", i, group[i].length, print_word(&group[i], buf), tr, trinv, log(mpq_get_d(tr)), log(mpq_get_d(trinv)));
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}
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mpq_clears(tr, trinv, NULL);
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}
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double max_slope(groupelement_t *group, mat *matrices, mpq_t s, mpq_t t, int *index)
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{
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double max = 0;
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double slope;
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mpq_t tr, trinv;
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char buf[100];
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mpq_inits(tr, trinv, NULL);
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for(int i = 0; i < MAX_ELEMENTS; i++) {
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if(group[i].length % 2 != 0 || !group[i].inverse)
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continue;
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mat_trace(tr, matrices[i]);
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mat_trace(trinv, matrices[group[i].inverse->id]);
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slope = log(mpq_get_d(trinv))/log(mpq_get_d(tr));
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if(slope > max)
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{
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*index = i;
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max = slope;
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}
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}
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mpq_clears(tr, trinv, NULL);
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return max;
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}
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int main(int argc, char *argv[])
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{
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groupelement_t *group;
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mat_workspace *ws;
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mat *matrices;
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mat tmp;
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mat gen[6];
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char buf[100], buf2[100], buf3[100];
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mpq_t s,t;
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mpq_t det, tr, trinv;
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mpq_t s, t, tmp;
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mpz_t accuracy;
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double t_;
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mat *matrices;
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groupelement_t *group;
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int index;
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// allocate stuff
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group = malloc(MAX_ELEMENTS*sizeof(groupelement_t));
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ws = mat_workspace_init(3);
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matrices = malloc(MAX_ELEMENTS*sizeof(mat));
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for(int i = 0; i < MAX_ELEMENTS; i++)
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mat_init(matrices[i], 3);
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for(int i = 0; i < 6; i++)
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mat_init(gen[i], 3);
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mat_init(tmp, 3);
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mpq_inits(s, t, tmp, NULL);
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mpz_init(accuracy);
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group = malloc(MAX_ELEMENTS*sizeof(groupelement_t));
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matrices = malloc(MAX_ELEMENTS*sizeof(mat));
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for(int i = 0; i < MAX_ELEMENTS; i++)
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mat_init(matrices[i], 3);
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mpq_inits(s, t, det, tr, trinv, NULL);
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// mpq_set_str(t, argv[1], 10);
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mpz_set_ui(accuracy, 100);
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for(int i = 0; ; i++) {
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mpq_set(t, tmp);
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continued_fraction_approximation(tmp, atof(argv[1]), i);
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if(mpz_cmp(mpq_numref(tmp),accuracy) > 0 && mpz_cmp(mpq_denref(tmp),accuracy) > 0)
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break;
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}
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mpq_canonicalize(t);
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gmp_fprintf(stdout, "\"t = %Qd = %.3f\"\n", mpq_get_d(t), t);
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mpq_set_ui(s,1,1);
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double t_ = atof(argv[1]);
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mpq_set_ui(t,(int)(t_*100),100);
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mpq_canonicalize(s);
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mpq_canonicalize(t);
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if(argc > 2 && strcmp(argv[2],"p") == 0) {
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gmp_fprintf(stdout, "%Qd\n", t);
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return 0;
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}
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// the real action
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generate_triangle_group(group, MAX_ELEMENTS, 3, 3, 4);
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initialize_triangle_generators(ws, gen, s, t);
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generate_triangle_group(group, MAX_ELEMENTS, 3, 3, 4);
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mat_identity(matrices[0]);
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for(int i = 1; i < MAX_ELEMENTS; i++) {
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if(group[i].length % 2 != 0)
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continue;
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if(!group[i].inverse)
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continue;
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// for(int i = 0; i < 10; i++) {
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// mpq_set_ui(t,100+i,100);
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// mpq_canonicalize(t);
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int parent = group[i].parent->id;
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int grandparent = group[i].parent->parent->id;
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int letter;
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enumerate(group, matrices, s, t);
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//printf("%f %f\n", mpq_get_d(t), max_slope(group, matrices, s, t, &index));
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output_invariants(group, matrices, s, t);
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// }
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if(group[parent].letter == 1 && group[i].letter == 2)
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letter = 0; // p = bc
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else if(group[parent].letter == 2 && group[i].letter == 0)
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letter = 1; // q = ca
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else if(group[parent].letter == 0 && group[i].letter == 1)
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letter = 2; // r = ab
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if(group[parent].letter == 2 && group[i].letter == 1)
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letter = 3; // p^{-1} = cb
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else if(group[parent].letter == 0 && group[i].letter == 2)
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letter = 4; // q^{-1} = ac
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else if(group[parent].letter == 1 && group[i].letter == 0)
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letter = 5; // r^{-1} = ba
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mat_multiply(ws, matrices[i], matrices[grandparent], gen[letter]);
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}
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for(int i = 0; i < MAX_ELEMENTS; i++) {
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if(group[i].length % 2 != 0)
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continue;
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if(!group[i].inverse)
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continue;
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mat_trace(tr, matrices[i]);
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mat_trace(trinv, matrices[group[i].inverse->id]);
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double lambda1, lambda2, lambda3;
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// int realevs = gsl_poly_solve_cubic(-mpq_get_d(tr), mpq_get_d(trinv), -1, &lambda3, &lambda2, &lambda1);
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// if(realevs != 3)
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// continue;
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// if(lambda1 < 0 || lambda2 < 0 || lambda3 < 0)
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// continue;
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// gmp_printf("%d %d %s %Qd %Qd\n", i, group[i].length, print_word(&group[i], buf), tr, trinv);
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printf("%d %d %s %f %f %f\n", i, group[i].length, print_word(&group[i], buf), log(mpq_get_d(tr)), log(mpq_get_d(trinv)));
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}
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// free stuff
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for(int i = 0; i < MAX_ELEMENTS; i++)
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mat_clear(matrices[i]);
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for(int i = 0; i < 6; i++)
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mat_clear(gen[i]);
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mat_clear(tmp);
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mpq_clears(s, t, det, tr, trinv, NULL);
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mat_workspace_clear(ws);
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return 0;
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for(int i = 0; i < MAX_ELEMENTS; i++)
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mat_clear(matrices[i]);
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free(matrices);
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free(group);
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mpq_clears(s, t, tmp, NULL);
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mpz_clear(accuracy);
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}
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@ -3,7 +3,7 @@ if(!exists("logs")) logs = log(1.0)
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file = sprintf("< ./singular_values %f", exp(logt))
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#title = sprintf("s = %f, t = %f", exp(logs), exp(logt))
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title = sprintf("t = %.3f", floor(exp(logt)*100)/100.0)
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title = sprintf("t = %.3f", exp(logt))
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# print title
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set zeroaxis
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@ -23,7 +23,7 @@ set parametric
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tr(a,b) = exp(a) + exp(b-a) + exp(-b)
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trinv(a,b) = exp(-a) + exp(a-b) + exp(b)
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plot file using 4:5 w p pt 7 ps 0.5 lc 1 t title, \
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plot file using 6:7 w p pt 7 ps 0.5 lc 1 t columnheader, \
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log(tr(t,t*2)),log(trinv(t,2*t)) w l lw 2 t "", \
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log(tr(t,t/2)),log(trinv(t,t/2)) w l lw 2 t ""
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