Tribonacci

ca.h

@@ -317,6 +317,8 @@ void ca_pi(ca_t res, ca_ctx_t ctx);
 void ca_pi_i(ca_t res, ca_ctx_t ctx);
 void ca_euler(ca_t res, ca_ctx_t ctx);
 
+void ca_tribonacci_constant(ca_t res, ca_ctx_t ctx);
+
 void ca_unknown(ca_t x, ca_ctx_t ctx);
 
 void ca_undefined(ca_t x, ca_ctx_t ctx);

ca/set_fexpr.c

@@ -79,6 +79,9 @@ _ca_set_fexpr(ca_t res, fexpr_vec_t inputs, ca_vec_t outputs, const fexpr_t expr
                 ca_add_ui(res, res, 1, ctx);
                 ca_div_ui(res, res, 2, ctx);
                 return 1;
+            case FEXPR_TribonacciConstant:
+                ca_tribonacci_constant(res, ctx);
+                return 1;
             case FEXPR_Infinity:
                 ca_pos_inf(res, ctx);
                 return 1;

ca/tribonacci.c

@@ -0,0 +1,24 @@
+/*
+    Copyright (C) 2022 Raoul Bourquin
+
+    This file is part of Calcium.
+
+    Calcium is free software: you can redistribute it and/or modify it under
+    the terms of the GNU Lesser General Public License (LGPL) as published
+    by the Free Software Foundation; either version 2.1 of the License, or
+    (at your option) any later version.  See <http://www.gnu.org/licenses/>.
+*/
+
+#include "ca.h"
+
+void
+ca_tribonacci_constant(ca_t res, ca_ctx_t ctx)
+{
+    qqbar_t tc;
+    qqbar_init(tc);
+    qqbar_tribonacci_constant(tc);
+
+    ca_set_qqbar(res, tc, ctx);
+
+    qqbar_clear(tc);
+}

doc/source/bibliography.rst

@@ -5,7 +5,7 @@ Bibliography
 
 (In the PDF edition, this section is empty. See the bibliography listing at the end of the document.)
 
-All referenced works: [BBK2014]_, [BF2020]_, [BFSS2006]_, [Boe2020]_, [Car2004]_, [Cho1999]_, [Coh1996]_, [Coh2000]_, [Fie2007]_, [GCL1992]_, [Har2010]_, [Har2015]_, [Har2018]_, [Joh2017]_, [JR1999]_, [Mos1971]_, [MP2006]_, [RF1994]_, [Ric1992]_, [Ric1995]_, [Ric1997]_, [Ric2007]_, [Ric2009]_, [Ste2002]_, [Ste2010]_, [Str1997]_, [Str2012]_, [vdH1995]_, [vdH2006]_, [vHP2012]_, [Zip1985]_.
+All referenced works: [BBK2014]_, [BF2020]_, [BFSS2006]_, [Boe2020]_, [Car2004]_, [Cho1999]_, [Coh1996]_, [Coh2000]_, [Fie2007]_, [GCL1992]_, [Har2010]_, [Har2015]_, [Har2018]_, [Joh2017]_, [JR1999]_, [Mos1971]_, [MP2006]_, [RF1994]_, [Ric1992]_, [Ric1995]_, [Ric1997]_, [Ric2007]_, [Ric2009]_, [Ste2002]_, [Ste2010]_, [Str1997]_, [Str2012]_, [Wol1998]_, [vdH1995]_, [vdH2006]_, [vHP2012]_, [Zip1985]_.
 
 .. [BBK2014] \D. H. Bailey, J. M. Borwein and A. D. Kaiser. "Automated simplification of large symbolic expressions". Journal of Symbolic Computation Volume 60, January 2014, Pages 120-136. https://doi.org/10.1016/j.jsc.2013.09.001
 
@@ -61,6 +61,8 @@ All referenced works: [BBK2014]_, [BF2020]_, [BFSS2006]_, [Boe2020]_, [Car2004]_
 
 .. [Str2012] \A. Strzebonski. "Real root isolation for exp-log-arctan functions". Journal of Symbolic Computation 47 (2012) 282–314. https://doi.org/10.1016/j.jsc.2011.11.004
 
+.. [Wol1998] \D. A. Wolfram. "Solving Generalized Fibonacci Recurrences". The Fibonacci Quarterly 36 (1998) 129-145.
+
 .. [vHP2012] \M. van Hoeij and V. Pal. "Isomorphisms of algebraic number fields". Journal de Théorie des Nombres de Bordeaux, Vol. 24, No. 2 (2012), pp. 293-305. https://doi.org/10.2307/43973105
 
 .. [vdH1995] \J. van der Hoeven, "Automatic numerical expansions". Proc. of the conference Real numbers and computers (1995), 261-274. https://www.texmacs.org/joris/ane/ane-abs.html

doc/source/ca.rst

@@ -349,6 +349,11 @@ Special values
     Sets *res* to Euler's constant `\gamma`. This creates an element
     of the (transcendental?) number field `\mathbb{Q}(\gamma)`.
 
+.. function:: void ca_tribonacci_constant(ca_t res, ca_ctx_t ctx)
+
+    Sets *res* to the Tribonacci constant `T_c`. This creates an element
+    of the algebraic number field `\mathbb{Q}(T_c)`.
+
 .. function:: void ca_unknown(ca_t res, ca_ctx_t ctx)
 
     Sets *res* to the meta-value *Unknown*.

doc/source/fexpr_builtin.rst

@@ -326,6 +326,10 @@ Particular numbers
 
     ``GoldenRatio`` is the golden ratio `\varphi`.
 
+.. macro:: TribonacciConstant
+
+    ``TribonacciConstant`` is the Tribonacci constant `T_c`.
+
 .. macro:: Euler
 
     ``Euler`` is Euler's constant `\gamma`.

doc/source/qqbar.rst

@@ -199,6 +199,10 @@ Special values
 
     Sets *res* to the golden ratio `\varphi = \tfrac{1}{2}(\sqrt{5} + 1)`.
 
+.. function:: void qqbar_tribonacci_constant(qqbar_t res)
+
+    Sets *res* to the Tribonacci constant `T_c = \tfrac{1}{3}(1 + \sqrt[3]{19 + 3\sqrt{33}} + \sqrt[3]{19 - 3\sqrt{33}})`.
+
 Input and output
 -------------------------------------------------------------------------------
 
@@ -811,7 +815,7 @@ Symbolic expressions and conversion to radicals
     * Arithmetic operations with algebraic operands
     * Square roots of algebraic numbers
     * Powers with algebraic base and exponent an explicit rational number
-    * NumberI, GoldenRatio, RootOfUnity
+    * NumberI, GoldenRatio, TribonacciConstant, RootOfUnity
     * Floor, Ceil, Abs, Sign, Csgn, Conjugate, Re, Im, Max, Min
     * Trigonometric functions with argument an explicit rational number times Pi
     * Exponentials with argument an explicit rational number times Pi * NumberI

examples/tribonacci.c

@@ -0,0 +1,64 @@
+/* This file is public domain. Author: Raoul Bourquin. */
+
+#include "ca.h"
+
+
+void main_fexpr()
+{
+    fexpr_t T;
+    fexpr_init(T);
+
+    flint_printf("Evaluating Tc as fexpr:\n");
+
+    fexpr_set_symbol_str(T, "TribonacciConstant");
+
+    fexpr_print(T);
+    printf("\n\n");
+
+    fexpr_clear(T);
+}
+
+
+void main_ca()
+{
+    ca_ctx_t ctx;
+    ca_t T;
+    ca_ctx_init(ctx);
+    ca_init(T, ctx);
+
+    flint_printf("Evaluating Tc as ca:\n");
+
+    ca_tribonacci_constant(T, ctx);
+
+    ca_print(T, ctx);
+    printf("\n\n");
+
+    ca_clear(T, ctx);
+}
+
+
+void main_qqbar()
+{
+    qqbar_t T;
+    qqbar_init(T);
+
+    flint_printf("Evaluating Tc as qqbar:\n");
+
+    qqbar_tribonacci_constant(T);
+
+    qqbar_printn(T, 50);
+    printf("\n");
+
+    qqbar_clear(T);
+}
+
+
+int main(int argc, char *argv[])
+{
+  main_fexpr();
+  main_ca();
+  main_qqbar();
+
+  flint_cleanup();
+  return EXIT_SUCCESS;
+}

fexpr/numerical_enclosure.c

@@ -129,6 +129,52 @@ fexpr_get_acb_raw(acb_t res, const fexpr_t expr, slong prec)
             return 1;
         }
 
+        if (op == FEXPR_TribonacciConstant)
+        {
+            /* Subexpressions */
+            arb_t r33, r33p, r33m;
+
+            /* Init */
+            arb_init(r33);
+            arb_init(r33p);
+            arb_init(r33m);
+
+            /* r33 := 3*sqrt(33) */
+            arb_sqrt_ui(r33, 33, prec);
+            arb_mul_ui(r33, r33, 3, prec);
+
+            /* r33p := cbrt(19 + r33) */
+            arb_add_ui(r33p, r33, 19, prec);
+            arb_root_ui(r33p, r33p, 3, prec);
+
+            /* r33m := cbrt(19 - r33) */
+            arb_sub_si(r33m, r33, 19, prec);
+            arb_neg(r33m, r33m);
+            arb_root_ui(r33m, r33m, 3, prec);
+
+            /* res := 1 */
+            arb_one(acb_realref(res));
+
+            /* res += r33p */
+            arb_add(acb_realref(res), acb_realref(res), r33p, prec);
+
+            /* res += r33m */
+            arb_add(acb_realref(res), acb_realref(res), r33m, prec);
+
+            /* res /= 3 */
+            arb_div_ui(acb_realref(res), acb_realref(res), 3, prec);
+
+            /* zero imag part */
+            arb_zero(acb_imagref(res));
+
+            /* Free */
+            arb_clear(r33);
+            arb_clear(r33p);
+            arb_clear(r33m);
+
+            return 1;
+        }
+
         acb_indeterminate(res);
         return 0;
     }

fexpr_builtin.h

@@ -433,6 +433,7 @@ typedef enum
     FEXPR_Tanh,
     FEXPR_Theta,
     FEXPR_Theta_,
+    FEXPR_TribonacciConstant,
     FEXPR_True,
     FEXPR_Tuple,
     FEXPR_Tuples,

fexpr_builtin/table.c

@@ -415,6 +415,7 @@ const fexpr_symbol_info fexpr_builtin_table[FEXPR_BUILTIN_LENGTH] = {
     { FEXPR_Tanh, "Tanh", "\\tanh", NULL, },
     { FEXPR_Theta, "Theta", "\\Theta", NULL },
     { FEXPR_Theta_, "Theta_", "\\Theta", fexpr_write_latex_subscript },
+    { FEXPR_TribonacciConstant, "TribonacciConstant", "T_c", NULL, },
     { FEXPR_True, "True", "\\operatorname{True}", NULL, },
     { FEXPR_Tuple, "Tuple", "", fexpr_write_latex_collection, },
     { FEXPR_Tuples, "Tuples", "", NULL, },

qqbar.h

@@ -207,6 +207,8 @@ void qqbar_i(qqbar_t res);
 
 void qqbar_phi(qqbar_t res);
 
+void qqbar_tribonacci_constant(qqbar_t res);
+
 /* Random generation */
 
 void qqbar_randtest(qqbar_t res, flint_rand_t state, slong deg, slong bits);

qqbar/set_fexpr.c

@@ -598,6 +598,12 @@ qqbar_set_fexpr(qqbar_t res, const fexpr_t expr)
             return 1;
         }
 
+        if (fexpr_is_builtin_symbol(expr, FEXPR_TribonacciConstant))
+        {
+            qqbar_tribonacci_constant(res);
+            return 1;
+        }
+
         return 0;
     }
 

qqbar/tribonacci.c

@@ -0,0 +1,63 @@
+/*
+    Copyright (C) 2022 Raoul Bourquin
+
+    This file is part of Calcium.
+
+    Calcium is free software: you can redistribute it and/or modify it under
+    the terms of the GNU Lesser General Public License (LGPL) as published
+    by the Free Software Foundation; either version 2.1 of the License, or
+    (at your option) any later version.  See <http://www.gnu.org/licenses/>.
+*/
+
+#include "qqbar.h"
+
+void
+qqbar_tribonacci_constant(qqbar_t res)
+{
+    /* Subexpressions */
+    arb_t r33, r33p, r33m;
+
+    fmpz_poly_zero(QQBAR_POLY(res));
+    fmpz_poly_set_coeff_si(QQBAR_POLY(res), 3, 1);
+    fmpz_poly_set_coeff_si(QQBAR_POLY(res), 2, -1);
+    fmpz_poly_set_coeff_si(QQBAR_POLY(res), 1, -1);
+    fmpz_poly_set_coeff_si(QQBAR_POLY(res), 0, -1);
+
+    /* Init */
+    arb_init(r33);
+    arb_init(r33p);
+    arb_init(r33m);
+
+    /* r33 := 3*sqrt(33) */
+    arb_sqrt_ui(r33, 33, QQBAR_DEFAULT_PREC);
+    arb_mul_ui(r33, r33, 3, QQBAR_DEFAULT_PREC);
+
+    /* r33p := cbrt(19 + r33) */
+    arb_add_ui(r33p, r33, 19, QQBAR_DEFAULT_PREC);
+    arb_root_ui(r33p, r33p, 3, QQBAR_DEFAULT_PREC);
+
+    /* r33m := cbrt(19 - r33) */
+    arb_sub_si(r33m, r33, 19, QQBAR_DEFAULT_PREC);
+    arb_neg(r33m, r33m);
+    arb_root_ui(r33m, r33m, 3, QQBAR_DEFAULT_PREC);
+
+    /* res := 1 */
+    arb_one(acb_realref(QQBAR_ENCLOSURE(res)));
+
+    /* res += r33p */
+    arb_add(acb_realref(QQBAR_ENCLOSURE(res)), acb_realref(QQBAR_ENCLOSURE(res)), r33p, QQBAR_DEFAULT_PREC);
+
+    /* res += r33m */
+    arb_add(acb_realref(QQBAR_ENCLOSURE(res)), acb_realref(QQBAR_ENCLOSURE(res)), r33m, QQBAR_DEFAULT_PREC);
+
+    /* res /= 3 */
+    arb_div_ui(acb_realref(QQBAR_ENCLOSURE(res)), acb_realref(QQBAR_ENCLOSURE(res)), 3, QQBAR_DEFAULT_PREC);
+
+    /* zero imag part */
+    arb_zero(acb_imagref(QQBAR_ENCLOSURE(res)));
+
+    /* Free */
+    arb_clear(r33);
+    arb_clear(r33p);
+    arb_clear(r33m);
+}