Tetranacci

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_tetranacci_constant(ca_t res, ca_ctx_t ctx)
+
+    Sets *res* to the Tetranacci constant `T_t`. This creates an element
+    of the algebraic number field `\mathbb{Q}(T_t)`.
+
 .. 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:: TetranacciConstant
+
+    ``TetranacciConstant`` is the Tetranacci constant `T_t`.
+
 .. 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_tetranacci_constant(qqbar_t res)
+
+    Sets *res* to the Tetranacci constant `T_t`.
+
 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, TetranacciConstant, 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/tetranacci.c

@@ -0,0 +1,65 @@
+/* This file is public domain. Author: Raoul Bourquin. */
+
+#include <stdlib.h>
+#include "ca.h"
+
+
+void main_fexpr()
+{
+    fexpr_t T;
+    fexpr_init(T);
+
+    flint_printf("Evaluating Tt as fexpr:\n");
+
+    fexpr_set_symbol_str(T, "TetranacciConstant");
+
+    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 Tt as ca:\n");
+
+    ca_tetranacci_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 Tt as qqbar:\n");
+
+    qqbar_tetranacci_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;
+}

src/ca.h

@@ -314,6 +314,8 @@ 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_tetranacci_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);

src/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_TetranacciConstant:
+                ca_tetranacci_constant(res, ctx);
+                return 1;
             case FEXPR_Infinity:
                 ca_pos_inf(res, ctx);
                 return 1;

src/ca/tetranacci.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_tetranacci_constant(ca_t res, ca_ctx_t ctx)
+{
+    qqbar_t tc;
+    qqbar_init(tc);
+    qqbar_tetranacci_constant(tc);
+
+    ca_set_qqbar(res, tc, ctx);
+
+    qqbar_clear(tc);
+}

src/fexpr/numerical_enclosure.c

@@ -129,6 +129,92 @@ fexpr_get_acb_raw(acb_t res, const fexpr_t expr, slong prec)
             return 1;
         }
 
+        if (op == FEXPR_TetranacciConstant)
+        {
+            /* Subexpressions */
+            arb_t t1, t2, l1, p1;
+
+            /* Init */
+            arb_init(t1);
+            arb_init(t2);
+            arb_init(l1);
+            arb_init(p1);
+
+            /* t1 := 3*sqrt(1689) */
+            arb_sqrt_ui(t1, 1689, prec);
+            arb_mul_ui(t1, t1, 3, prec);
+
+            /* l1 := (cbrt(t1 - 65) - cbrt(t1 + 65)) / (12 * cbrt(2)) */
+            arb_zero(l1);
+
+            /* First part of l1 */
+            arb_sub_ui(t2, t1, 65, prec);
+            arb_root_ui(t2, t2, 3, prec);
+
+            arb_add(l1, l1, t2, prec);
+
+            /* Second part of l1 */
+            arb_add_ui(t2, t1, 65, prec);
+            arb_root_ui(t2, t2, 3, prec);
+
+            arb_sub(l1, l1, t2, prec);
+
+            /* Denominator of l1 */
+            arb_set_ui(t2, 2);
+            arb_root_ui(t2, t2, 3, prec);
+            arb_mul_ui(t2, t2, 12, prec);
+
+            /* Combine l1 */
+            arb_div(l1, l1, t2, prec);
+
+            /* p1 := sqrt(l1 + 11/48) */
+            arb_set_ui(p1, 11);
+            arb_div_ui(p1, p1, 48, prec);
+
+            arb_add(p1, p1, l1, prec);
+            arb_sqrt(p1, p1, prec);
+
+            /* t2 := 1/4 - p1 */
+            arb_one(t2);
+            arb_mul_2exp_si(t2, t2, -2);
+            arb_add(t2, t2, p1, prec);
+
+            /* Invert p1 */
+            arb_inv(p1, p1, prec);
+
+            /* Final result */
+            arb_one(acb_realref(res));
+            arb_div_ui(acb_realref(res), acb_realref(res), 6, prec);
+
+            arb_set_ui(t1, 7);
+            arb_div_ui(t1, t1, 24, prec);
+            arb_mul(t1, t1, p1, prec);
+            arb_add(acb_realref(res), acb_realref(res), t1, prec);
+
+            arb_sqr(t1, t2, prec);
+            arb_add(acb_realref(res), acb_realref(res), t1, prec);
+
+            arb_mul_ui(t1, t2, 2, prec);
+            arb_mul(t1, t1, l1, prec);
+            arb_mul(t1, t1, p1, prec);
+            arb_sub(acb_realref(res), acb_realref(res), t1, prec);
+
+            arb_sqrt(acb_realref(res), acb_realref(res), prec);
+
+            arb_add(acb_realref(res), acb_realref(res), t2, prec);
+
+            /* zero imag part */
+            arb_zero(acb_imagref(res));
+
+            /* Free */
+            arb_clear(t1);
+            arb_clear(t2);
+            arb_clear(l1);
+            arb_clear(p1);
+
+            return 1;
+        }
+
         acb_indeterminate(res);
         return 0;
     }

src/fexpr_builtin.h

@@ -431,6 +431,7 @@ typedef enum
     FEXPR_SymmetricPolynomial,
     FEXPR_Tan,
     FEXPR_Tanh,
+    FEXPR_TetranacciConstant,
     FEXPR_Theta,
     FEXPR_Theta_,
     FEXPR_True,

src/fexpr_builtin/table.c

@@ -413,6 +413,7 @@ const fexpr_symbol_info fexpr_builtin_table[FEXPR_BUILTIN_LENGTH] = {
     { FEXPR_SymmetricPolynomial, "SymmetricPolynomial", "e", fexpr_write_latex_subscript_call, },
     { FEXPR_Tan, "Tan", "\\tan", NULL, },
     { FEXPR_Tanh, "Tanh", "\\tanh", NULL, },
+    { FEXPR_TetranacciConstant, "TetranacciConstant", "T_t", NULL, },
     { FEXPR_Theta, "Theta", "\\Theta", NULL },
     { FEXPR_Theta_, "Theta_", "\\Theta", fexpr_write_latex_subscript },
     { FEXPR_True, "True", "\\operatorname{True}", NULL, },

src/qqbar.h

@@ -202,6 +202,8 @@ void qqbar_i(qqbar_t res);
 
 void qqbar_phi(qqbar_t res);
 
+void qqbar_tetranacci_constant(qqbar_t res);
+
 /* Random generation */
 
 void qqbar_randtest(qqbar_t res, flint_rand_t state, slong deg, slong bits);

src/qqbar/set_fexpr.c

@@ -609,6 +609,12 @@ qqbar_set_fexpr(qqbar_t res, const fexpr_t expr)
             return 1;
         }
 
+        if (fexpr_is_builtin_symbol(expr, FEXPR_TetranacciConstant))
+        {
+            qqbar_tetranacci_constant(res);
+            return 1;
+        }
+
         return 0;
     }
 

src/qqbar/tetranacci.c

@@ -0,0 +1,105 @@
+/*
+    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 "fmpz_poly.h"
+#include "qqbar.h"
+
+void
+qqbar_tetranacci_constant(qqbar_t res)
+{
+    /* Subexpressions */
+    arb_t t1, t2, l1, p1;
+
+    fmpz_poly_zero(QQBAR_POLY(res));
+    fmpz_poly_set_coeff_si(QQBAR_POLY(res), 4, 1);
+    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(t1);
+    arb_init(t2);
+    arb_init(l1);
+    arb_init(p1);
+
+    /* t1 := 3*sqrt(1689) */
+    arb_sqrt_ui(t1, 1689, QQBAR_DEFAULT_PREC);
+    arb_mul_ui(t1, t1, 3, QQBAR_DEFAULT_PREC);
+
+    /* l1 := (cbrt(t1 - 65) - cbrt(t1 + 65)) / (12 * cbrt(2)) */
+    arb_zero(l1);
+
+    /* First part of l1 */
+    arb_sub_ui(t2, t1, 65, QQBAR_DEFAULT_PREC);
+    arb_root_ui(t2, t2, 3, QQBAR_DEFAULT_PREC);
+
+    arb_add(l1, l1, t2, QQBAR_DEFAULT_PREC);
+
+    /* Second part of l1 */
+    arb_add_ui(t2, t1, 65, QQBAR_DEFAULT_PREC);
+    arb_root_ui(t2, t2, 3, QQBAR_DEFAULT_PREC);
+
+    arb_sub(l1, l1, t2, QQBAR_DEFAULT_PREC);
+
+    /* Denominator of l1 */
+    arb_set_ui(t2, 2);
+    arb_root_ui(t2, t2, 3, QQBAR_DEFAULT_PREC);
+    arb_mul_ui(t2, t2, 12, QQBAR_DEFAULT_PREC);
+
+    /* Combine l1 */
+    arb_div(l1, l1, t2, QQBAR_DEFAULT_PREC);
+
+    /* p1 := sqrt(l1 + 11/48) */
+    arb_set_ui(p1, 11);
+    arb_div_ui(p1, p1, 48, QQBAR_DEFAULT_PREC);
+
+    arb_add(p1, p1, l1, QQBAR_DEFAULT_PREC);
+    arb_sqrt(p1, p1, QQBAR_DEFAULT_PREC);
+
+    /* t2 := 1/4 - p1 */
+    arb_one(t2);
+    arb_mul_2exp_si(t2, t2, -2);
+    arb_add(t2, t2, p1, QQBAR_DEFAULT_PREC);
+
+    /* Invert p1 */
+    arb_inv(p1, p1, QQBAR_DEFAULT_PREC);
+
+    /* Final result */
+    arb_one(acb_realref(QQBAR_ENCLOSURE(res)));
+    arb_div_ui(acb_realref(QQBAR_ENCLOSURE(res)), acb_realref(QQBAR_ENCLOSURE(res)), 6, QQBAR_DEFAULT_PREC);
+
+    arb_set_ui(t1, 7);
+    arb_div_ui(t1, t1, 24, QQBAR_DEFAULT_PREC);
+    arb_mul(t1, t1, p1, QQBAR_DEFAULT_PREC);
+    arb_add(acb_realref(QQBAR_ENCLOSURE(res)), acb_realref(QQBAR_ENCLOSURE(res)), t1, QQBAR_DEFAULT_PREC);
+
+    arb_sqr(t1, t2, QQBAR_DEFAULT_PREC);
+    arb_add(acb_realref(QQBAR_ENCLOSURE(res)), acb_realref(QQBAR_ENCLOSURE(res)), t1, QQBAR_DEFAULT_PREC);
+
+    arb_mul_ui(t1, t2, 2, QQBAR_DEFAULT_PREC);
+    arb_mul(t1, t1, l1, QQBAR_DEFAULT_PREC);
+    arb_mul(t1, t1, p1, QQBAR_DEFAULT_PREC);
+    arb_sub(acb_realref(QQBAR_ENCLOSURE(res)), acb_realref(QQBAR_ENCLOSURE(res)), t1, QQBAR_DEFAULT_PREC);
+
+    arb_sqrt(acb_realref(QQBAR_ENCLOSURE(res)), acb_realref(QQBAR_ENCLOSURE(res)), QQBAR_DEFAULT_PREC);
+
+    arb_add(acb_realref(QQBAR_ENCLOSURE(res)), acb_realref(QQBAR_ENCLOSURE(res)), t2, QQBAR_DEFAULT_PREC);
+
+    /* zero imag part */
+    arb_zero(acb_imagref(QQBAR_ENCLOSURE(res)));
+
+    /* Free */
+    arb_clear(t1);
+    arb_clear(t2);
+    arb_clear(l1);
+    arb_clear(p1);
+}