notebook/curve_v1_newton.ipynb

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    "# Newton's method\n",
    "def f_solve(f, df, x0, n, delta):\n",
    "    for i in range(n):\n",
    "        print(i, x0)\n",
    "        x1 = x0 - f(x0) / df(x0)\n",
    "        if abs(x1 - x0) <= delta:\n",
    "            return x1\n",
    "        x0 = x1\n",
    "\n",
    "    assert False, \"Cannot find solution\" "
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     "text": [
      "0 1\n",
      "1 1.5\n",
      "2 1.4166666666666667\n",
      "3 1.4142156862745099\n",
      "x 1.4142135623746899\n",
      "f(x) 1.4142135623746899\n"
     ]
    }
   ],
   "source": [
    "# Simple example\n",
    "def f(x):\n",
    "    return x**2 - 2\n",
    "\n",
    "def df(x):\n",
    "    return 2*x\n",
    "\n",
    "x0 = 1\n",
    "n = 1000\n",
    "delta = 0.001\n",
    "\n",
    "x = f_solve(f, df, x0, n, delta)\n",
    "print(\"x\", x)\n",
    "print(\"f(x)\", x)"
   ]
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     "text": [
      "0 200\n",
      "1 209.45945945945945\n",
      "2 209.8951422053809\n",
      "3 209.89598435931026\n",
      "d 209.89598436244353\n",
      "f_D(d) -5.820766091346741e-11\n",
      "0 90\n",
      "1 81.11111111111111\n",
      "2 80.20523708421798\n",
      "3 80.19562927252291\n",
      "y 80.19562819149829\n",
      "f_y(y) -1.1641532182693481e-10\n"
     ]
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   "source": [
    "### Curve v1 examples ###\n",
    "# StableSwap invariant\n",
    "def f(x, y, A, D):\n",
    "    k = A*x*y/(D/2)**2\n",
    "    return x*y + k*(x+y)*D - (D/2)**2 - k*D**2\n",
    "\n",
    "n = 100\n",
    "delta = 0.0001\n",
    "\n",
    "x = 120\n",
    "y = 90\n",
    "A = 10\n",
    "D = 200\n",
    "\n",
    "### get_D ###\n",
    "def f_D(d):\n",
    "    return f(x, y, A, d)\n",
    "\n",
    "def df_dD(d):\n",
    "    return -4*A*x*y*(x + y)/d**2 - d/2\n",
    "\n",
    "# Calculate D, given x, y and A\n",
    "d = f_solve(f_D, df_dD, D, n, delta)\n",
    "print(\"d\", d)\n",
    "print(\"f_D(d)\", f_D(d))\n",
    "\n",
    "### get_y ###\n",
    "def f_y(y):\n",
    "    return f(x, y, A, D)\n",
    "\n",
    "def df_dy(y):\n",
    "    return -4*A*x + 4*A*x*y/D + 4*A*x*(x + y)/D + x\n",
    "\n",
    "# Calculate y, given x, D and A\n",
    "y = f_solve(f_y, df_dy, y, n, delta)\n",
    "print(\"y\", y)\n",
    "print(\"f_y(y)\", f_y(y))"
   ]
  }
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