constant_c.ipynb

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     "data": {
      "text/plain": [
       "(a*c + a/(a - 1) - exp(c*(a - 1)),\n",
       " -a/(a - 1)**2 - c*exp(c*(a - 1)) + c + 1/(a - 1))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sp\n",
    "a = sp.Symbol('a')\n",
    "c = sp.Symbol('c')\n",
    "f = (a*c  - sp.exp((a - 1)*c)) + a / (a - 1)\n",
    "df = f.diff(a)\n",
    "f, df"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
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      "text/plain": [
       "matrix(\n",
       "[['2.1369288434405983790865170951767170599904779730789449344794844361536205601773777424850797757178744377791613191684345515993224217927500645297697720151273214104825006580890633351903047069526592911249008787095329540018528815759529365889422114001441246630676080501223506784902879492880036329858560929964335662124026497564620860355272457674531031969580611216253150746613695782880428433635931431295253628038740275088034409943030315488104062364169189301199231361987377688743503812607758229121259834372619955'],\n",
       " ['0.23713459431207597508706354607900278401265681434774266673126097338724911991228990266934243036798935194223645533810387885661440940351790095831018323258137352829262520560896803124938071351209375320453534670353177206912396151977812791123316428707544035276178661527257442086120225024946965458265036217141467384666365025494015422983264318011547916962649523773004295500254021250037985649171052751506746561720702451705917395535699987901839312542940686216580985327445186171036176784393182467933133666448004691']])"
      ]
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     "execution_count": 19,
     "metadata": {},
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   ],
   "source": [
    "from mpmath import mp\n",
    "mp.dps = 500\n",
    "\n",
    "def g(c, a):\n",
    "    return (a*c + a/(a - 1) - mp.exp(c*(a - 1)), -a/(a - 1)**2 - c*mp.exp(c*(a - 1)) + c + 1/(a - 1))\n",
    "\n",
    "# Provide an initial guess for c and a \n",
    "initial_guess = (mp.mpf('2.1'), mp.mpf('0.5'))\n",
    "\n",
    "M = mp.findroot(g, initial_guess)\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "c6c44fd3-b031-4e77-a600-e8b756229815",
   "metadata": {},
   "outputs": [],
   "source": [
    "# c = 1/2*(sqrt(-4*a + 1) - 1)/(a^2 - a)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "da7ed082-4377-41e0-8951-fec77e9aca0f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def new_g(a):\n",
    "    return 1/2*(a - 1)*(a*(mp.sqrt(-4*a + 1) - 1)/(a**2 - a) + 2*a/(a - 1) - 2*mp.exp(1/2*(a - 1)*(mp.sqrt(-4*a + 1) - 1)/(a**2 - a)))\n"
   ]
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   "cell_type": "code",
   "execution_count": 22,
   "id": "487011ec-2de7-41c6-9181-df4e6e7d4e32",
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    {
     "data": {
      "text/plain": [
       "mpc(real='0.23713459431207597508706354607900278401265681434774266673126097338724911991228990266934243036798935194223645533810387885661440940351790095831018323258137352829262520560896803124938071351209375320453534670353177206912396151977812791123316428707544035276178661527257442086120225024946965458265036217141467384666365025494015422983264318011547916962649523773004295500254021250305677129524158833577915913382148218637768189869230150976722986751505898906016816749784240116647277261694014694987458287430627001834', imag='9.0950494792575572707138980937408016523425350698951843679694332350785039736358254456467319091641444317185374422056420904232753166755354639876228131771856030100547712630314097895489256220786532926261678692186535647319600426589761493537833007746167033238993273349466230632295360068868552237264493769200625877396517558937306955349614180593567852331122926022048026983080185706550198586816842798788656254203280314073642469527382108338440455046268386499736502272847879847925510561687776839717929636228052181438e-558')"
      ]
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     "execution_count": 22,
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    }
   ],
   "source": [
    "a = mp.findroot(new_g, mp.mpf('0.23'))\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "615109f6-3523-45af-9523-e501e27734e7",
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "mpc(real='2.1369288434405983790865170951767170599904779730789449344794844361536205601773777424850797757178744377791613191684345515993224217927500645297697720151273214104825006580890633351903047069526592911249008787095329540018528815759529365889422114001441246630676080501223506784902879492880036329858560929964335662124026497564620860355272457674531031969580611216253150746613695782988307850651144091668887428835106618882995878596226476299223141111493418156813379737696768501404949569449468568998360536258609302577', imag='1.6514315383478467918984856362468155740989767493981537895165860394696497887014141343667909899537884056395702350747026546562733265109375856539392673279344881902350922366834548341851697831181477919975948444729663209099981840587385080630428356381678117032932339530553325258176922470913654489755988632878323169770619290386848563457060349148128329844250473292037978902121863163383730527109398916333534187029548420076238746992169045499493536414730666064500610863016047151164947128937720208658786221081333013834e-556')"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = 1/2*(mp.sqrt(-4*a + 1) - 1)/(a**2 - a)\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "source": []
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