SRF-I.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h1 align=\"center\"> Geometric Methods in ML - ENSAE 2016/2017 </h1>\n",
    "<h2 align=\"center\"> Kernels approximation with Random Features </h2>\n",
    "<h4 align=\"right\"> <i>Author: Hicham Janati </i></h4>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<h2 align=left><font color=\"red\"> Polynomial kernels: SRF Implementation I (Kernel approx) </font></h2>\n",
    "\n",
    "In this notebook, we adapt the <a href=\"https://github.com/felixyu/SRF\"> Matlab functions </a> (provided by Felix Yu in the paper <a href = \"https://research.google.com/pubs/pub44290.html\"> (Pennington et al, 2015) </a> ) as two Python classes: \n",
    "- <font color=red> **KernelApprox:**</font> approximates the kernel function by a kernel verifying Bochner theorem assumptions and yields an empirical sampling distribution p. \n",
    "- **SRFF:** (next notebook) Spherical Random Fourier Features object performing the low-dimensional embedding given the sampling distribution p.\n",
    "<hr>\n",
    "\n",
    "<div style=\"  \n",
    "    padding: 10px;\n",
    "    border: 5px solid gray;\n",
    "    margin: 0; \"> \n",
    "Schoberg's theorem characterizes positive-definite radial functions by their integral representation: \n",
    "$$ f(r) = \\int_{\\mathbb{R}_+} e^{-r^2 t^2}d\\mu(t) $$ where $\\mu$ is a finite non-negative Borel measure on $\\mathbb{R}_+$  \n",
    "</div>\n",
    "Given that any finite sum of gaussians verifies the Bochner theorem, approximating the polynomial kernel by a sum of gaussians seems a good idea:\n",
    "$$ \\hat{K}(z) = \\sum_{i=1}^{N} c_i e^{-\\sigma_i ^2 z^2} $$ \n",
    "It's Fourier tranform is a probability distribution if $c$ is positive and is given by:\n",
    "$$ p(w) = \\sum_{i=1}^N c_i \\left(\\frac{1}{\\sqrt{2}\\sigma_i}\\right)^d e^{-\\frac{w^2}{4\\sigma_i^2}} $$ \n",
    "However, for a good approximation, we let $c \\in \\mathbb{R}^N$ and clip the negative values of p. \n",
    "Let $\\theta$ be the 2N-concatenated vector $(\\sigma, c)$. \n",
    "The class solves the following minimization problem: \n",
    "\n",
    "$$ \\min_{\\theta}{\\frac{1}{2} \\int_{0}^{2} \\left[K(z) - \\hat{K}_{\\theta}(z)\\right]^2 dz} $$\n",
    "\n",
    "Where: $\\hat{K}_{\\theta}(z)$ can be approximated on a grid:\n",
    "$$ \\hat{K}_{\\theta}(z) = \\int_{\\mathbb{R}_+} w p(w) \\left(\\frac{w}{z}\\right)^{d/2 -1} \\mathcal{J}_{d/2 -1}(wz) dw $$ \n",
    "\n",
    "Avoiding overflows can be an absolute nightmare when computing theses quantities ... so as diving into the Matlab code ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np # Thinly−wrapped numpy\n",
    "import pandas as pd\n",
    "from math import factorial\n",
    "%matplotlib inline \n",
    "from matplotlib import pyplot as plt \n",
    "data = '../data/'\n",
    "\n",
    "from sklearn.base import BaseEstimator\n",
    "from sklearn.exceptions import NotFittedError\n",
    "from scipy.optimize import check_grad, fmin_l_bfgs_b\n",
    "from scipy import special"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class KernelApprox(BaseEstimator):\n",
    "    def __init__(self, dim, kernelfunc, eps = 1e-10, grid_size = 500, N=10, verbose = True):\n",
    "        self.dim = dim\n",
    "        self.eps = eps\n",
    "        self.grid_size = grid_size\n",
    "        self.kernelfunc = kernelfunc\n",
    "        self.N = N\n",
    "        self.fitted = False\n",
    "        self.verbose = verbose\n",
    "\n",
    "    @staticmethod\n",
    "    def scaled_airy(order, x):\n",
    "        #Airy function scaled by exponential prefactor:\n",
    "        #If order == 0, \n",
    "        #  y = np.exp(2/3*z**(3/2)) * pi**(3/2) * z**(1/4) * AiryAi(z)\n",
    "        #If order == 1,\n",
    "        #  y = np.exp(2/3*z**(3/2)) * pi**(3/2) * z**(-1/4) * AiryAi'(z)\n",
    "        #See http://dlmf.nist.gov/9.7.\n",
    "\n",
    "        y = 0\n",
    "        if order == 0: #AiryAi\n",
    "            for k in range(7):\n",
    "                y = y + (-3)**k*special.gamma(k+1/6)*special.gamma(k+5/6) *1/2**(2*k+2)/factorial(k)/x**(3*k/2)\n",
    "\n",
    "        elif order == 1: #AiryAi'\n",
    "            for k in range(7):\n",
    "                y = y + (-3)**k*special.gamma(k-1/6)*special.gamma(k+7/6) *1/2**(2*k+2)/factorial(k)/x**(3*k/2)\n",
    "\n",
    "\n",
    "        if x.any() < 0: #Should never happen!\n",
    "            y1 = 0\n",
    "            if order == 0: #AiryAi\n",
    "                for k in range(7):\n",
    "                    y1 = y1 + (3)**k*special.gamma(k+1/6)*special.gamma(k+5/6) *1/2**(2*k+2)/factorial(k)/x**(3*k/2)\n",
    "                y1 = y1 * np.exp(2/3*x**(3/2))/np.pi**(3/2)/x**(1/4)\n",
    "                y = y + np.sqrt(-1)*y1\n",
    "            elif order == 1:\n",
    "                for k in range(7):\n",
    "                    y1 = y1 + (3)**k*special.gamma(k-1/6)*special.gamma(k+7/6) *1/2**(2*k+2)/factorial(k)/x**(3*k/2)\n",
    "\n",
    "                y1 = -y1 * np.exp(-2/3*x**(3/2))*x**(1/4)/np.pi**(3/2)\n",
    "                y = y + np.sqrt(-1)*y1\n",
    "        y = np.real(y)\n",
    "        return y\n",
    "    \n",
    "    @staticmethod\n",
    "    def bessel_coeff_uk(k, x):\n",
    "        #Precomputed Debeye polynomials\n",
    "        if k ==0:\n",
    "             uk = 1\n",
    "        if k ==1:\n",
    "             uk = x/8 - (5*x**3)/24\n",
    "        if k ==2:\n",
    "             uk = (9*x**2)/128 - (77*x**4)/192 + (385*x**6)/1152\n",
    "        if k ==3:\n",
    "             uk = (75*x**3)/1024 - (4563*x**5)/5120 + (17017*x**7)/9216 - (85085*x**9)/82944\n",
    "        if k ==4:\n",
    "             uk = (3675*x**4)/32768 - (96833*x**6)/40960 + (144001*x**8)/16384 - (7436429*x**10)/663552 + (37182145*x**12)/7962624\n",
    "        if k ==5:\n",
    "             uk = (59535*x**5)/262144 - (67608983*x**7)/9175040 + (250881631*x**9)/5898240 - (108313205*x**11)/1179648 + (5391411025*x**13)/63700992 - (5391411025*x**15)/191102976\n",
    "        if k ==6:\n",
    "             uk = (2401245*x**6)/4194304 - (388895895*x**8)/14680064 + (1441372804469*x**10)/6606028800 - (33010308331*x**12)/47185920 + (4445922195*x**14)/4194304 - (1169936192425*x**16)/1528823808 + (5849680962125*x**18)/27518828544\n",
    "        if k ==7:\n",
    "             uk = (57972915*x**7)/33554432 - (25388505925*x**9)/234881024 + (1007390378503*x**11)/838860800 - (1602251736839*x**13)/301989888 + (10559432785187*x**15)/905969664 - (36927006432745*x**17)/2717908992 + (1774793203908725*x**19)/220150628352 - (1267709431363375*x**21)/660451885056\n",
    "        if k ==8:\n",
    "             uk = (13043905875*x**8)/2147483648 - (928090660435*x**10)/1879048192 + (667955999804539*x**12)/93952409600 - (276439228010667*x**14)/6710886400 + (3542717254441859*x**16)/28991029248 - (39803268297948155*x**18)/195689447424 + (75358832548684685*x**20)/391378894848 - (512408152157076175*x**22)/5283615080448 + (2562040760785380875*x**24)/126806761930752\n",
    "        if k ==9:\n",
    "             uk = (418854310875*x**9)/17179869184 - (472414367256615*x**11)/188978561024 + (1359491937582325*x**13)/30064771072 - (3739063570455884033*x**15)/11274289152000 + (817138105244771959*x**17)/644245094400 - (17618708259302571707*x**19)/6262062317568 + (35348759075759093965*x**21)/9393093476352 - (3128960418491082175*x**23)/1043677052928 + (1330723971151926826475*x**25)/1014454095446016 - (6653619855759634132375*x**27)/27390260577042432\n",
    "        if k ==10:\n",
    "             uk = (30241281245175*x**10)/274877906944 - (20993386079260455*x**12)/1511828488192 + (45660648644355162105*x**14)/148159191842816 - (35892416277828185849*x**16)/12884901888000 + (1232816120293110459821*x**18)/92771293593600 - (5227733363217471800551*x**20)/139156940390400 + (19941766574064397067317*x**22)/300578991243264 - (16705838021516291703055*x**24)/225434243432448 + (91891691150784941691275*x**26)/1803473947459584 - (4318199286388002551911375*x**28)/219122084616339456 + (4318199286388002551911375*x**30)/1314732507698036736\n",
    "        if k ==11:\n",
    "             uk = (1212400457192925*x**11)/2199023255552 - (26416375998266454375*x**13)/314460325543936 + (241770821762631191867*x**15)/107752139522048 - (329641577686894230674187*x**17)/13469017440256000 + (527174389121818780771231*x**19)/3710851743744000 - (23186185730591085896097833*x**21)/46756731971174400 + (164293183874328160710877*x**23)/148434069749760 - (11694306169843138084657687*x**25)/7213895789838336 + (201734750392525792544487385*x**27)/129850124217090048 - (365967912305800454531456575*x**29)/389550372651270144 + (3424332034105686023665720375*x**31)/10517860061584293888 - (1556514560957130010757145625*x**33)/31553580184752881664\n",
    "        if k ==12:\n",
    "             uk = (213786613951685775*x**12)/70368744177664 - (1383228778609739182725*x**14)/2515682604351488 + (4288535808487871046652577*x**16)/246536895226445824 - (1247431715337149394208117*x**18)/5541538603991040 + (1440134927609056360249131941*x**20)/923589767331840000 - (12275126748382902959849354203*x**22)/1870269278846976000 + (40295057240679430076066704447*x**24)/2244323134616371200 - (1058890883945614567899138703*x**26)/32061759065948160 + (171527591207538652048777279465*x**28)/4155203974946881536 - (323782209780772997264458503055*x**30)/9349208943630483456 + (38826657063629086145255937175*x**32)/2077601987473440768 - (1480867953294613492234348347625*x**34)/252428641478023053312 + (7404339766473067461171741738125*x**36)/9087431093208829919232\n",
    "        if k ==13:\n",
    "             uk = (10278202593831046875*x**13)/562949953421312 - (311689729229640242227335*x**15)/80501843339247616 + (25668144329807404111206969*x**17)/179299560164687872 - (17485727773335283402112401979*x**19)/8068480207410954240 + (1172682485145099862891889874709*x**21)/66498463247892480000 - (18549286571206971359638512589*x**23)/211106232532992000 + (3692240508888356366982787102811*x**25)/12824703626379264000 - (14897903722108127256835851172369*x**27)/23084466527482675200 + (18618233657875371822100705069009*x**29)/18467573221986140160 - (328606042646275236446664121476415*x**31)/299174686196175470592 + (367634232582365988089137540027775*x**33)/448762029294263205888 - (179099209477310436044105950399825*x**35)/448762029294263205888 + (8323958765469022439849272062000125*x**37)/72699448745670639353856 - (3201522602103470169172796946923125*x**39)/218098346237011918061568\n",
    "        if k ==14:\n",
    "             uk = (1070401384414690453125*x**14)/9007199254740992 - (18797752383521406538249035*x**16)/644014746713980928 + (123552672971010529264790525*x**18)/99079191802150912 - (60197559206626892846461221113*x**20)/2758454771764428800 + (1533615778845746926778555049075167*x**22)/7447827883763957760000 - (1909654265469256217367321730519213*x**24)/1595963117949419520000 + (420670869682837226127426675930539*x**26)/91197892454252544000 - (2275295753169799866029447924232463*x**28)/184675732219861401600 + (103485029425656867671478423988690253*x**30)/4432217573276673638400 - (4678514217333130937255657203225879*x**32)/147740585775889121280 + (438926968126721480011081447330894535*x**34)/14360384937416422588416 - (662918789842359228359591343469410875*x**36)/32310866109186950823936 + (1177322813516754964196492125503858125*x**38)/129243464436747803295744 - (4198476740398490779853205916194986125*x**40)/1744786769896095344492544 + (2998911957427493414180861368710704375*x**42)/10468720619376572066955264\n",
    "        if k ==15:\n",
    "             uk = (60013837619516978071875*x**15)/72057594037927936 - (3735271835560741658500637055*x**17)/15924728282382073856 + (118145843799243877057467427205*x**19)/10304235947423694848 - (4682053290157804644291649610772557*x**21)/20390497672882657689600 + (4476579701403857364279102891783*x**23)/1801439850948198400 - (1517060956256085600229087638761327663*x**25)/91197892454252544000000 + (7325285829673636954246260387489676301*x**27)/98493723850592747520000 - (1527341067295812807125649376475592779*x**29)/6566248256706183168000 + (18546344338818795607290212453008613027*x**31)/35457740586213389107200 - (18242178522176433211173936393164309389*x**33)/21274644351728033464320 + (65544319685459694180664515606842619431*x**35)/63823933055184100392960 - (919693328928875517974470387276902815315*x**37)/1033947715493982426365952 + (187054812240653376693658577595543666125*x**39)/344649238497994142121984 - (2825480904467831144445100711243097375*x**41)/12764786611036820078592 + (4537353791587797535655643250859295719375*x**43)/83749764955012576535642112 - (4537353791587797535655643250859295719375*x**45)/753747884595113188820779008\n",
    "        if k ==16:\n",
    "             uk = (57673297952355815927071875*x**16)/9223372036854775808 - (255005467589750245013902496625*x**18)/127397826259056590848 + (23930632228676872808287094850927*x**20)/215596321361480384512 - (411326657717984844291245101099269569*x**22)/163123981383061261516800 + (12139335575858921085347526474466207309*x**24)/391497555319347027640320 - (19184188463249985491959692568453667557*x**26)/81064793292668928000000 + (66886935239112343759822339412549592082043*x**28)/55156485356331938611200000 - (10352066634614181764059489715007707655223*x**30)/2363849372414225940480000 + (1609057486347529247953695407110092411199*x**32)/140079962809731907584000 - (18949942802726836829409263606971205617559*x**34)/850985774069121338572800 + (59074304522438219672139306944094150384209*x**36)/1838129271989302091317248 - (31659227230675746511992457214201335968895*x**38)/919064635994651045658624 + (298380340782844687194470219764567108053325*x**40)/11028775631935812547903488 - (125147594895406727667181525777666404273125*x**42)/8271581723951859410927616 + (849525181693753993883643210633357171800375*x**44)/148888471031133469396697088 - (7845084705655301939148607180735722298799375*x**46)/6029983076760905510566232064 + (39225423528276509695743035903678611493996875*x**48)/289439187684523464507179139072\n",
    "        if k ==17:\n",
    "             uk = (3694483615889146090857721875*x**17)/73786976294838206464 - (5601202382478651012510097287895875*x**19)/309831513462025628942336 + (25307936122315048864599612652365261*x**21)/22422017421593959989248 - (17075739714414991994043276803439819669*x**23)/591596305815902175100928 + (639250722250629496365483268167565271593*x**25)/1597949205385089908736000 - (6202733385292850084270145070389220684392643*x**27)/1797692856058226147328000000 + (4918554203860156635144716898340926135544589*x**29)/245139934917030838272000000 - (109487110251520541349783293821947174450442753*x**31)/1323755648551966526668800000 + (22656976680462433461046473301644789924750967*x**33)/90771815900706276114432000 - (17023703028998683474035793235643067182905467*x**35)/30257271966902092038144000 + (352014074661025026479632974329618842445974511*x**37)/367625854397860418263449600 - (6046767369418617392447022239134638130160647*x**39)/4901678058638138910179328 + (434324663126461865364496896478471302338035*x**41)/363087263602825104457728 - (227437475147602591711240197307248667449645125*x**43)/264690615166459501149683712 + (4754087392220751898398489247314331285540440325*x**45)/10719969914241609796562190336 - (1667275152429410778583358653001802852281228375*x**47)/10719969914241609796562190336 + (76858294861304993097838904549667871361337476875*x**49)/2315513501476187716057433112576 - (22605380841560292087599677808725844518040434375*x**51)/6946540504428563148172299337728\n",
    "        if k ==18:\n",
    "             uk = (502860269940467106811189921875*x**18)/1180591620717411303424 - (32848477905191480676470504898494775*x**20)/190665546745861925502976 + (7216070211653983528525765701391767079653*x**22)/599833810062481617632362496 - (1244871457408461970370166238367252101607*x**24)/3619177400285519188852736 + (67512932186883166194556881593301716464061*x**26)/12654466434564752408576000 - (2698766759858833014205911295589437107851741*x**28)/52296519448966578831360000 + (39732393413258582818456597433543988061348917673*x**30)/117667168760174802370560000000 - (146957279777702083489469692961702969742402769*x**32)/93386641873154605056000000 + (196172231632052896021939551786629194331093001899*x**34)/36308726360282510445772800000 - (18261434948825344307602068729927718533955986373*x**36)/1307114148970170376047820800 + (14416334392120207468368228108578563189046136643*x**38)/522845659588068150419128320 - (2731124460448440343256369615269965213481857679*x**40)/65355707448508518802391040 + (22868999656866719261134680839956677441101178409*x**42)/470561093629261335377215488 - (10120723781510555707788922845057965284862967265*x**44)/235280546814630667688607744 + (9764679006095184354177598214026976455096819542925*x**46)/343039037255731513489990090752 - (3509140234776489527398109461648354044083449543375*x**48)/257279277941798635117492568064 + (2044604580045901059668365698711801118797401693875*x**50)/457385383007642017986653454336 - (49826780450967195819487209825993506486664725449375*x**52)/55572324035428505185378394701824 + (249133902254835979097436049129967532433323627246875*x**54)/3000905497913139280010433313898496\n",
    "        if k ==19:\n",
    "             uk = (36232405765710498380237842265625*x**19)/9444732965739290427392 - (5270618082084864640591779010940423175*x**21)/3050648747933790808047616 + (643617691230342300867772202166782802324525*x**23)/4798670480499852941058899968 - (511290603091990756321621785602479678715718017*x**25)/119966762012496323526472499200 + (9578510926265808749239543652362821145078653469*x**27)/130290386410278690798698496000 - (3608793655049366427755026382191879580906402267*x**29)/4555607916443310867087360000 + (778605359192630424174485689370678541153366154099*x**31)/134476764297342631280640000000 - (187973203756033464466274463861125534417385919868989*x**33)/6212826510537229565165568000000 + (18892625343549330867979679263691816799239745614837*x**35)/161372117156811157536768000000 - (897888445598605915923390318015931467120927078392341*x**37)/2614228297940340752095641600000 + (16222238070489694130918937921555356858923228707359*x**39)/20913826383522726016765132800 - (70239668257654763841328806708670410338848690183*x**41)/51639077490179570411765760 + (6991065680795630961804716828763946745922759601591*x**43)/3764488749034090683017723904 - (66667297710177390946655195253136110176945908415243*x**45)/33880398741306816147159515136 + (488393019248226745499796510949450606738795530834905*x**47)/304923588671761345324435636224 - (8088398159551059878168644656424684012396121108257975*x**49)/8232936894137556323759762178048 + (43398697174771671575635378868549090707872913819769875*x**51)/98795242729650675885117146136576 - (13349386123251183201469531253100521966852501723899375*x**53)/98795242729650675885117146136576 + (613716454814562950908623963426762019396249423359951875*x**55)/24007243983305114240083466511187968 - (161504330214358671291743148270200531420065637726303125*x**57)/72021731949915342720250399533563904\n",
    "        if k ==20:\n",
    "             uk = (11021897833929133607268351617203125*x**20)/302231454903657293676544 - (443863585329019676901014600158486627725*x**22)/24405189983470326464380928 + (7051504650692012922107270586229951846285125*x**24)/4516395746352802768055435264 - (281455295925390008139045560174186364411534291*x**26)/5132267893582730418244812800 + (92819048753175303652301382783593067554729561856361*x**28)/88722541529943377286281727836160 - (12674074492013646204309038268611734847410602546809*x**30)/1015249764235937850379468800000 + (131810321642212959067823751018561574030743411653989*x**32)/1301602261840945962024960000000 - (146418779187950135225508434919692120653775229217054181*x**34)/248513060421489182606622720000000 + (1824336367202196000530904722626314573557902316050369351*x**36)/715717614013888845907073433600000 - (3906663644300258072745811444749701215638380977439083*x**38)/464751697411616133705891840000 + (179753625149125126857292240802977667122589767348030603*x**40)/8365530553409090406706053120000 - (21595789303723522142722550461795057852700907459698657*x**42)/501931833204545424402363187200 + (9079828598096525946757702627510256710073940544191563*x**44)/133848488854545446507296849920 - (22830571636776305811972242610676904910054857771278061*x**46)/271043189930454529177276121088 + (1199349512131949785092521372248630847266863258754124905*x**48)/14636332256244544575572910538752 - (50196165852977381848005361305078288352521092865332843*x**50)/813129569791363587531828363264 + (111549656209419888821395674550399025985589438923447496675*x**52)/3161447767348821628323748676370432 - (105189027214710231484783695637837900781660279240767956125*x**54)/7113257476534848663728434521833472 + (182894486402114387773498357808723553155777102174932206875*x**56)/42679544859209091982370607131000832 - (442037351796699683325500996815538854496719650456891653125*x**58)/576173855599322741762003196268511232 + (442037351796699683325500996815538854496719650456891653125*x**60)/6914086267191872901144038355222134784\n",
    "        if k ==21:\n",
    "             uk = (882276678992136837800861860405640625*x**21)/2417851639229258349412352 - (3601863375850721941134465726404739628869375*x**23)/17962219827834160277784363008 + (11636012637930893116357496749137014206091373255*x**25)/614229821503981176455539195904 - (104392943374522986327161123219037126149263496233*x**27)/142623023569035876885961113600 + (318882827614495782877146446223016905753960001860199*x**29)/20753810884197281236557129318400 - (1088234168339950246779000834589819245239646315685409887*x**31)/5388008594935832628721562419200000 + (6778092592066390402910468646387625117318852478375083*x**33)/3748614514101924370631884800000 - (1259954758557860262541614757440759697441640955358141041991*x**35)/108241244094693066201995673600000000 + (103948063410295300109566013588489288279056607856996909191*x**37)/1859006789646464534823567360000000 - (5097681450828076156452601417693508837778855430567550639*x**39)/24786757195286193797647564800000 + (54131922623521119144353295846786658324673623820924249*x**41)/91802804426985902954250240000 - (53554837271745590089748337619111480761419974252975404401*x**43)/40154546656363633952189054976000 + (346461683842702502080560162932309920326032941097720383733*x**45)/144556367962909082227880597913600 - (33063635010120972399234777612458335698852059269923207817*x**47)/9637091197527272148525373194240 + (457269943742018458843771837212180633436501622187169963231*x**49)/117090658049956356604583284310016 - (1233349808938159584983440018590335385963922198574167509427*x**51)/351271974149869069813749852930048 + (6917264816013883432252595158356080640736850880957442585535*x**53)/2810175793198952558509998823440384 - (299803193949148812529649487428299273970786604853599377125325*x**55)/227624239249115157239309904698671104 + (532055746636321797087804484520373031253282935480154886480875*x**57)/1024309076621018207576894571144019968 - (16193637736972925421725874572655878501311880188802637533125*x**59)/113812119624557578619654952349335552 + (1337162989185016542059640515367005034852576942632097250703125*x**61)/55312690137534983209152306841777078272 - (955116420846440387185457510976432167751840673308640893359375*x**63)/497814211237814848882370761575993704448\n",
    "        if k ==22:\n",
    "             uk = (148302689041496455735799416353639046875*x**22)/38685626227668133590597632 - (332073446135546784224083686241998203006956875*x**24)/143697758622673282222274904064 + (82693125029477684924414098677282792398976910258065*x**26)/345702996008828935078035236847616 - (5585436433495548583153190526622481251117545627293*x**28)/551821443918015174569682272256 + (3477975368433094977787233513447821349320389475554763627*x**30)/14942743836622042490321133109248000 - (9945430488741205467473056975346043783432579353299612743*x**32)/2964830126313897319508161331200000 + (48733082165485087839813906498850723040121004025263922351*x**34)/1477853786039542663877914263552000 - (1820614930483987715332914309003780860256853497313678258511689*x**36)/7793369574817900766543688499200000000 + (3088728341815904916175956479933000312873293448749599621431177*x**38)/2493878263941728245293980319744000000 - (5003315868003919000134550065164480303827913595359354920511*x**40)/991470287811447751905902592000000 + (134112497310507682599931168369714925532958635573865160407393*x**42)/8328350417616161116009581772800000 - (7276898585248876923939680022059269689182972124391109499983*x**44)/178464651806060595343062466560000 + (955527463467585833042403049867886649064622120741114150784077*x**46)/11564509437032726578230447833088000 - (93389385578941122112910632847954580705633153743195587194977*x**48)/693870566221963594693826869985280 + (81579349026649916265853483855559562986377791119753384783027*x**50)/462580377481309063129217913323520 - (520633711000711955991906527488128574368216764019251143321357*x**52)/2810175793198952558509998823440384 + (62645916944334201901559967175697152521821099379989653710983317*x**54)/404665314220649168425439830575415296 - (20532817151688513679944050045562468464219483272549311742502905*x**56)/202332657110324584212719915287707648 + (1672957394323402512429007240647035474716660439381236592705589875*x**58)/32777890451872582642460626276608638976 - (467252291432876308261007267953586199590743739511431348622501125*x**60)/24583417838904436981845469707456479232 + (485393221446707713376888500542257952234422236065605889656093125*x**62)/98333671355617747927381878829825916928 - (3178054378724445744320891322022980394977474656367171708563984375*x**64)/3982513689902518791058966092607949635584 + (1444570172147475338327677873646809270444306661985078049347265625*x**66)/23895082139415112746353796555647697813504\n",
    "        if k ==23:\n",
    "             uk = (13057084578653492298477992092005176953125*x**23)/309485009821345068724781056 - (63881841061417460560126117044175857304644748425*x**25)/2299164137962772515556398465024 + (7766306081078717102552658788250080000424812510523015*x**27)/2474505655642143956348041695330304 - (147603846418781123497889146283372464079417235402042369*x**29)/1018914093499706334966840698077184 + (33417952196478088203151885428753580717861699927440711989*x**31)/9195534668690487686351466528768000 - (1577399784357289865813270322474400475345934888249906234479*x**33)/27586604006071463059054399586304000 + (20178746713532090646701331350405706974306188457660170520743*x**35)/32841195245323170308398094745600000 - (4755251866546922263545054499633841071688710349393325704950224433*x**37)/997551305576691298117592127897600000000 + (778782353044251754377015353369914847185676925697129708247315789*x**39)/28069625481595209687291553382400000000 - (398231759233031087573950632984346712117120800538303725392763*x**41)/3198817718700308796272541696000000 + (29263361929733875772983354882323555601563016206864853870145293*x**43)/66626803340929288928076654182400000 - (2222665408717572286527821325962617352818612268394229356871386431*x**45)/1798923690205090801058069662924800000 + (143821733770330388990402148726558299076185911915108961158434563*x**47)/51397819720145451458801990369280000 - (1424377065090533189747864509426425487277571135349043597113191949*x**49)/277548226488785437877530747994112000 + (254496731723314102009663829791441844730487973887175715883138507*x**51)/33305787178654252545303689759293440 - (184415376387096941953156894893207349478290967008369401715451053*x**53)/19983472307192551527182213855576064 + (145667467785608001202143207257917528177866106016227681820477712167*x**55)/16186612568825966737017593223016611840 - (68209307548588717219996066619296639328974086572510362528327405441*x**57)/9711967541295580042210555933809967104 + (13994212724840723785179369406340877734881477812680981106144178015*x**59)/3237322513765193347403518644603322368 - (1613382568684986823317540920622197800647927899700130570975856778125*x**61)/786669370844941983419055030638607335424 + (5121861816684832095900688933166392225557549096995241231076824734375*x**63)/7080024337604477850771495275747466018816 - (1264540559307330111789532303846710258748418896742567010994001984375*x**65)/7080024337604477850771495275747466018816 + (5264302621339829627933723707143702343353142337606021427431305390625*x**67)/191160657115320901970830372445181582508032 - (1144413613334745571289939936335587465946335290783917701615501171875*x**69)/573481971345962705912491117335544747524096\n",
    "        if k ==24:\n",
    "             uk = (9614366611415188162445961510413145296484375*x**24)/19807040628566084398385987584 - (376446109033272332065136012354514176092773017775*x**26)/1081959594335422360261834571776 + (265662732561310733128838422112356738891269018733254375*x**28)/6216939829051336882064501449424896 - (1485194077025985600703178275664372145672887225927198592673*x**30)/692861583579800307777451674692485120 + (1667465687322450726531991712454791896830351120646120031562429*x**32)/28529594617991777379071539546161152000 - (220709387077761065839773023610937950778181894692206200661557*x**34)/220692832048571704472435196690432000 + (1161867287597955672621146405592783311268771361450741705215894899*x**36)/99311774421857267012595838510694400000 - (140407677060382566581340823226704136203549599960351155480899721*x**38)/1418739634597960957322797693009920000 + (96732190459707245530314445401874578661513592331682751098556408838469*x**40)/153696793748112437043303083409408000000000 - (2084283857893795728757556431272675009894339299079790324507316494457*x**42)/673671011558285032494997281177600000000 + (49740600573130715887766197733835056154364702518396425497894507769*x**44)/4145667763435600199969214038016000000 - (76587575255677638683521855144680233604116263900421855908945151309*x**46)/2055912788805818058352079614771200000 + (461747375807618028852687814942488783827193771331808256400526094883*x**48)/4934190693133963340044991075450880000 - (26252585194995166985647375289214953748135467414800759389703469683*x**50)/137060852587054537223471974318080000 + (4272000556519249365888077402473036359613532749064952766322887131959*x**52)/13322314871461701018121475903717376000 - (9485641831123649298426990826278048216604445143512898547722601992963*x**54)/21582150091767955649356790964022149120 + (56649413097136698258246007199183604021904037500067428712561583148559*x**56)/115104800489429096796569551808118128640 - (34788538415518232180028669059113593162757910360340299301727047145591*x**58)/77695740330364640337684447470479736832 + (152796051299184506372142132217465401802798019666093043153248365807133*x**60)/466174441982187842026106684822878420992 - (132945080203286968535094966090039081336074063701445653902862713731905*x**62)/699261662973281763039160027234317631488 + (19339646949385204336135748268274681675081827804773618916185177170034375*x**64)/226560778803343291224687848823918912602112 - (1619896902764524903453332206778828215257311530510742739955957593328125*x**66)/56640194700835822806171962205979728150528 + (28231812702346969527073139755829052296001262817839174349613351015625*x**68)/4195569977839690578234960163405905788928 - (4549501878450947544106027222908494412123061314982386431002263358671875*x**70)/4587855770767701647299928938684357980192768 + (22747509392254737720530136114542472060615306574911932155011316793359375*x**72)/330325615495274518605594883585273774573879296\n",
    "        if k ==25:\n",
    "             uk = (923363769360314671121310143460078474274359375*x**25)/158456325028528675187087900672 - (313718822263146600653563948312712364044784337455525*x**27)/69245414037467031056757412593664 + (1578353575980774463221335426000146878099442239008551275*x**29)/2617658875390036581921895347126272 - (2149017802495908277373843891739538946845209902884821098219*x**31)/65596362942466301328042762101063680 + (1506693574371792599184092138050129436061478476570003173741993*x**33)/1557200615938547848917884700312207360 - (70460773857631569242328281394349308782579427372149984515374365461*x**35)/3927351553433249354256135677993287680000 + (2348845808293897121545739108238566083496893147341204539075121829*x**37)/10318106433439716053256710494617600000 - (21579838239746481999353095507697653227173144953133253141610508303*x**39)/10318106433439716053256710494617600000 + (659015229271827668111328398387897405444917845746504784947773728433487*x**41)/45539790740181462827645358047232000000000 - (515093804301993260985774931898619296700648995773689828827854751271304357*x**43)/6639701489918457280270693203286425600000000 + (31916737268071571910775109157262100766857253281149621135932595454713067*x**45)/97008625664393044679279608489574400000000 - (1003836233779334522033312946222089355663054826854858776655471255353263*x**47)/895464236902089643193350232211456000000 + (612523744723408283880475635042679785372716832082722166698324094964747*x**49)/197367627725358533601799643018035200000 - (833213730438616410106528354982453257047370228781146891952535149651467*x**51)/118420576635215120161079785810821120000 + (155471967442635036391453420743479529304066514009270892729921394968349*x**53)/11842057663521512016107978581082112000 - (174459677921770752620550302107747100637669056452013754900552956639392751*x**55)/8632860036707182259742716385608859648000 + (212601235497642911300959670205020728284723640938813522857309746612055471*x**57)/8287545635238894969353007730184505262080 - (8217364122169231843408202653667966656013128259299959010439554841354147*x**59)/306946134638477591457518804821648343040 + (85116311756759180911975871557625209479110626684578589313016527252836475*x**61)/3729395535857502736208853478583027367936 - (1583950654864744963336809562811601985507753519002735502506157771802955875*x**63)/100693679468152573877639043921741738934272 + (1737440404114174277725194251743193432812573747534086514871356890904067875*x**65)/201387358936305147755278087843483477868544 - (6663100715859830968116791144203454965643660511930135717213624181788128125*x**67)/1812486230426746329797502790591351300816896 + (3188657436131490167395412947892054602134149336705613932191657270295421875*x**69)/2718729345640119494696254185887026951225344 - (2149587370563475126281649092281742682969166036558825643150111369011796875*x**71)/8156188036920358484088762557661080853676032 + (98291988083932721690410718150938021773918739710194458841803899864105859375*x**73)/2642604923962196148844759068682190196591034368 - (19658397616786544338082143630187604354783747942038891768360779972821171875*x**75)/7927814771886588446534277206046570589773103104\n",
    "        if k ==26:\n",
    "             uk = (184743781854321419968194437164589547045200671875*x**26)/2535301200456458802993406410752 - (33933260100144421439148977814291289768859322014623125*x**28)/553963312299736248454059300749312 + (1218804849708592824313094830409160054755199131046508241445*x**30)/138394486629316716679001075743719424 - (284612378715933892342485100576243995653281797200517740589117*x**32)/550846476386425089413129281743093760 + (1888404232187900063928729744516017766488346244664294903299853425*x**34)/114609965333077121680356313942978461696 - (1586394174235529077535305530198564170700150653683219376017237268616673*x**36)/4807078301402297209609510069863784120320000 + (102394378270724568558492244625789852900689741760452196994679174248793*x**38)/22621544947775516280515341505241337036800 - (18574585233549891825390146659981819729438336193812658718851409894481*x**40)/412724257337588642130268419784704000000 + (418322321334084606738072821645752773930193478533287258397632857231482279883*x**42)/1239410944784778692317196064613466112000000000 - (637219808617932801536255664380413017724367362668167363996470287370207243*x**44)/324283345050962504530925186973696000000000 + (95323611872499163407924838721569705647980797983431404276871645263253347621*x**46)/10506780379651185146801976057947750400000000 - (3627055513703314218697032203251210023766962070062762570613452874015406663*x**48)/107455708428250757183202027865374720000000 + (978092783924874641151422894946109699895261770058890526456064902651795353*x**50)/9551618526955622860729069143588864000000 - (1210298773191349813197854360416054814123356430691124820867128065429396971*x**52)/4736823065408604806443191432432844800000 + (8994278025223758924080517047060862885313880826772814801619196798078364939*x**54)/17052563035470977303195489156758241280000 - (6935805075312757907901036394951796369751163843227567362287006905187397197*x**56)/7673653365961939786437970120541208576000 + (853370692513943967735941941484581931313745144276097620317458198411415195143*x**58)/663003650819111597548240618414760420966400 - (1512833106951070632196307286694423678400773644201556122016704089328303290911*x**60)/994505476228667396322360927622140631449600 + (197067886612238771533667970396589880554110694970406989473236041289957413863*x**62)/132600730163822319509648123682952084193280 - (960106118328539390146261388403644377353603253013590293729763442988389950925*x**64)/805549435745220591021112351373933911474176 + (2500623904999177839094274543853278392917623293113014539154462100114432199625*x**66)/3222197742980882364084449405495735645896704 - (24084647329180462860570023098633100307581034456317453618114126134945206875*x**68)/59670328573720043779341655657328437886976 + (14237581584180635068532237274153664892187933348231009250522599815786238996875*x**70)/86999339060483823830280133948384862439211008 - (29262467964326174128661163972977558012987241922335486245912167758738567015625*x**72)/587245538658265810854390904151597821464674304 + (50343573069936639120083020939436882596545759654223987948293968722746203359375*x**74)/4697964309266126486835127233212782571717394432 - (92020959244177814046562514332908175984742724116684052367696811052775905546875*x**76)/63422518175092707572274217648372564718184824832 + (35392676632376082325600967051118529224901047737186173987575696558759963671875*x**78)/380535109050556245433645305890235388309108948992\n",
    "        if k ==27:\n",
    "             uk = (19220195675140328470024376814641927320369210640625*x**27)/20282409603651670423947251286016 - (11618011506443975947412600466565128276447229807284999375*x**29)/13528367205635664172772816607772672 + (3386674925635360943751669835907108902786531101519696513919635*x**31)/25464585539794275868936197936844374016 - (952316764897307491508342519469846909625054926757462221223667895*x**33)/113355730279151107630615650347624955904 + (134148989830719394371122120871798349684910873907176150251417104633*x**35)/465083917293646290876808230493245931520 - (43502954423812894126709563383506391967449746638318848184286424296372737*x**37)/6992113892948795941250196465256413265920000 + (30619398105537168815357378741777464146130716192722633979054637765040899*x**39)/332957804426133140059533165012210155520000 - (53659773650883421256254900824162696134119827196135287271771016404610703*x**41)/54404870004270120924760321080426496000000 + (3168763244368123974280616789890000188505775130530496844861963851850544216009*x**43)/396611502331129181541502740676309155840000000 - (22455840354619781995515820127053935018976510113296902313783801825959634474745961*x**45)/446187940122520329234190583260847800320000000000 + (11765250754342762054143817797245162755260219791592925115803114419661064502649*x**47)/46696801687338600652453226924212224000000000 - (5395851173334097149085705801615611675977493941518697335072667532556560282444047*x**49)/5295417311344197313988195933205666201600000000 + (4967442466115962665963480846987776013443698729733638162472473146762261799391*x**51)/1473678287016010384226770667867996160000000 - (542611340104170730860433829899801198435069798626501397637892703598273506331*x**53)/58947131480640415369070826714719846400000 + (14247977024610315889125328944908400217447704523656685229764098882708278894491*x**55)/682102521418839092127819566270329651200000 - (145700452740184287501624433197927767912250097556510683799197297163375709723833*x**57)/3683353615661731097490225657859780116480000 + (20511113943131115904047268115852759281084893420796012186742942517902026966961*x**59)/327409210281042764221353391809758232576000 - (264073067932790047559624459559927191330352537285442978352117692748992103991391*x**61)/3182417523931735668231554968390850020638720 + (525434213771504849726524148481183254437103410690206919135971405025799676709705*x**63)/5728351543077124202816798943103530037149696 - (482485783524253010620579030454611868908672124355772557434349063753958553326205*x**65)/5728351543077124202816798943103530037149696 + (1643637297674350120338749984620685788378252574323772968077455009766140548227575*x**67)/25777581943847058912675595243965885167173632 - (3042360193128042487276472595418078139986669110964601800595051814401371515328625*x**69)/77332745831541176738026785731897655501520896 + (1504509932079087620295266988766272291926381482623340115268511775633339786749375*x**71)/77332745831541176738026785731897655501520896 - (141374413068374034405153792452319834991141768121810341982307482819467865833890625*x**73)/18791857237064505947340508932851130286869577728 + (247016105147191970460110349158635174299625092157595687967341339001362495350484375*x**75)/112751143422387035684043053597106781721217466368 - (5659076963656925937563948174067555799614708079894694372566423702761725619921875*x**77)/12527904824709670631560339288567420191246385152 + (178924137447314046588843128830224612643564756730570983976790176383155120346796875*x**79)/3044280872404449963469162447121883106472871591936 - (894620687236570232944215644151123063217823783652854919883950881915775601733984375*x**81)/246586750664760447041002158216872531624302598946816\n",
    "        if k ==28:\n",
    "             uk = (8305870273899927660260534266327404306302408883984375*x**28)/649037107316853453566312041152512 - (1349463284458456304232957001188549157914464129473593706125*x**30)/108226937645085313382182532862181376 + (24510435234027110651673117298780346868283946112112619258772905235*x**32)/11815567690464544003186395842695789543424 - (38154026130873918064894214971776883829530989729193944408829841773845*x**34)/271146906827729449452432635631518894522368 + (96813225714243228848504305476912302286015627168171012947359876581631193*x**36)/18662887433159438360304560673232972740034560 - (13145636678023423952094111075043544796759695654895928742888454528313669*x**38)/109387737347465607614225295812011443093504 + (355896995005908189984192298193614300071931461980538358951731412006669477547*x**40)/186456370478634558433338572406837687091200000 - (4398399891330955101450820008016240235810789682405784632810485920955206336609*x**42)/199774682655679884035719899007326093312000000 + (2978786441195345902132046197767026009848785095244615103134372992067039733617599*x**44)/15511916535617496878067662746451202539520000000 - (51165742930715875990753369562443301199497068721971532374895449525265190268342681*x**46)/39225313417364424548060710616338268160000000000 + (30187160408574861775300086936783116324445986910844207687891789716812458010053693*x**48)/4279125100076119041606622976327811072000000000 - (7343736805962335481875291372107418825379891644724240204941567081845645731655537*x**50)/237729172226451057867034609795989504000000000 + (5127035945661991318180752674913266575614240498356367785322626666904662528201633179*x**52)/46214551080822085649351528144340359577600000000 - (35009185706730048663685199229620479577706809468710798121252494272980453606779781*x**54)/106104836665152747664327488086495723520000000 + (48634929451148944774838095164634162197030317108696157113634156089680921639536709*x**56)/59418708532485538692023393328437605171200000 - (50155444685730187702797951354208201138420607943190220048573333708096691958169837*x**58)/29466828925293848779921805262878240931840000 + (4209249517537655170422704568748231520718302649635709923421813290308732704968736173*x**60)/1414407788414104741436246652618155564728320000 - (309736167727209284239625713308432819851347325393966199765102744804979940734745597*x**62)/70720389420705237071812332630907778236416000 + (2484356584912688393829861971854238358000723044360464996164571508256052057384369657*x**64)/458268123446169936225343915448282402971975680 - (773708524798919640916360863632410913018718716805562971445273661295268725229562095*x**66)/137480437033850980867603174634484720891592704 + (33092575462325778823958338978874920140701066329505704137124522836691996156635325*x**68)/6789157384387702758893983932567146710695936 - (2161571470914349270301980820004202774277659465207198151817390369726196874430509975*x**70)/618661966652329413904214285855181244012167168 + (22809679289972840816753090108101660321529399214202018001888410553695556727356262125*x**72)/11135915399741929450275857145393262392219009024 - (16126353437804021351282406144474686347642053300115512301380049854195001467745323125*x**74)/16703873099612894175413785718089893588328513536 + (1287407436564911303610956234607942151210315509909699170906896055128022924419769234375*x**76)/3608036589516385141889377715107417015078958923776 - (269424840670644110748794131384446673355939059097067591060915346325251666378680546875*x**78)/2706027442137288856417033286330562761309219192832 + (106705814996531963240304373078281589482909859465982548747802260840836909660861671875*x**80)/5412054884274577712834066572661125522618438385664 - (4870493945453335662194898809887544180770476242962872754832205391325865530960157734375*x**82)/1972694005318083576328017265734980252994420791574528 + (3478924246752382615853499149919674414836054459259194824880146708089903950685826953125*x**84)/23672328063817002915936207188819763035933049498894336\n",
    "        if k ==29:\n",
    "             uk = (930543879996581550627119856251646089350914705657421875*x**29)/5192296858534827628530496329220096 - (20104674159678211114535974166078927263042484091369941041105125*x**31)/107361122143924630875125072599283924992 + (8930686704534156691085901249315695791807089595353736951481944125*x**33)/267499345706426517180448375136925515776 - (13269679961145366477748812690870434639579366270924879038226551257885747*x**35)/5470094120350715849822988823174989872103424 + (16901451588840293688291578121096195126353015637981953207085342612818720023*x**37)/176449117549871053588334028183293560451235840 - (10755079549153951765608206449228261247676714512027051237710454490230243939*x**39)/4524336347432591117649590466238296421826560 + (35865031001072419465486807628869181171724328456522535477155234937320017039*x**41)/886304791342291424519731776146584371200000 - (2395987619287955611652122389786286365836040781551474764977095945676447761431*x**43)/4786045873248373692406551591191555604480000 + (26178587496876137917380149605221352538527484598877754125240247900132109932109128279*x**45)/5584289952822298876104358588722432914227200000000 - (4779045515680704508972440919721541817651825370466658896367464085007462971115909581237*x**47)/139607248820557471902608964718060822855680000000000 + (375328603622517869352743578985456834755982741593321518750390873887885033599531669609*x**49)/1882815044033492378306914109584236871680000000000 - (96716816197647061019034835463722601206423663399206837686425678134702672298801319011*x**51)/102699002401826856998558951431867465728000000000 + (21463164787342640509255070810767260837249503045175856582980519016879865157697766021*x**53)/5868514422961534685631940081820998041600000000 - (61674609950829127764809266551547286332658985356011125549807789339915867982338271188629*x**55)/5228846350858727404898058612902509255065600000000 + (378287707349826982933618233573467310259080873512089181772498264652150801218017505379*x**57)/11883741706497107738404678665687521034240000000 - (103091179343156566784289859663675693415142779851827081386786937243262736543135401543*x**59)/1426049004779652928608561439882502524108800000 + (62760690128676169298994968040967922533530790382032877612948260510213343737661747321*x**61)/452610492292513517259598928837809780713062400 - (183379944420011355174315933727088294805040419155833771449542324305686644162583009919*x**63)/814698886126524331067278071908057605283512320 + (629970171755112356170266870574568597109036873603572873629318121996250566618007375151*x**65)/2036747215316310827668195179770144013208780800 - (3951248040801819098515102965988756252664721934806172057261807732059214033604300441791*x**67)/10998434962708078469408253970758777671327416320 + (4639184807760358916132384538819104802184459929768458654249662140378956691939407188735*x**69)/13198121955249694163289904764910533205592899584 - (1267302573671322435742412211402300631662345358174557779874373375103493994863544544975*x**71)/4399373985083231387763301588303511068530966528 + (17466485261409159632000899740994302345507905181773950963561940573347764273312840918875*x**73)/89087323197935435602206857163146099137752072192 - (87778068340998553343099717319342965477917218428009310959930972146549758578176958681125*x**75)/801785908781418920419861714468314892239768649728 + (17563818537120018606892550469306498885008271254344484210618038320366508498499831066875*x**77)/356349292791741742408827428652584396551008288768 - (1510615869518228228044360067247253663909761190650090222295008353252851003806762247296875*x**79)/86592878148393243405345065162578008361895014170624 + (1820713869532094154882940908063494911758072184534543182731699646820234030825223529265625*x**81)/389667951667769595324052793231601037628527563767808 - (3121353329831388923350032254657930442016324277394521917747585233924155655165216502109375*x**83)/3507011565009926357916475139084409338656748073910272 + (20342661640459882107941750929240304173312344845072215819004169860884904361240304525703125*x**85)/189378624510536023327489657510558104287464395991154688 - (3507355455251703811714094987800052443674542214667623417069684458773259372627638711328125*x**87)/568135873531608069982468972531674312862393187973464064\n",
    "        if k ==30:\n",
    "             uk = (215948216417873358515533614640798669135368939359565703125*x**30)/83076749736557242056487941267521536 - (2495842812524883878245316674854105000805688243868419294426774625*x**32)/858888977151397047001000580794271399936 + (57469643912016650848385518270593236938811163087959597767429584989596875*x**34)/103755506217843066071814872161109573655068672 - (1882255478011682768207196415511523915699554502662537740126181907753820257*x**36)/43760752962805726798583910585399918976827392 + (108963664406794005207198555687118289274033207458577167856856515288207006434475*x**38)/60039753064969457167657125323168688836207181824 - (113246529493683393579183310079300188147787760548663138206644481977020919193539*x**40)/2352654900664947381177787042443914139349811200 + (40785310968772003213275097012600834826488056804670585229026166803736212897554595147*x**42)/46555421015081362600614285897592070180595302400000 - (29813384905720014202961580666800899011163779158729667782335995346199522983333*x**44)/2571257856201812484320980097831849164800000 + (799055789048547421401384045014887718944784251205831073052814148749710074074475824401*x**46)/6872972249627444770589979801504532817510400000000 - (11088348409495616704743196394071677209476585585771375261371343623774637630317765754537*x**48)/12183905351612288456954964193576217267404800000000 + (9627514480944089303943264758538369833027022165951124271385835088986131156696920658781943*x**50)/1692209076612817841243745026885585731584000000000000 - (118733314651507895123551591160225400739444831334268343244640136864182914048657952694637*x**52)/4107960096073074279942358057274698629120000000000 + (10732444140388542710647711543227976502190425001392960486933233475294131469477038984745477*x**54)/88731938075178404446754934037133490388992000000000 - (1958382568436297526813173411339232641586560238776197180226013686041599231377513457023989*x**56)/4647863422985535471020496544802230448947200000000 + (20610569414420531238679412335824188885751610696802268733637681606447528048842702727746553*x**58)/16732308322747927695673787561288029616209920000000 - (371624689944343931001056529619939198457017770707727884611622583663143867490637280104253*x**60)/122232771838255965309305266275643073495040000000 + (4479834831949331704683369176565025872310220279997723262250835922758165371169664048712293*x**62)/704060765788354360181598333747704103331430400000 - (368939660660191714118892375583323043681547576435028667644923808445351300019150534828591*x**64)/32587955445060973242691122876322304211340492800 + (1117463542774473056625799105422672217979807601312116983234413176682541074003870206364989*x**66)/65175910890121946485382245752644608422680985600 - (888761242744927431966581049975428394793636660844227195475420918053868505230718025439*x**68)/40232043759334534867519904785583091618938880 + (25515413293359036725565270037522676740072879833003110292025586960681600550523274805971239*x**70)/1055849756419975533063192381192842656447431966720 - (21244785926358408955181646176375535375451301338992781099059214725848321033571682986383365*x**72)/950264780777977979756873143073558390802688770048 + (11013725163807497010848861215554030646752037225228200996505451398078023066895558192004075*x**74)/633509853851985319837915428715705593868459180032 - (72246840741718421867628057875736759451725542621783786908434811185169525756757092154930875*x**76)/6414287270251351363358893715746519137918149197824 + (924270724713312352865872096974052780727194110861605214654586380629325899859773295704766875*x**78)/153942894486032432720613449177916459310035580747776 - (199131853616930253423525173742888993718628725015571729942938505415424234027780753025269375*x**80)/76971447243016216360306724588958229655017790373888 + (10953775049071505075078069091178089060705950408402489030853039981805386612705596155289140625*x**82)/12469374453368627050369689383411233204112882040569856 - (19039510471373779306925923559847234894074700506756750769939956048705467573026550365663734375*x**84)/84168277560238232589995403338025824127761953773846528 + (1552744780128744180551276422084498361899361350746697166993931000075058740405654458161328125*x**86)/37408123360105881151109068150233699612338646121709568 - (21973581927151924380388805098567328559621006974892660707941573134214469969512156526470703125*x**88)/4545086988252864559859751780253394502899145503787712512 + (21973581927151924380388805098567328559621006974892660707941573134214469969512156526470703125*x**90)/81811565788551562077475532044561101052184619068178825216\n",
    "        if k ==31:\n",
    "             uk = (25920752041642153775364534841239091866216381398611096171875*x**31)/664613997892457936451903530140172288 - (639623519099220536607540538109625786677031934366670429611478334375*x**33)/13742223634422352752016009292708342398976 + (341571695327370850713041756493690472480966853305674752326884743495861625*x**35)/36088871727945414285848651186472895184371712 - (578425546819180545689584452171790928571612730438015354009312410793233360596577*x**37)/737981337964755776731319068112184233625217138688 + (31423887688567313187863914842857437594876782253037036831055053305292787974779*x**39)/891128060333498436625708724648143804619030528 - (64044717049402577115706309867749837514838277674881642239958420354063014152481531*x**41)/64213639641678563815676069864351538862253670400 + (220593092405513106248961951752901489901171131547226175138001474652112532094248363559*x**43)/11392385377808145201091495843175471291251556352000 - (269666267355392954371054019017899288149546634561797302386256734147654837462185495649767*x**45)/985879503848781796248302524890185015589076992000000 + (17897487939881989480030430408585171853557707183777763465816377061535700773410346762263*x**47)/6109308666335506462746648712448473615564800000000 - (73611112934744130625669877157518337754247694673099640532677525955230762535752139592073867*x**49)/3002114278637267875793703177297179934688542720000000 + (145669023384397370139455509008221353289561502037651353218032992933911124492209503947658591591*x**51)/887684532760323873292433105531981543768064000000000000 - (9334808751889839720017412238102869718766774243759533740034042195597365629479446135380298211*x**53)/10443347444239104391675683594493900514918400000000000 + (14260406273962258701317172436272111871829461371396082655949793130403451139695775452664328571*x**55)/3549277523007136177870197361485339615559680000000000 - (352920522772898417969041203401545673075801027728405948120914549623729679650080217936919300311*x**57)/23425231651847098773943302585803241462693888000000000 + (35700553376350063221049625433414645777259514669217849607747101952293343209644404224471983*x**59)/751169846139076439760888330473087749324800000000 - (556143881377343368289644670101239946216321961288804722691663662791892279442591549188328891*x**61)/4380822542683093796685500743319047754062233600000 + (2924953206775987175842562521266218063554930858829190836572396308052269361723152017251774271*x**63)/10138475027352302786615016005966939087972597760000 - (1134727715057145909293772280641684257422515869898039138317670631672199519094131592702144523*x**65)/2027695005470460557323003201193387817594519552000 + (96795845146362604110237114482351629208159522977318241844780448493526130050482362783577797*x**67)/104281457424195114376611593204231373476289576960 - (2060207914448805790309544185114277898332979826655316083789429840612023253408633937117787043*x**69)/1564221861362926715649173898063470602144343654400 + (1498958054444347684895661274141607245072110149520779354286489407392471504476418940991886229*x**71)/938533116817756029389504338838082361286606192640 - (25100492778250369960313914032431003962317113315602656261264021985780349059198579530300932393*x**73)/15204236492447647676109970289176934252843020320768 + (66061373478520702974678625392829279469555066646790077690424049910803860887403091364146904355*x**75)/45612709477342943028329910867530802758529060962304 - (5430059589015227282003604661554397720862281132014174200644624142624631628644681002146975075*x**77)/5068078830815882558703323429725644750947673440256 + (815606857828127509698302875897559136241799871674133509163053190931990207373633450905228707125*x**79)/1231543155888259461764907593423331674480284645982208 - (3742909407087477171263671228043553705644715100278451626058421993434877101247077710429489693125*x**81)/11083888402994335155884168340809985070322561813839872 + (1546434312860829423272749293125915054639424065374861005122233145535598801066245193299309728125*x**83)/11083888402994335155884168340809985070322561813839872 - (122614027057203727725192329877337821959986783734448355457001410305054988131597941615427120009375*x**85)/2693384881927623442879852906816826372088382520763088896 + (91202812321602738055305009545038516884545584402161020160054835451559479969128155264428824578125*x**87)/8080154645782870328639558720450479116265147562289266688 - (16143126078758799297282067581502923579826188049797722252952366104488283478165228687055683203125*x**89)/8080154645782870328639558720450479116265147562289266688 + (147157078166136437575463827745105399363781883710856148761084715279834305385822912257774298828125*x**91)/654492526308412496619804256356488808417476952545430601728 - (23735012607441360899268359313726677316739013501750991735658825045134565384810147138350693359375*x**93)/1963477578925237489859412769069466425252430857636291805184\n",
    "        if k ==32:\n",
    "             uk = (102879464853277708334421838784877955617012817771087440706171875*x**32)/170141183460469231731687303715884105728 - (84515655256042452964103260161216675813485994174906395944398979865625*x**34)/109937789075378822016128074341666739191808 + (2210368251915771134533881892430081034796158589251800420322244685784171236625*x**36)/13280704795883912457192303636622025427848790016 - (86600993650418583138318225523540702248091082480648173627168491915778677878263125*x**38)/5903850703718046213850552544897473869001737109504 + (580438725939177178319659777887503581765124774319888563085056498504470818657473265029*x**40)/826539098520526469939077356285646341660243195330560 - (133670272870426818235403829091930027384560756264638611801935426964927890855329073641*x**42)/6323936372987378560605891570089241206572292505600 + (2041628349245808379666911357716590947146166883917988538346567708332787474553823677797*x**44)/4673799129357187774806767525405321555385253888000 - (389709088330228215440588164217379967053765209595870675760381705753147810979965674985139*x**46)/59301022787896649398394136835499850561748992000000 + (24635973017206340865801353963881609298091762785916683906955917923357769208170364351492553*x**48)/328436433902196827437259834781229941572763648000000 - (451278256113176658157643245631016500850564234283175652478321166662862033902476502084335359*x**50)/673875258953371015890842464039771029110784000000000 + (3330772408349738283245492305894914584006638138329524180299561526455261836285126411156731848769381*x**52)/695944673684093916661267554737073530314162176000000000000 - (1781621244292053214572947357578831124077987528480867035989361714733857470973501023557418874405371*x**54)/63913286358743318877055183598302671151300608000000000000 + (8476030955460564703739629110380501391303300609592524043378496188020214854509976836228153353400027*x**56)/63161365342758103360854534379499110314226483200000000000 - (14455169659720758401398621614701799703244406115238712393054702069500456572513645981101661981259*x**58)/26771693316396684313078060098060847385935872000000000 + (294220953651592381482195464002233706974783521232222075733197934088971950158526181469901846398467*x**60)/160630159898380105878468360588365084315615232000000000 - (2567519387996044857524973533387147540133648402082109148338418838346443484632247193363208352423*x**62)/486758060298121532965055638146560861562470400000000 + (588717342193445085452754968104092023833792546928161642383537756508592732951033922699606771676189*x**64)/45420368122538316484035271706731887114117237964800000 - (2212541789266825808961285503132814964113672104771856003203070998904525531922121839739776959737*x**66)/81107800218818422292920128047735512703780782080000 + (11845247770049645585495782184712754835110833140355985645945986177781300834819042380908019107*x**68)/240319408055758288275318897919216333937128243200 - (4789349116259842190464263928536634864647014640060572441296796477870012052504838525968594562027*x**70)/62568874454517068625966955922538824085773746176000 + (138056612211579533562052658138579069063945911140517619235734800142189403799600852462993083322619*x**72)/1351487688217568682320886247926838600252712917401600 - (7904294475204964669341808091255550481831951249074278563170016600465775431298218517413198069679*x**74)/67574384410878434116044312396341930012635645870080 + (83624533981058445646129241760934562790525893879283080118768190791351001481441328834110077767911*x**76)/729803351637487088453278573880492844136464975396864 - (104626195419215175852830101740914189737630118178380253264567643976975974537437134733349537770025*x**78)/1094705027456230632679917860820739266204697463095296 + (590428276506377763659696137426446317313079063899185275296095898719252974768713677263641685784525*x**80)/8757640219649845061439342886565914129637579704762368 - (3532781785925504963206862458166128531842475377645181032506778779122326360287265012346322235117875*x**82)/88671107223954681247073346726479880562580494510718976 + (10358160251550277704615864252468730027156691964565659306056447519938078895427372467966113424588125*x**84)/532026643343728087482440080358879283375482967064313856 - (2055212067264542595697606229966386306669410791079832249881445871294245762809920335794352953971875*x**86)/266013321671864043741220040179439641687741483532156928 + (627307291200434023157857151457036706638501001458012834252453679356378210616761589249913747797578125*x**88)/258564948665051850516465879054415331720484721993256534016 - (337416999866371645786664415137726671582960132270126277522056448287400980095683155602018407542109375*x**90)/581771134496366663662048227872434496371090624484827201536 + (115364672098115469161911284629242578268955007974198078697818307259103647117151578726698376045703125*x**92)/1163542268992733327324096455744868992742181248969654403072 - (169548689059996617447833597921675146744393469048408034364505250827414254369852805068094342943359375*x**94)/15707820631401899918875302152555731402019446861090334441472 + (847743445299983087239167989608375733721967345242040171822526254137071271849264025340471714716796875*x**96)/1507950780614582392212029006645350214593866898664672106381312\n",
    "        if k ==33:\n",
    "             uk = (13171689060760555082210068753518465529753913790389225363138671875*x**33)/1361129467683753853853498429727072845824 - (16007043198900159912954000262234499150821381581457864510369649663499375*x**35)/1223655391447694714614295088324638488395776 + (319742661427492904709211202437976652561380615261133983315729857944281529291375*x**37)/106245638367071299657538429092976203422790320128 - (65766125599527540422416202299964268638706958655694047304374158931727042539888588625*x**39)/233421667492455645347611928717268718919870333321216 + (5572276522987968386484941909444126643256774484524018988018402683585154472254553640377*x**41)/388959575774365397618389344134421807840114444861440 - (36661751325972018833217197892932478121120614633329241306115645664232183007183568024268455*x**43)/80047880694364398829864527022864008053495552752484352 + (659844591020062446554552394647946901272650200516565550580895369740158900452697161228791493*x**45)/65620139776174916358287016056690714637608964587520000 - (2254398608638686197545838820354565003918278600105850724178378441713948788343520321312197987*x**47)/14021397388071563324420302576215964666155761664000000 + (66663653896432064986723908418719708197463577500000085302073552144867159127367222895356371581*x**49)/34157389125828470053475022817247913923567419392000000 - (237539496852592944032360695837081421809803228360523547851874066233109279230814126917591231651709*x**51)/12809020922185676270053133556467967721337782272000000000 + (499123172824920405415999502356521626084644586205221325907232157169535396385823039662571290142789*x**53)/3534957072681111957644533611362913169849712640000000000 - (31396045821540070748248549550614633117822332524350498716123867210939147863440177672051910928180401939*x**55)/35791440360896258571150902815049495844728340480000000000000 + (44708107541347588028184380275567626642973550391690252461753452823085918959690657594550343480167208469*x**57)/9911475792248194681241788471859860387770925056000000000000 - (22871649443938199486542269379897612181677213575779727863472895210311526846498354983715917216460997*x**59)/1177834318746071857545072902181801590941286400000000000 + (329987481588611905657968733570951056584846013627515357311511383379084715615704079870495994344683*x**61)/4672877378861966716464534126206984270999715840000000 - (12864628195152684412022800854226374052223800241362352139334247918777854519114714971280951631969387*x**63)/58878254973660780627453129990208001814596419584000000 + (16678972825037480510578012211321571718883365865097430907944742053346151327046366349964918088140223*x**65)/28838328966690994593038267750305960072455389184000000 - (68268034963334394674955175642036397946763659722771239593056430942870641182039062722228199531960637*x**67)/51908992140043790267468881950550728130419700531200000 + (10030338272904494772879658390737208147167922487380499361571280790319711013399609714595618005858773*x**69)/3893174410503284270060166146291304609781477539840000 - (3391342880210446846636362621986943173376861127975734874397941788000979647427822677905138570234707*x**71)/778634882100656854012033229258260921956295507968000 + (68735264570934261952841940539349935930355619328161814539414863824310465444073625673485925460053863*x**73)/10811901505740549458567089983414708802021703339212800 - (259695117588467369403571892310531612482782976152193773248272045120432627043816456142590950883362123*x**75)/32435704517221648375701269950244126406065110017638400 + (3129440506971357233809266931369510382534959892714219952917895428407546131760429158349803203085881*x**77)/360396716858018315285569666113823626734056777973760 - (141570847772828917820124429172681204102017177931399573772600605642709984968401042377938481609818517*x**79)/17515280439299690122878685773131828259275159409524736 + (12153860696071278004026291810623872927685644797630040938365970828383310008680738288529955422308015975*x**81)/1891650287444366533270898063498237452001717216228671488 - (2730793538649693346194994247038822027052836290739157741335314872289793920379876178642876753441514875*x**83)/630550095814788844423632687832745817333905738742890496 + (10434225124090867257831457138712168976096315701573823929282204178257739937317217579685215749143918875*x**85)/4256213146749824699859520642871034267003863736514510848 - (14684910868858498618728321960784696943055096348291819998088825624288295708134010355375987493570051875*x**87)/12768639440249474099578561928613102801011591209543532544 + (100903176871158732443606651504993505688013674068653273651502524635155857275069940643075381919878990625*x**89)/229835509924490533792414114715035850418208641771783585792 - (2473515016749269770875187666685384820726553151940655411539704307024579919092543107207761327930409609375*x**91)/18616676303883733237185543291917903883874899983514470449152 + (285431961114074312369885264496875079969525002215562215581558622663979203866585093683746501863002734375*x**93)/9308338151941866618592771645958951941937449991757235224576 - (582501831210312986226536162074362467506130860678352629057530605198687087526164453851746114342578125*x**95)/114917754962245266896207057357517925209104320885891792896 + (6446410706750131391984081226580010754368584086689521874572854141709117365396173501494015013049466796875*x**97)/12063606244916659137696232053162801716750935189317376851050496 - (2930186684886423359992764193900004888349356403040691761169479155322326075180078864315461369567939453125*x**99)/108572456204249932239266088478465215450758416703856391659454464\n",
    "        if k ==34:\n",
    "             uk = (3478100717279654809649470507914375986062665823826896038537029296875*x**34)/21778071482940061661655974875633165533184 - (2242995390564074395769834867110596898289516526218666319604876586081574375*x**36)/9789243131581557716914360706597107907166208 + (28895253430930409153080571645858957932817748532385786425604457502197420164878453125*x**38)/516778785017434801534266919108236253448452117102592 - (10365213800813796902570072008718317280132711168741979101064155687324680565539365013575*x**40)/1867373339939645162780895429738149751358962666569728 + (2284778452680810082660615538923117283070201739919206753917659275534558980953293260501237327*x**42)/7635625194829073220777882070977579224694682517304770560 - (28712890293350807007751749155743920466622550181639795206066422346045379189050169149161637*x**44)/2831047946750288199111035438474412309584281264455680 + (237491454972783287858739193426156754763555506008740982938074651171770380804931305509365269233*x**46)/1005767033713918175999973116300120162615468343951360000 - (2321572288425667758009930523257436120537320441114660614232011656102220479105604406746554307*x**48)/579640542741331613028667053868449448399931965440000 + (5538893099580232519535991935905239231909985615519059698611771634663282034709109820778636249397*x**50)/107340841248394743153456861827490638592579993600000000 - (693742500413094426435105876860057157040944621772401786336249618218752562866967102979426555493069289*x**52)/1332138175907310332085525889872668643019129356288000000000 + (660859162968730169110183577204507892539790014819498457183494448368619529378798303354101180601013974651*x**54)/156782416087552677545450354731167924909174455009280000000000 - (886612203460448820828014619383319482329143079374824520377657055068144407424702250095901253130948452261*x**56)/31814613654130007618800802502266218528647413760000000000000 + (18517678050160723089068181389302046776652403894351621648048110690508203282071696210313471232497666342267*x**58)/121269821458095558452840706008638291803314847744000000000000 - (1402237952283651325803616707733952371924539740581328090983272612241634449577259041437082663101251222739*x**60)/1998953100957619095376495153988543271483211776000000000000 + (198224776853964977282527887493309865951074330638435890575605880811795812322104677810026048229261115609*x**62)/72689203671186148922781641963219755326662246400000000000 - (21284628299086214825615874044955171907828699748965909129063858446172110394998120860208507403166307541*x**64)/2355130198946431225098125199608320072583856783360000000 + (1741986949332642454778835837986211414880404671791727676341930853044469438043360751373498056494903324973*x**66)/67827749729657219282826005748719618090415075360768000000 - (107427177387205728639814084894644937370308755993170682623823434260347704595529466117831165212403191*x**68)/1708938012840947827735601051869982819108467507200000 + (2318999131818483897699783360693156593374063487848110411897572085183917640987728681555182782459923637047*x**70)/17441421359054713529869544335385044651821019378483200000 - (9752158689957853222051140466388187341128106451991807107906605471131161783296460214275499861001829431*x**72)/40044079650890923920618851790424847414895197552640000 + (37165237711209521521300360315717369181758317953226586547237688552342216195737688035260854485793167077*x**74)/96105791162138217409485244297019633795748474126336000 - (138137610252338525361480794460898458944187443235892021260895080233100559652794287154346283334236665231*x**76)/259485636137773187005610159601953011248520880141107200 + (197612260974721476302995193259718710996857732074821805478038870863352957836300186210141587404577487401*x**78)/311382763365327824406732191522343613498225056169328640 - (50884240148946597806485512035844378285091645013211746045087961159320863101831982607518503407391157211*x**80)/77845690841331956101683047880585903374556264042332160 + (17540922543916725563623032898786599673672528055527451099161228969635720709918339405886043937232235585107*x**82)/30266404599109864532334369015971799232027475459658743808 - (19975626607355719705132009751571055982212878249055730540270067190689009631525767042520981638601347425075*x**84)/45399606898664796798501553523957698848041213189488115712 + (954917320316131239133459868508439973883698337781828937061729420993737785990281477431790589715665091875*x**86)/3362933844345540503592707668441311025780830606628749312 - (15755204044023285548526976416783139928523966679263943530599774600533155215162014492609430237158842678125*x**88)/102149115521995792796628495428904822408092729676348260352 + (2302625136706932278479113011691989657418252441363612449398035344017996871596375541404705772068863297666875*x**90)/33096313429126636866107632518965162460222044415136836354048 - (423386061524358176144821533237803992503520430125794090775648407844594191628027688181815896425808032821875*x**92)/16548156714563318433053816259482581230111022207568418177024 + (2225716976267508299737124101489851133893447128032194873864750383884287807899726509655546026604281542109375*x**94)/297866820862139731794968692670686462141998399736231527186432 - (372376360898951558969004484876697106956747139322943788032468719386183213104569444744393229900772739453125*x**96)/223400115646604798846226519503014846606498799802173645389824 + (5720200803281729534396456110939707570408899409906976164860730633325963142338211095922155884505433716796875*x**98)/21446411102074060689237745872289425274223884781008669957423104 - (23674150301871368894725539028195699494929790122726965015192689887511201291809929176350338589287210017578125*x**100)/868579649633999457914128707827721723606067333630851133275635712 + (6962985382903343792566335008292852792626408859625577945644908790444470968179390934220687820378591181640625*x**102)/5211477897803996747484772246966330341636404001785106799653814272\n",
    "        if k ==35:\n",
    "             uk = (473121071856241044249746545376581259132695771063995772556422756640625*x**35)/174224571863520493293247799005065324265472 - (23921594627420235831642207753935321617915443221769945486084928458352374766875*x**37)/5795231933896282168413301538305487881042395136 + (5211682830294553819878122730279709161919367703104682126593316201244793918361295216875*x**39)/4885908512892110850869432689750597305330820016242688 - (2603935665997188561895506198884633415307062630685178266341232000536770221006047824168475*x**41)/23229933313213386372803773325438261868567760639885312 + (7015344623146557352180740261551754425798502297457023234801920817975921295456869250065651915*x**43)/1096882380929006184037441882641980421132610739327991808 - (1028804618315041442307826145774329919772858526956212276773193339009706222783053918555522911999*x**45)/4483918520482867326293213480210542878378581164661145600 + (442651134529522357605166178408644524708739692136844401035078330136681888986966064793725718503*x**47)/78277823412450364952378860049983750495417915146240000 - 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    "        if k ==36:\n",
    "             uk = (265000369247479011562552481693705125254213264659289187717436346247265625*x**36)/5575186299632655785383929568162090376495104 - (3543230830890101904728017131351726845004588024972003812258941340086918109765625*x**38)/46361855471170257347306412306443903048339161088 + (60394375613855732728036346468650823060165728064953657242084717283659300837872408043216375*x**40)/2892457839632129623714704152332353604755845449615671296 - (5810743941725756860445062925566217715367541655625086207674340492033019458080590674721309425*x**42)/2505628669094188709452765619378306314651032940743491584 + (31812780584297212170301499079475412606252277242482179113059566818900334209888688876150553478205*x**44)/227517711339580777503376151660610873004143010598468190208 - (9253422368748207153907360927369542845415100597431486068139346541504862691780767353189763810907*x**46)/1745701402009028409608660821180870696762775712458342400 + (29686790159115289211070369811228078264190588080571059188337879099004680710219822004044664584721*x**48)/215013075907270361300773473576529528633538357546188800 - 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    "        if k ==37:\n",
    "             uk = (38167215343778801422076815539074448985937905064036542739086980787883203125*x**37)/44601490397061246283071436545296723011960832 - (2155918201410855036865031718510173312414601809375731765517953262700704800775390625*x**39)/1483579375077448235113805193806204897546853154816 + (9698200862577835381466341054487977696401291767818798489676602893587623204229856968277850125*x**41)/23139662717057036989717633218658828838046763596925370368 - (124306038034390010615376617821794970435554647508007515336713060906465024791784480733331805900625*x**43)/2530390175941707750816769420793338988819349030981427265536 + (31011265329498145073731013559970134209752569048756662962617543275192070360342908094833073565848171*x**45)/9922707926810103586598856678875674203277462913842870747136 - (340774123248232836705107344025996401018231592788997186243288227893209502474984493021122259292315043*x**47)/2726018661211566919395290296394415989911390910396393062400 + (3185648422869113905891627970346605424922125035252831677658581242298157880645985896187304033150944133*x**49)/927136383312149797928935218061995327467817397739166105600 - 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(16183687088697667571880778322961104017360512299970948399022601516615443971466968320212261793002678891169691798821875*x**99)/915046873688493256074143823884348811700219083989703251516719104 + (257357738925392045561101648914917685812803681230847614810540991040433254839211788465593617516250593813234536957328125*x**101)/43922249937047676291558903546448742961610516031505756072802516992 - (1836596765416999518318756830857452026805442464404311895899112653619568390453658420046700464771707058187098937836328125*x**103)/1185900748300287259872090395754116059963483932850655413965667958784 + (558526181849898918142200696754099050467480658771792473303709124603284233251660560274627100472766713754831221450390625*x**105)/1778851122450430889808135593631174089945225899275983120948501938176 - (81544993889186548568619097492599546419405114865987983198379541323545542127854738977375382907095140514385958021484375*x**107)/1778851122450430889808135593631174089945225899275983120948501938176 + (3708775152811827155294072873174630402957232083960335450799538813334864630033519138188149765050509942675692816591796875*x**109)/864521645510909412446753898504750607713379787048127796780971941953536 - (501185831461057723688388226104679784183409741075721006864802542342549274328853937592993211493312154415634164404296875*x**111)/2593564936532728237340261695514251823140139361144383390342915825860608\n",
    "        if k ==38:\n",
    "             uk = (11299504542566092526272741442489146081363195578168713310913908785886474609375*x**38)/713623846352979940529142984724747568191373312 - (336565190370068211553362512783250083141362474089765033879006076129965919090714453125*x**40)/11868635000619585880910441550449639180374825238528 + (220135937942901463634762641665333930463666006998335075708685888181023894668057597485719914625*x**42)/25533420929166385643826353896451121476465394313848684544 - (139008775163516998474497097338089319576605215335181689250170185908000245873444225933924081275821125*x**44)/130533231145130855007651277707132245768060212080972937560064 + (1074045614339462955532777864718967170798418610963394136288199362258998098304309928247164014997297063*x**46)/15045983299414940181877403765242202179504916086494618189824 - (1793767496851708501830545076919313390258431576903335999306159103448847645786369022693584495233557*x**48)/596258353785168430763153038172394474895177779444187136 + (1091446964427712690070662112336354586701033858113548644866595348332980776030047206396651278504057139715383*x**50)/12526642690084157269795391390704292424454066173898066493440000 - (14510593318264305836604460650925738441163513498269502122217420788247564269668883136849302851389902818515749*x**52)/7884490400233075554288217680874323900532100018790098534400000 + (83943236530588414658435998145692753066711537331882316333256722573036673935612798630112867055664547503650955593*x**54)/2838416544083907199543758365114756604191556006764435472384000000 - (149222123723477326252257351124234530942786503279363501839859290076848383448941359413075033638816794558408581020647*x**56)/399978728844122016084776508748702114643452962133342420992000000000 + (81745276858140746509598380976847483722706386128105779821032124001520093709136773434247866624582984013683146289553923*x**58)/21598851357582588868577931472429914190746459955200490733568000000000 - (406506633986490267171803167366546868419318999684855268844264763869110770106572865461822098293058820054597686076619*x**60)/12919166205456387787726371254045378034492274715268218880000000000 + (6455259017966666161061197611986979933117329761434135009618049636296663224980871557260862984861574280172763287918229*x**62)/29658324622259843406167059811858076295895947463884800000000000000 - (4569530690612070572069273499620768118703157197631585542939961717505297978191981682423480254905605700497223670335461337*x**64)/3603486441604570973849297767140756269951357616862003200000000000000 + (306624506426529021841924286753397543067926789349607984141818296916776541615381917543666205245508361776424620342613027*x**66)/48785662594031114722882800539751777193187610812900966400000000000 - (1532098158385322398901712572839328060220460324423129732021862060221095723110902691528988188159475463366873074443*x**68)/57341551364197334190226324018630007876060908093440000000000 + (11773747182096312571084246944451975736331177665060772037111795796271733143650317113322531659968339944652286829669091*x**70)/120046743225230357486246125730079561104221051897774080000000000 - (2658184701359218014923600856456509578573152110929573055906234687816646649517127108390756011034411423454946604636103*x**72)/8504769272224774597422101618778231296887963735031808000000000 + (2954707128562653400713832291370506038523377869540251381839424992462370381945137975496219559263849786524719042406569*x**74)/3401907708889909838968840647511292518755185494012723200000000 - (1939752566487430632075325465619649465118546837513020695265961576678284132860006423549462012653309370328900865133579*x**76)/918515081400275656521586974828048980063900083383435264000000 + (9929759707436523992407927768047465127907661984568923860599423893979345306880834796617500730213071521706158745886749*x**78)/2204436195360661575651808739587317552153360200120244633600000 - (6647227382812503954948921908566466341490708114144733065216862662358234223329080488679365217293067148183154305936517*x**80)/787298641200236277018503121281184840054771500042944512000000 + (355103135148506995178141564159087993545566609340415454013237836579279886591385791162718818543245933693893391915789509*x**82)/25508475974887655375399501129510388817774596601391402188800000 - (19317972289327149623240118943871921722567275153504340277073066623087511329090847762401133184101227788683186254273443*x**84)/956567849058287076577481292356639580666547372552177582080000 + (525725177560488176302543007635144239262094938384113354858058297594157731483990179584153587666999261575482806202063*x**86)/20406780779910124300319600903608311054219677281113121751040 - (14310641886921333398009023760483431468267916037450379200566784409103909826995782128984026063119964442618992969287681*x**88)/495884772951816020497766301957681958617538157931048858550272 + (252798457038202237031904566223174186006220262214156663980085651046344161852886848293446773358422847369999412009876367*x**90)/8925925913132688368959793435238275255115686842758879453904896 - (54155665473931155929214199072882591956080822953403101783285568858247993682027074764455432024290996503404390979578865*x**92)/2231481478283172092239948358809568813778921710689719863476224 + (13067906019759217644776141176085940427632958664542277151049370082791781166644826788790834422731990099956237745642050075*x**94)/722999998963747757885743268254300295664370634263469235766296576 - (12610927576897373995141784953676197530209529438069662436293629401362437068153224389046016732708856883367839928373961375*x**96)/1084499998445621636828614902381450443496555951395203853649444864 + (4635031563353592635619421624553606948289939385139185843177136625577077088509920009728349797598853483039532279823523125*x**98)/722999998963747757885743268254300295664370634263469235766296576 - (21938826220754184069486889798676676947537392995931791906348048760408480196238266293286620833311963575302573515338058125*x**100)/7320374989507946048593150591074790493601752671917626012133752832 + (2469795918519895605693428961290735848596872496363203922208397253305146489659172489058597346324692543555618819654757903125*x**102)/2108267996978288461994827370229539662157304769512276291494520815616 - (44003553250322683916970895768027347570769425256445803207518060146808656529642546396560330253984183666566841319968390625*x**104)/117125999832127136777490409457196647897628042750682016194140045312 + (5478932057615727610453638881970967625963572535315459196971047131874552838994861812171934997732030967310781934792784765625*x**106)/56923235918413788473860338996197570878247228776831459870352062021632 - (2425579798512359385758779365820426454807855126655221672897321561275740160526805552173969623985627903313784105886673828125*x**108)/128077280816431024066185762741444534476056264747870784708292139548672 + (4129082705491655517011769916997387020870442559682491160321428732910820974042014622979809544067367287363479273190248046875*x**110)/1536927369797172288794229152897334413712675176974449416499505674584064 - (5074105594878040606165980078728999071029676900598814617700633899184437363160183034978981871800590913734763407261982421875*x**112)/20748519492261825898722093564114014585121114889155067122743326606884864 + (1335290946020537001622626336507631334481493921210214373079114183995904569252679746047100492579102872035464054542626953125*x**114)/124491116953570955392332561384684087510726689334930402736459959641309184\n",
    "        if k ==39:\n",
    "            uk = (1717814421355752886878745743910721720933394527768264133856629876705664306640625*x**39)/5708990770823839524233143877797980545530986496 - (4418752638288076379714192588150485249180051427854956970366357347320545392345713696484375*x**41)/7785824560406448337877249657094963302325885356474368 + (3723227874243054805360487333115534307749535521872625666898727075783395275886131312926444690625*x**43)/20497417493310039686393474408223737655847652182743580672 - (3181654718372763552237929651245514007115347159075072314647849392772017825566411856781760427795044475*x**45)/134743980536909269685317447955749414986384735051326903287808 + (27153847194753467144842080164856079691301695921314496197381033996968995040368019614447552663590643933709*x**47)/16304021644966021631923411202645679213352552941210555297824768 - (14050879793365252109151047642422548673500788139356307265481789931958437350958566342622409052105379225115*x**49)/190258885592601824235352976644353008205352487287286784851968 + (676649917417843255594036403330696439994122558245738937183691819127496278678649249136803459675272145669264749*x**51)/300639424562019774475089393376903018186897588173553595842560000 - (4211809315021295104587752159671238850134974000433684159691676362112815490325112178233368893460005960802375459*x**53)/84179038877365536853025030145532845092331324688595006835916800 + (1057116139809833547464855792742919826710881683355663802999383338021986337332905392824817471096647166483725592430233*x**55)/1248903279396919167799253680650492905844284642976351607848960000000 - (34760994274312274022371967241993711265224201817459976458110783317295258079253481879224258657231277590400883988853649459*x**57)/3091035616507374940303152859609969941964604491366470229426176000000000 + (32605720662287360465718152031411914844499012995903728587220681067148463992076209450267184169398979657892755839647*x**59)/271229483428944787342929609662184099779412900789721497600000000 - (32991633674667947684511677700560222461092333730638663576987318774244384292187130693648473885822222468601182154629750863*x**61)/31306663124785952079057650195257599717038734073653603860480000000000 + (9461066263061400113283834550744822395206049692372045780492534572007454918137043046369172689945049274663831901828063*x**63)/1229990038734360225740560304517378140143396733222230425600000000 - (79791833558018924110867175351888388224069768266097124970058448549543529048299575772649709330668169160603036232299331883*x**65)/1685841610107401625192653926147722231556190697947136000000000000000 + (1182649993304337927300275890544802745021421454851678587216628220528632324160197924534965605553844830849864057432228055743*x**67)/4768523411446650311259221105389271454973225117050470400000000000000 - (127714916338353794921353180253961406696130482992148625057106549006006707259304972504372513322901391573467273229912368209*x**69)/114444561874719607470221306529342514919357402809211289600000000000 + (6022586072786765197032246337674763138611281865327889296041430416329902263129663067536533906017983364348929179111971121*x**71)/1387206810602661908729955230658697150537665488596500480000000000 - (685453558448137234013891242896927252372067095048701525454976591222530965908339527338978900359593600498482615429600258459*x**73)/46674173765969562990652493683854933357321144977854562304000000000 + (17713897030073166046843509860442752067857338524863298255471750885953709038938204132940896776515222526426587203546688099*x**75)/408228925066789180676260877701355102250622259281526784000000000 - (7137424421948577132636483412593849538969559275887515303980370916702178542914117192397747690593330484573325000509813721*x**77)/63502277232611650327418358753544127016763462554904166400000000 + (3224106314535532104884795251155162816109097946283906442407558687548197414487687262405250274080382879400943701289867621*x**79)/12596778259203780432296049940498957440876344000687112192000000 - (174629124499177423571795715145333243653311375728294390623365064182590886040519887649014875759634731140171927228448722503*x**81)/340113012998502071671993348393471850903661288018552029184000000 + (20601059365688248959535703968495019549126040639070772895771010837401072634371027731585383605830411241847895053484290741*x**83)/22674200866566804778132889892898123393577419201236801945600000 - (868769427003366709562074300261206854182516357142487668444189643350586491746064407000851105352052375681989297757407950791*x**85)/612203423397303729009588027108249331626590318433393652531200000 + (359269215906462309447824982372860482053441729944512048190563518504638157911143490000857766179677538353330236971986898157*x**87)/183661027019191118702876408132474799487977095530018095759360000 - (10480962606290828580245153841527118501937197788614224112670360110201529283860081850721189398560538660028588072990620179*x**89)/4407864648460586848869033795179395187711450292720434298224640 + (181696602916394737721961507581322505481648000879674141106875069362523036821503030045432001271543168329185438862448456163*x**91)/71407407305061506951678347481906202040925494742071035631239168 - (56913501567402785497214151444661700563461470045024804758082302516060876683560654660788580107642388765234736330663285905*x**93)/23802469101687168983892782493968734013641831580690345210413056 + (15599015120348335028779153381271647999489589086931113495933440780147064199834707942004087372329995795376419845885431745*x**95)/7934156367229056327964260831322911337880610526896781736804352 - (24428071198680118700779078355868349593214967254606594258716354852295734448973081967621855219250812667272663832240809858475*x**97)/17351999975129946189257838438103207095944895222323261658391117824 + (136236634057181713011439455720849837160067800938481205558565877531314981247313813269498162388517146326809173506947827068375*x**99)/156167999776169515703320545942928863863504057000909354925520060416 - (24155121094720139867693201277793261302206865601684253340904325126243257763077765769816141879065301467599141160795730198125*x**101)/52055999925389838567773515314309621287834685666969784975173353472 + (3535360211558615003390583037036225693599880687748602491797247020697734291804796888211374896825646683199840125295616991946875*x**103)/16866143975826307695958618961836317297258438156098210331956166524928 - (4011901482870775429632133326036549247983743681998972468153014721507670914718026701096638155660529951712799995733251692440625*x**105)/50598431927478923087875856885508951891775314468294630995868499574784 + (1246731123596729799631903134737387038505712751160279552962069614053977503259797656483039629037984289136258689584170108953125*x**107)/50598431927478923087875856885508951891775314468294630995868499574784 - (25101357580221127315678710377070578872282100326877187975312264452633180501071748229072213672174190453368953993769780526953125*x**109)/4098472986125792770117944407726225103233800471931865110665348465557504 + (43177051167189312861091276731780315605539918507175841620727056813873120479291735205087611163014577675479714150647878544921875*x**111)/36886256875132134931061499669536025929104204247386785995988136190017536 - (661789184399307442054668685598610077077121075685715126145605961256437206625633815660301509138649127701045338161113623046875*x**113)/4098472986125792770117944407726225103233800471931865110665348465557504 + (14249423801363558559115694687407537022785814230802681660876459103093896020323046641967028196508638388639251111646189267578125*x**115)/995928935628567643138660491077472700085813514679443221891679677130473472 - (5480547615909060984275267187464437316456082396462569869567868885805344623201171785371933921734091687938173504479303564453125*x**117)/8963360420657108788247944419697254300772321632114988997025117094174261248;\n",
    "        return uk\n",
    "\n",
    "    def bessel_coeff_ak(self, k, zeta, y):\n",
    "        ak = 0\n",
    "        y = y.astype(np.complex)\n",
    "        for s in range(2*k+1):\n",
    "            mu = -2*special.gamma(3*s+3/2)/9**s/np.sqrt(np.pi)/(6*s-1)/special.gamma(2*s+1)\n",
    "            ak = ak + mu*zeta**(-3*s/2) * \\\n",
    "                        self.bessel_coeff_uk(2*k-s,(1-y**2)**(-1/2))\n",
    "        return ak\n",
    "\n",
    "\n",
    "    def bessel_coeff_bk(self, k, zeta, y):\n",
    "        bk = 0\n",
    "        y = y.astype(np.complex)\n",
    "        for s in range(2*k+2):\n",
    "            l = special.gamma(3*s+1/2)/9**s/np.sqrt(np.pi)/special.gamma(2*s+1)\n",
    "            bk = bk + l * zeta**(-3*s/2) * self.bessel_coeff_uk(2*k-s+1, (1 - y**2)**(-1/2))\n",
    "\n",
    "        bk = -zeta**(-1/2) * bk\n",
    "        return bk\n",
    "\n",
    " \n",
    "    def scaled_bessel(self, v, z):\n",
    "        # Easier to treat data as a 1D array.\n",
    "        shapez = z.shape\n",
    "        z = z.reshape(1,-1)\n",
    "        J = np.zeros(z.shape, dtype=complex)\n",
    "\n",
    "        # Use built-ins if possible\n",
    "        pref = special.gamma(v+1) * 2**(v) * z**(-v)\n",
    "        easyind = (pref < 1e100)\n",
    "\n",
    "        J[easyind] = pref[easyind]*special.jv(v,z[easyind])\n",
    "\n",
    "        #Use taylor expansion when it converges.\n",
    "        smallind = (z<2*np.sqrt(v)) & ~easyind\n",
    "        y = z[smallind]\n",
    "        J[smallind] = 1 - y**2/(4*(1+v)) + y**4/(32*(1+v)*(2+v)) - y**6/(384*(1+v)*(2+v)*(3+v)) + y**8/(6144*(1+v)*(2+v)*(3+v)*(4+v)) - y**10/(122880*(1+v)*(2+v)*(3+v)*(4+v)*(5+v))\n",
    "        #Use uniform asymptotics, combined with asymptotic expansion of Airy\n",
    "        #function to isolate exponential decay factor.\n",
    "        y = z/v\n",
    "        medind = (y<2*np.exp(-1)) & ~smallind & ~easyind\n",
    "        y = y[medind]\n",
    "        #Assuming y < 2/e, so zeta > 0.\n",
    "        zeta = (3/2)**(2/3) * (np.log((1+np.sqrt(1-y**2))/y) - np.sqrt(1-y**2))**(2/3)\n",
    "        Ja = np.zeros(y.shape)\n",
    "        Jb = np.zeros(y.shape)\n",
    "        #Higher values may lead to vmerical instability, though in \n",
    "        #principle it should improve accuracy\n",
    "        for k in range(5): \n",
    "            Ja = Ja + self.bessel_coeff_ak(k, zeta, y)/v**(2*k)\n",
    "            Jb = Jb + self.bessel_coeff_bk(k, zeta, y)/v**(2*k)\n",
    "\n",
    "        gammaSer = 1/12/v - 1/360/v**3 + 1/1260/v**5 - 1/1680/v**7 + 1/1188/v**9\n",
    "        gammaSerExp = np.exp(gammaSer)\n",
    "        expA = np.sqrt(2)/np.pi/zeta**(1/4) * np.exp(-v*(1+2/3*zeta**(3/2)-np.log(2/y)))*gammaSerExp\n",
    "        expB = np.sqrt(2)/np.pi*zeta**(1/4) * np.exp(-v*(1+2/3*zeta**(3/2)-np.log(2/y)))*gammaSerExp/v\n",
    "        Ja = Ja * self.scaled_airy(0, v**(2/3) * zeta)*(expA)\n",
    "        Jb = Jb * self.scaled_airy(1, v**(2/3) * zeta)*(expB)\n",
    "        y22 = 1-np.power(y,2)\n",
    "        power = np.power(4 * zeta/y22,1/4)\n",
    "\n",
    "        J[medind] = power * (Ja+Jb)\n",
    "\n",
    "        #Use expansion of prefactor, since it will be exponentially decaying\n",
    "        y = z/v\n",
    "        largeind = ~smallind & ~medind & ~easyind\n",
    "        y = y[largeind]\n",
    "        J[largeind] = np.exp(gammaSer + v*(np.log(2/y)-1) + 1/2*np.log(2*np.pi/v)) *( special.jv(v, z[largeind]))\n",
    "        J = np.reshape(J.T,shapez)\n",
    "\n",
    "        return np.real(J)\n",
    "\n",
    "    @staticmethod\n",
    "    def Lambert_W(x,branch=1):\n",
    "        # Lambert_W  Functional inverse of x = w*exp(w).\n",
    "        # w = Lambert_W(x), same as Lambert_W(x,0)\n",
    "        # w = Lambert_W(x,0)  Primary or upper branch, W_0(x)\n",
    "        # w = Lambert_W(x,-1)  Lower branch, W_{-1}(x)\n",
    "        #\n",
    "        # See: http://blogs.mathworks.com/cleve/2013/09/02/the-lambert-w-function/\n",
    "\n",
    "        # Copyright 2013 The MathWorks, Inc.\n",
    "\n",
    "        # Effective starting guess\n",
    "        if branch != -1 :\n",
    "            w = np.ones(x.shape)  # Start above -1\n",
    "        else:\n",
    "            w = -2*np.ones(x.shape);  # Start below -1\n",
    "        v = np.float('inf')*w\n",
    "\n",
    "        # Haley's method\n",
    "        while np.any(abs(w - v)/abs(w) > 1.e-8):\n",
    "            v = w.copy()\n",
    "            e = np.exp(w);\n",
    "            f = w*e - x;  # Iterate to make this quantity zero\n",
    "            w = w - f/((e*(w+1) - (w+2)*f/(2*w+2)))\n",
    "        return w\n",
    "\n",
    " \n",
    "    def find_cutoff(self, dim, eps, sigma):\n",
    "    #Simple approximate inversion function to find value beyond which the \n",
    "    #pdf has negligibly small mass.\n",
    "        if dim < 200:\n",
    "            x = -2/(dim-1) * (eps * sigma * special.gamma(dim/2))**(2/(dim-1))\n",
    "        else:\n",
    "            x = -2/(dim-1) * (eps * sigma)**(2/(dim-1)) * (4*np.pi/dim)**(1/(dim-1)) * (np.exp(-1)*(dim/2 + 1/(6*dim - 1/5*dim)))**(1/(1-1/dim));\n",
    "        cutoff = sigma * np.sqrt(-2*(dim-1)*self.Lambert_W(x,-1))\n",
    "        return cutoff\n",
    "\n",
    "    def _fit(self, theta0 = None):\n",
    "        \n",
    "        def approx_kernel_cost_grad(theta, self=self):\n",
    "            # Calculates the cost and  gradient of weighted sum of zero-centered\n",
    "            # Gaussians as it approximates kernelfunc. There is an additional\n",
    "            # restriction that the pdf be non-negative and the normalized, which we\n",
    "            # impose in Fourier space before transforming back and calculating the\n",
    "            # error in x-spacbessele.\n",
    "            #\n",
    "            # -theta(1:end/2) should be the estimate of std deviation (should be >0)\n",
    "            # -theta(end/2+1:end) should be corresponding weights (can be negative)\n",
    "            # -params is a structure containing the following fields:\n",
    "            #   -num_points_x: how many points to sample in x-space -num_points_w: how\n",
    "            #   many points to sample in w-space -eps: a cutoff parameter for\n",
    "            #   integrating the pdf -dim: the dimension of vectors\n",
    "            #\n",
    "            # Output: f is total cost, df is gradient wrt theta\n",
    "\n",
    "            # Should be equal to num_gaussians\n",
    "            N = len(theta)//2\n",
    "            sigma = abs(theta[:N])\n",
    "            c = theta[N:]\n",
    "            d, eps, grid_size = self.dim, self.eps, self.grid_size\n",
    "            # The pdf is a sum of exp(-x**2)*x**n terms, which have a peak before\n",
    "            # decaying. The following line finds the position on the tail of the decay\n",
    "            # which has value < eps. Everything farther out should be smaller, so we\n",
    "            # can neglect it. We then weight each term and find the maximum one, which\n",
    "            # is how far we have to integrate before knowing there is nothing\n",
    "            # interesting left.\n",
    "            w_max = max(self.find_cutoff(d,eps/abs(c),sigma))\n",
    "\n",
    "            # Grid in x-space\n",
    "            w = np.linspace(0,w_max,grid_size)\n",
    "\n",
    "            # Use different grid for pdf\n",
    "            z = np.linspace(0,2,grid_size)\n",
    "\n",
    "            # Initialize gradient vector\n",
    "            df = np.zeros((2*N));\n",
    "\n",
    "            # Argument of gaussians\n",
    "            X = (1/2*(1/sigma))[:,np.newaxis] * w[np.newaxis,:] \n",
    "\n",
    "            # For large dim, evaluate the series to avoid overall. \n",
    "            # Also bring the prefactor into the exponent for cancellations.\n",
    "            if d > 100 :\n",
    "                logGammaSer = d/2*(np.log(d/2)-1) - 1/2 * np.log(d/4/np.pi) + 1/6/d \\\n",
    "                -1/45/d**3 + 8/315/d**5 - 8/105/d**7 + 128/297/d**9;\n",
    "                A = np.exp(-X**2 + (d-1)*np.log(X) - logGammaSer) * ((1/sigma)[:,np.newaxis] * np.ones((1,grid_size)))\n",
    "            else:\n",
    "                A = 1/special.gamma(d/2) * np.exp(-X**2 + (d-1)*np.log(X)) * ((1/sigma)[:,np.newaxis] * np.ones((1,grid_size)))\n",
    "\n",
    "            # Weighted sum of gaussians in w-space, i.e. the unnormalized pdf\n",
    "            f1 = c.dot(A) \n",
    "\n",
    "            # The integral of positive part: w(2) is grid spacing, i.e. 'dw'\n",
    "            normalization = 1/(eps + w[1] * sum(f1[f1>0])) \n",
    "\n",
    "            pdf = normalization * f1\n",
    "            #pdf = f1.copy()\n",
    "            pdf[f1<0] = 0\n",
    "\n",
    "            # Because we made the pdf positive, it's no longer integrable analytically.\n",
    "            # But we need to integrate it to do the inverse Fourier transform, to get\n",
    "            # the predicted function in x-space (our approximate kernel). This is one\n",
    "            # numerical integral. After we get the approximate kernel, we want to\n",
    "            # calculate the L2-norm between it and the actual kernel. This is another\n",
    "            # numerical integral. So we have a 2D numerical integration to perform:\n",
    "\n",
    "            # The grid for the 2D integral.\n",
    "            wz = w[:,np.newaxis]*z[np.newaxis,:]\n",
    "\n",
    "            # The integrand of the inverse Fourier transform\n",
    "            Jwz = self.scaled_bessel(d/2-1,wz)*((f1>0)[:,np.newaxis].dot(np.ones((1,grid_size))))\n",
    "\n",
    "            # The difference between our approximate kernel and the exact one\n",
    "            f2 = w[1]*(pdf.dot(Jwz))- self.kernelfunc(z)\n",
    "\n",
    "            # Total cost, i.e. squared L2-norm between approx. kernel and the exact one\n",
    "            f = 1/2 * z[1] * (f2.dot(f2.T)) \n",
    "            # Now we calculate some derivatives. Vectorizing this code made it really\n",
    "            # ugly, and surely it could be cleaned up / made more efficient.\n",
    "\n",
    "\n",
    "            ee = (c/sigma)[:,np.newaxis]*np.ones((1,grid_size))\n",
    "            X2 = (-d + 2*X**2)\n",
    "            rr =  (-normalization**2)*w[1]*X2*A\n",
    "            df[:N] = w[1]*z[1]*(( (ee*rr).dot(f1>0)[:,np.newaxis]*c[np.newaxis,:].dot(A) + ee*X2*(A*normalization)).dot(Jwz)).dot(f2)\n",
    "            rr =  (-normalization**2)*w[1]*A\n",
    "            df[N:] = w[1]*z[1]*(( (rr).dot(f1>0)[:,np.newaxis] * c[np.newaxis,:].dot(A) + (A*normalization)).dot(Jwz)).dot(f2)\n",
    "            return f, df\n",
    "        \n",
    "        if theta0 is None:\n",
    "            theta0 = 1/np.sqrt(self.N)*(2*np.random.rand(2*self.N)-1)\n",
    "            theta0[:self.N] = abs(theta0[:self.N])\n",
    "        if self.verbose:\n",
    "            print(\"Launching L-BFGS ...\")\n",
    "        self.opt = fmin_l_bfgs_b(approx_kernel_cost_grad,x0=theta0, m=500, maxls = 500, pgtol=1e-10)\n",
    "        self.thetaopt = self.opt[0]\n",
    "        self.warnflag = self.opt[2]['warnflag']\n",
    "        if self.warnflag != 0:\n",
    "            warnings.warn(\"L-BFGS descent algorithm did not converge because: {}\".format(self.opt[2]['task']))\n",
    "        else:\n",
    "            if self.verbose:\n",
    "                print(\"Converged.\")\n",
    "                \n",
    "    def fit(self, theta0 = None, eval_grid_factor = 10):\n",
    "        \n",
    "        def approx_kernel_eval(theta, self=self):\n",
    "            # Calculates the cost and  gradient of weighted sum of zero-centered\n",
    "            # Gaussians as it approximates kernelfunc. There is an additional\n",
    "            # restriction that the pdf be non-negative and the normalized, which we\n",
    "            # impose in Fourier space before transforming back and calculating the\n",
    "            # error in x-space.\n",
    "            #\n",
    "            # -theta(1:end/2) should be the estimate of std deviation (should be >0)\n",
    "            # -theta(end/2+1:end) should be corresponding weights (can be negative)\n",
    "            # -params is a structure containing the following fields:\n",
    "            #   -num_points_x: how many points to sample in x-space -num_points_w: how\n",
    "            #   many points to sample in w-space -eps: a cutoff parameter for\n",
    "            #   integrating the pdf -dim: the dimension of vectors\n",
    "            #\n",
    "            # Output: f is total cost, df is gradient wrt theta\n",
    "\n",
    "            # Should be equal to num_gaussians\n",
    "            N = len(theta)//2\n",
    "            sigma = abs(theta[:N])\n",
    "            c = theta[N:]\n",
    "\n",
    "            d, eps, grid_size = self.dim, self.eps, self.grid_size\n",
    "            # The pdf is a sum of exp(-x**2)*x**n terms, which have a peak before\n",
    "            # decaying. The following line finds the position on the tail of the decay\n",
    "            # which has value < eps. Everything farther out should be smaller, so we\n",
    "            # can neglect it. We then weight each term and find the maximum one, which\n",
    "            # is how far we have to integrate before knowing there is nothing\n",
    "            # interesting left.\n",
    "            w_max = max(self.find_cutoff(d,eps/abs(c),sigma))\n",
    "\n",
    "            # Grid in x-space\n",
    "            w = np.linspace(0,w_max,grid_size)\n",
    "\n",
    "            # Use different grid for pdf\n",
    "            z = np.linspace(0,2,grid_size)\n",
    "\n",
    "            # Initialize gradient vector\n",
    "            df = np.zeros((2*N));\n",
    "\n",
    "            # Argument of gaussians\n",
    "            X = (1/2*(1/sigma))[:,np.newaxis] * w[np.newaxis,:] \n",
    "\n",
    "            # For large dim, evaluate the series to avoid overall. \n",
    "            # Also bring the prefactor into the exponent for cancellations.\n",
    "            if d > 100 :\n",
    "                logGammaSer = d/2*(np.log(d/2)-1) - 1/2 * np.log(d/4/np.pi) + 1/6/d \\\n",
    "                -1/45/d**3 + 8/315/d**5 - 8/105/d**7 + 128/297/d**9;\n",
    "                A = np.exp(-X**2 + (d-1)*np.log(X) - logGammaSer) * ((1/sigma)[:,np.newaxis] * np.ones((1,grid_size)))\n",
    "            else:\n",
    "                A = 1/special.gamma(d/2) * np.exp(-X**2 + (d-1)*np.log(X)) * ((1/sigma)[:,np.newaxis] * np.ones((1,grid_size)))\n",
    "\n",
    "            # Weighted sum of gaussians in w-space, i.e. the unnormalized pdf\n",
    "            f1 = c.dot(A) \n",
    "\n",
    "            # The integral of positive part: w(2) is grid spacing, i.e. 'dw'\n",
    "            normalization = 1/(eps + w[1] * sum(f1[f1>0])) \n",
    "\n",
    "            pdf = normalization * f1\n",
    "            pdf[f1<0] = 0\n",
    "\n",
    "            # The grid for the 2D integral.\n",
    "            wz = w[:,np.newaxis]*z[np.newaxis,:]\n",
    "\n",
    "            # The integrand of the inverse Fourier transform\n",
    "            Jwz = self.scaled_bessel(d/2-1,wz)*((f1>0)[:,np.newaxis].dot(np.ones((1,grid_size))))\n",
    "\n",
    "            fz = w[1]*(pdf.dot(Jwz))\n",
    "            # The difference between our approximate kernel and the exact one\n",
    "            f2 = fz - self.kernelfunc(z)\n",
    "\n",
    "            # Total cost, i.e. squared L2-norm between approx. kernel and the exact one\n",
    "            f = 1/2 * z[1] * (f2.dot(f2.T)) \n",
    "\n",
    "            return pdf*w[1], w_max, fz\n",
    "\n",
    "        self._fit(theta0)\n",
    "        if self.verbose:\n",
    "            print(\"Estimating PDF and Kernel approximation on the given grid ...\")\n",
    "        self.grid_size *= eval_grid_factor\n",
    "        self.__pdf_opt, self.w_max, self.__K_approx = approx_kernel_eval(self.thetaopt)\n",
    "        self.wpdf = np.linspace(0,self.w_max,self.grid_size)\n",
    "        self.grid_size /= eval_grid_factor\n",
    "        wxpdf = list(zip(self.wpdf, self.__pdf_opt))\n",
    "        self.__cdf = [(i, sum(p for j,p in wxpdf if j < i)) for i,_ in wxpdf]\n",
    "        self.fitted = True\n",
    "        print(\"Done.\")\n",
    "        \n",
    "    def get_pdf(self):\n",
    "        if not self.fitted:\n",
    "            raise NotFittedError(\"Fit method must be run before\")\n",
    "        \n",
    "        return self.__pdf_opt\n",
    "    \n",
    "        \n",
    "    def get_cdf(self):\n",
    "        if not self.fitted:\n",
    "            raise NotFittedError(\"Fit method must be run before\")\n",
    "        \n",
    "        return self.__cdf\n",
    "    \n",
    "    def get_kernel_approx(self):\n",
    "        if not self.fitted:\n",
    "            raise NotFittedError(\"Fit method must be run before\")\n",
    "        \n",
    "        return self.__K_approx\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulation: Kernel approximation on a grid "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Launching L-BFGS ...\n",
      "Converged.\n",
      "Estimating PDF and Kernel approximation on the given grid ...\n",
      "Done.\n"
     ]
    }
   ],
   "source": [
    "DIM, EPS, GRID_SIZE = 256, 1e-20, 500\n",
    "A,B, N = 2,20, 10\n",
    "\n",
    "def polynomial_kernel_scalar(z, a = A, degree = B):\n",
    "    return (1- (z/a)**2)**degree\n",
    "\n",
    "Kapp = KernelApprox(dim = DIM, kernelfunc = polynomial_kernel_scalar, eps = EPS, grid_size=GRID_SIZE, N = 10)\n",
    "Kapp.fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ff09UVBQAJUuWpHfv3vz000+cc845gDfPSatWrfjggw+S1u/ZsyfnnHMOjz/+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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x119c59550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z = np.linspace(0,2,10*GRID_SIZE)\n",
    "plt.figure(figsize=(20,4))\n",
    "plt.subplot(121)\n",
    "plt.plot(z,polynomial_kernel_scalar(z),linewidth=2, label = \"Exact Kernel\")\n",
    "plt.plot(z,Kapp.get_kernel_approx(),'r--',linewidth=3, label = \"Approximated Kernel\")\n",
    "plt.title(\"Kernel Approximation, dim = {}, a = {}, p = {}\".format(DIM,A,B))\n",
    "plt.legend()\n",
    "plt.subplot(122)\n",
    "wmax = Kapp.get_cdf()[-1][0]\n",
    "w = np.linspace(0,wmax,10*GRID_SIZE)\n",
    "plt.plot(w,Kapp.get_pdf())\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Launching L-BFGS ...\n",
      "Converged.\n",
      "Estimating PDF and Kernel approximation on the given grid ...\n",
      "Done.\n"
     ]
    }
   ],
   "source": [
    "DIM, EPS, GRID_SIZE = 1000, 1e-20, 500\n",
    "A,B, N = 2,2, 10\n",
    "\n",
    "def polynomial_kernel_scalar(z, a = A, degree = B):\n",
    "    return (1- (z/a)**2)**degree\n",
    "\n",
    "Kapp = KernelApprox(dim = DIM, kernelfunc = polynomial_kernel_scalar, eps = EPS, grid_size=GRID_SIZE, N = 10)\n",
    "Kapp.fit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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yXH+9//rAAb8YZf58vy1aBKGp6zdvhvfe8xtA6dJw+eU+t+Hll/ui8SIiIiIi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TAAAg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\nk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hscpqSsYozTgenlVo0PIWwNnAdYRFWSGonXXxWo24FvAXvmOxBJkirxM0q19R6pBtpm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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x119fd7240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z = np.linspace(0,2,10*GRID_SIZE)\n",
    "plt.figure(figsize=(20,4))\n",
    "plt.subplot(121)\n",
    "plt.plot(z,polynomial_kernel_scalar(z),linewidth=2, label = \"Exact Kernel\")\n",
    "plt.plot(z,Kapp.get_kernel_approx(),'r--',linewidth=3, label = \"Approximated Kernel\")\n",
    "plt.title(\"Kernel Approximation, dim = {}, a = {}, p = {}\".format(DIM,A,B))\n",
    "plt.legend()\n",
    "plt.subplot(122)\n",
    "wmax = Kapp.get_cdf()[-1][0]\n",
    "w = np.linspace(0,wmax,10*GRID_SIZE)\n",
    "plt.plot(w,Kapp.get_pdf())\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# In the Next Notebook: \n",
    "- Implementation of the second Python class SRFF that performs Random Fourier Features sampling and the low dimentional embedding Z(X). \n",
    "\n",
    " <a href=\"SRF-II.ipynb\"> Go to the next notebook </a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}