Uteshev A.Yu., Kalinina E., Goncharova M. Computing the Nearest Singular Polynomial. JSC, 2025, pp.
Example 1.
$$ f(w) = w^6+\frac{199}{100} \, w^5-\frac{45101}{5000}\, w^4-\frac{1194}{625}\, w^3+\frac{40109}{5000}\,w^2-\frac{199}{2500}\, w-\frac{1}{625} $$ $$ \equiv \left( w-1 \right) \left( w+\frac{3}{100} \right) \left( w-\frac{1}{100} \right) \left( w+\frac{1}{50} \right) \left( w+2 \right) \left( w+1 \right) $$
Distance polynomial for the case where the nearest singular polynomial possesses 1 double zero $$ \mathcal F(z)= $$ $$ 5431473255157470703125\times 10^{128} \,z^{15} -7790687979128015041351318359375\times 10^{120} \,z^{14} $$ $$ +5373955582758269714842349357306957244873046875\times 10^{107}\,z^{13} $$ $$ -681605086030890135552520125382397096058862209320068359375\times 10^{97}\,z^{12} $$ $$ +79937490590634871365708168269300054932092125853036622846126556396484375\times 10^{84}\,z^{11} $$ $$ -1387255945936866517035905184282820854748150539215840011902816230436229705810546875\times 10^{75} \,z^{10} $$ $$ -641731661948516540544177876561401368191819171121027110981232623358901552340190410614013671875\times 10^{63} \,z^9 $$ $$ +{\scriptstyle 181702586720153037058578430017358981436661917520841565320273217595302896435703317654826073978424072265625}\times 10^{53} \,z^8 $$ $$ +{\scriptstyle 521374098266699169245380513770913097462105106023452545306874841505556214474456318830036366907057963357084827423095703125} \times 10^{40} \,z^7 $$ $$ -{\scriptstyle 119745542885185060628493781073889910697304242483202076843289635895397426961817863539631361543623098435328119906561244964599609375}\times 10^{32}\,z^6 $$ $$ +{\scriptstyle 29949524115691692155389396637041217903815688748079593635475845458825778146761228232212264831281328478971374937572190798321148034619140625}\times 10^{24}\,z^5 $$ $$ -{\scriptstyle 55625673973762627715725810079539021834282381814097964430351328836864625268150339577452834752915833202687887517969713905013108528126437567892578125}\times 10^{14}\,z^4 $$ $$ +{\scriptstyle 2679022014250267786121262651088790233428194531546267375849424976195181165430312580239296464178351827951321492907081720904111925578011538916068277648125}\times 10^8 \,z^3 $$ $$ -{\scriptstyle 3633466419739783142122029937709662004657271550952648427524011788234068299279269955451503355831757547462897500469578224407443389662251195601182057322705078125}\,z^2 $$ $$ +{\scriptstyle 544789323664503644894250447426366801016742549586345079383657110540650456158298377378870741225732896487962930766358005939136498663535683193187211875}\,z $$ $$ -{\scriptstyle 1606785855467258821561647792314101112780992637286544570643184465508649504317013655230934853926200921362848619282823225358817952896385277} $$ possesses 5 real zeros $$ z_{\ast} \approx 3.00978880063127356116074882487084082158793552993692099669664\times 10^{-12}, $$ $$ 1.46926736443971770450555021474619009009033673852518601533147\times 10^{-10}, $$ $$ 0.0239570729283112131147548878588768795017411565620311866823148, $$ $$ 0.0417347691049052146838851750629150166723483487015679944643384, $$ $$ 0.130974281928809991824384479442729423129798998472119895818168 \, . $$ and $$ d(f,\mathbb D) = \sqrt{z_{\ast}}\approx 1.73487428957583942897120131823942247999648391628331277196331\times 10^{-6} $$ $$ f_{\ast}(w)=w^6+2.03999999999998186293661685969433620997273932880792360797987\,w^5 $$ $$ -0.919899999999284385267887761704963576135629735494476322423454\, w^4 $$ $$ -2.03980600002823524591594624929791153324474691529436026725999\, w^3 $$ $$ -0.0801119988859520686769197059065096150306550905106831040694123\, w^2 $$ $$ -0.000194043955798967711646493569850711420174809943386751408269955\, w $$ $$ +0.0000137343169971109228710783575767894732064716288491347511663592 $$ possesses zeros $$ -1.999999922326739090145137955104575660658,\ -1.00000092667458865069513361061775706082 $$ $$ 0.0106905834346081386756179577441377418224,\ 0.999999728958879465539136286439022230478 $$ and the double zero $$ -0.0253447316960708631555497690775817303946506_{(2)} $$ Distance polynomial for the case where the nearest singular polynomial possesses 2 double zeros $$ \widetilde{\mathcal F}(z)=2^{536}3^{15}5^{652}7^5 z^{54}+\dots+(680 \ \mbox{digits number in the free term}) $$ Real zeros $$ \widetilde{z_1} \approx 0.0487829278381575785242090628983582792780806809123839830669622, $$ $$ \widetilde{z_2} \approx 0.175620718619294827951071560356500798857508190445361582288331, $$ $$ \widetilde{z_3} \approx 0.213155064732428803028236810241046725951605429454424118446590, $$ $$ \widetilde{z_4} \approx 0.229804403653162570480595933733241790350888818229560614142877 $$ $$ \widetilde{f_1}(w)= w^6+2.06264250983837890745133558147801108090946400068774659120782\,w^5 $$ $$ -0.956286262834351658562711786803345370159954060795227411110019\, w^4 $$ $$ -1.98019816407837115644156264337231515327230820599058922164251\,w^3 $$ $$ -0.170046789927134838940117846394071511417376051066170071945698\, w^2 $$ $$ +0.186910487311894935067063084529852097306453181407307971437328\, w $$ $$ -0.0172168928425833374656635905782239598831458608800368466919052 $$ possesses only real zeros $$ -2.0968670270064029614768407169021569619,\ -0.619971835651679084681803982143049602762432 $$ and 2 double zeros $$ -0.6199718356516790846818039821430496027624320676814_{(2)}, \ 0.1477398854747188367035895831732061571930824507385_{(2)} $$ Polynomial $$ \widetilde{f_2}(w)= w^6+1.93689734159594230703608100417705432469727138250467863483748\,w^5 $$ $$ -0.739954309516456115807965307760263029415658786546392023182508\, w^4 $$ $$ -2.01092529860460727899822058349797064410126573097724125778101\, w^3 $$ $$ +0.104211293976337819955871495351413762503462492972282880538327\, w^2 $$ $$ +0.152378334471946154590841199302515457001327119595846371626549\, w $$ $$ +.273000314885664916655423416969111102043011091964303563854103 $$ possesses 2 imaginary zeros $$ -0.219976045005646820602225872824139899\pm \mathbf i 0.424158571230526685764852966227672645825 $$ and 2 double real zeros $$ -1.484901735705600617601582264540850863928_{(2)}, \ 0.7364291099132762846857676352723145_{(2)} $$ Polynomial $$ \widetilde{f_3}(w)= w^6+2.10401857768231915401175644283389718547059753805287496252303\,w^5 $$ $$ -0.830146807772319038616126797138704055517146480528701937459411\, w^4 $$ $$ -1.89711811702910230506447475322817682666153649084879035750802\, w^3 $$ $$ +0.0862906238468264551114314675774310104986357376633249561495320\, w^2 $$ $$ +0.384991442665046901111400379220487172890413395596813871646096\, w $$ $$ +0.0677139978878723670151009176335345097477689672473196056635407 $$ possesses 6 real zeros $$ -2.0722434513553163396549270517866325057896, -0.8258641931283037497556962409623710049045882329 $$ $$ -0.28966285241361539544103175211777822397997_{(2)},\ 0.686707385814265863140465177075331386591_{(2)} $$ Polynomial $$ \widetilde{f_4}(w)= =w^6+2.11687428414291374390030783290146782683705772640778826989450\, w^5 $$ $$ -.865221785302741852257648968085289895710340397418319585550707\, w^4 $$ $$ -1.86293418895596868419423884620971275497501856630552596218765\, w^3 $$ $$ +0.0027132543681125715657996883428038014732681574171129298469801\,w^2 $$ $$ +0.420138105989787220419707020987045198314865841214600191000535\, w $$ $$ +0.0780011101324402060640345662439134037191941552845250923624499 $$ possesses 6 real zeros $$ -2.130971277988524551468426927905083895146893195,\ -0.2328785252369129787769217496564148875 $$ $$ 0.6944128460527561686802245647102587070343023492_{(2)}, -0.5709250865114942755077041423818645590917_{(2)} $$
Example 3.
$$ f(w)= w^7-28\,w^6+298\,w^5-1452\,w^4+2980\, w^3-1616\, w^2-596\, w+10000 \equiv $$ $$\equiv (w-(7+\mathbf i))^2(w-(7-\mathbf i))^2(w^3+2\,w+4)+4\, w $$
Distance polynomial for the case where the nearest singular polynomial possesses 1 double zero $$ \mathcal F(z)= 40353607\, z^{18}-51195434458966140\, z^{17}+29749786493812900050170782\, z^{16} + \dots + $$ $$ +22613158545463473513982340432071157192874555631981195371753451140301111368417966143056570744832 $$ possesses $ 4 $ real zeros $$ z_{\ast} \approx 0.0000101228761991869317602662610701427562977224053328501359798 $$ $$ d(f,\mathbb D)=\sqrt{z_{\ast}}\approx $$ The nearest singular polynomial with 1 double zero $$ f_{\ast}(w)\approx w^7-28.00315013554385616053181563793652998254429832102880588327470317225804\, w^6 $$ $$ +297.9995577447979857701682384161086671318766499582530039373563560153135\,w^5 $$ $$ -1452.000062089285043659101667252029261218606700235321826844628831170667\,w^4 $$ $$ +2979.999991283134037372588602601389182867170622293520006174214449861483\,w^3 $$ $$ -1616.000001223782044792158306228922442202988863986230264349175742072364\,w^2 $$ $$ -596.0000001718097421225301581128683696083771839504584485315788012210831\,w $$ $$ +9999.999999975879211814041920191876318014754442441990042493326776438336 $$ Its zeros are as follows $$ -1.179343016136166369700388691381122569093942874319477974019789827562787 $$ $$ 0.5893428677735577228152954576935873190167561 \pm \mathbf i 1.74456872983616906845630727453542 $$ $$ 6.87901283078785684424355510737354135086080969 \pm \mathbf i 1.40248739786980377586519031723781613692160085896 $$ and double zero is $$ 7.1228908772785966980572515995916976059415547_{(2)} $$
Distance polynomial for the case where the nearest singular polynomial possesses 2 double zeros $$ \widetilde{ \mathcal F}(z) = $$
Polynomial possesses $8$ real zeros $$ \widetilde{z}_{\ast} \approx 0.0000508355930186617591058334087207439146162230707843233487 $$ $$ d(f(w), \widetilde{\mathbb D}) = \sqrt{\widetilde{z}_{\ast}} \approx $$ The nearest singular polynomial with a pair of double imaginary zeros $$ \widetilde{f}_{\ast}(w) \approx w^7-27.9991809068962809658749771225037265798120952364789880947508984\, w^6 $$ $$ +297.993195184519237320428572590608538575896059225119194662018043953\,w^5 $$ $$ -1452.00192146378368081307460625095890992002431660236346326063628476\, w^4 $$ $$ +2979.99959814288735976622254947397804222932097334356846432267587286\,w^3 $$ $$ -1616.00007408024319672719897568235257420873400112544249089698334448\, w^2 $$ $$ -596.000012703522494417779786954820285205857060846838266229419654276\, w $$ $$ +9999.999997924914007464160654775627061209335721556968577597697276766 $$ possesses zeros $$ 0.589334522241115815389235596565354747177118610293422805182471738\pm $$ $$ \pm \mathbf i \, 1.744561781915016356971431192792789745227078513037173803438780982, $$ $$ -1.179343794543465694842691841510787252067532475073241335290960295 $$ and double imaginary zeros $$ \big\{6.999963914239378757484799442720956353608053758596755893770717808 \pm $$ $$ \pm \mathbf i \, 1.003584654706403105138427255093379017425915480284113734934217292\big\}_{(2)}, $$
Conclusion. The nearest singular polynomial possesses $ 1 $ double zero.
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