# Computational results for the paper

Kalinina E., Uteshev A. On the Real Stability Radius for Some Classes of Matrices. LNCS, 2021, V. 12865, pp. 192-208

Ex

Example 3.

$$A=\left(\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -91 & -55 & -13 \end{array} \right); \quad \mbox{spectrum} \ \{- 7 , - 3 \pm 2 \mathbf i\} \ ;$$

$$\mathcal F(z) := 33076090700402342058246544\,z^6-377039198861306289080145178864\, z^5$$ $$+937864902703881321034450183916\,z^4-771868276098720970149792503999\,z^3+211070978787821517684022650624\,z^2$$ $$-510584100140452518540394496\,z+319295875259784560640000$$ has the following real zeros $$z_1\approx 0.739335899598383440691,\ 0.765571063601656535,\ z_3 \approx 0.98046795419204527546,\ 11396.6585485070048$$ $$d(A,\mathbb D) = \sqrt{z_1} \approx 0.859846439545098913524729075$$ $$U_{\ast}^{\top} = \left[ 0.6966761574560766967,0.71737869934856438,-0.003183292822 \right]$$ $$E_{\ast}= \left( \begin{array}{ccc} 0.198499246325849218 & -0.195124609543571 &-0.530440179782975170\\ 0.204397882182276117 & -0.200922964145057 & -0.546202826352530108 \\ -0.00090699418851755 & 0.000891574603775858 & 0.00242371781920822643 \end{array}\right) , \ \mbox{spectrum} \ \{ 0_{3}\} .$$ $$B_{\ast}=A+E_{\ast}= \left( \begin{array}{ccc} 0.19849924632584921831 & 0.804875390456428643 & -0.5304401797829751\\ 0.20439788218227611716 & -0.200922964145057444 & 0.45379717364746989\\ -91.00090699418851755174& -54.999108425396224141 & -12.997576282180791 \end{array}\right) ,$$ spectrum $\{-14.649555268913533896, 0.824777634456766_{2}\}$.

Ex

Example 4.

$$A=\left(\begin{array}{rrrr} 5& -36 & -57 & 85\\ 80 & 90 & 74 & 27\\ 9 & -91 & 81 & 65\\ -12& 78& 5 & -63 \end{array} \right); \quad \mbox{spectrum} \ \{-20.70628367\pm \mathbf 68.729333451 \mathbf i,\ 77.20628367744 \pm 32.470588942507 \mathbf i \}$$

$$\mathcal F(z)= 4448361079836520604065069210589460378731456940350930681856\,z^{12}$$ $$-570855722457161621736317871818174636181122049098567826998296576\,z^{11}$$ $$+31155223197262531244386904471884989549707583986373778960414669651968\,z^{10}$$ $$-950786358973606254764281608332832011565408877461871860116989940979212288\,z^9$$ $$+18087921440463510513576653011286530750620496420094902505837989836882125964544\,z^8$$ $$-230256486522695403968697926862945469730669298755789403750858099163745402987702272\,z^7$$ $$+2083390382968318560764783321340384271495131820650601924883486385938799810883712914688\,z^6$$ $$-13596188554913486145815888086257149673380937546840178424652637590674221144010030343208192\,z^5$$ $$+56424326734246271949081805060826853112072416348740569659837736069494467749593171233324858880\,z^4$$ $$-97354040432848455219213815520370071545769708577857812801963839553456139318874428669431511499264\,z^3$$ $$+40534706193400773638815258048976684390487592692154262157178785171895729107261935784297827401648384\,z^2$$ $$-6119695866908795314093428763939971317092873463919296220120304656709335285008028519899842734898066688\,z$$ $$+\underbrace{287238134962174593952714912654943336715870461861629431306789995226269028760429110276332899990829689600}_{102 \ digits}$$

$$z_1\approx 87.61471413892475714849222295865385305461, z_2\approx 2588.50966144335716993,$$ $$z_3\approx 17853.2563345611693,\ z_4 \approx 32194.078324415763652346$$ $$d(A,\mathbb D)=\sqrt{z_1} \approx 9.360273187195166321883022$$ $$E_{\ast}= \left( \begin{array}{cccc} 3.35032444017926146 & -0.177130061428 & -3.70404184955 & -0.3282161273\\ 2.48971280432635996 & -0.1316299331134& -2.752569366117 & -0.243905899072\\ 2.56586347332755790 & -0.1356559828047& -2.83675979898102& 0.25136603558093 \\ 3.898666398607990825& -0.2061206402556& -4.31027614063447& 0.38193470808475 \end{array}\right) , \ \mbox{spectrum} \ \{ 0_{4}\} .$$ $$U_{\ast}= \left( \begin{array}{r} -0.5350659090801356437374930907327623178367\\ -0.3976213255705048920988243067187755262401\\ -0.4097830214491304142601557808991998754956\\ -0.6226392452486625036761842291063429133157 \end{array} \right)$$ $$B_{\ast}=A+E_{\ast} , \ \mbox{spectrum} \ \{-12.58107674 \pm 67.3307079694510778354 \ \mathbf i, 69.0810767464521527_{2} \}$$

Ex

$n= 5$

$$\left[ \begin {array}{ccccc} 55&-28&16&30&-27\\ -15 &-59&-96&72&-87\\ 47&-90&43&92&-91 \\ -88&-48&53&-28&5\\ 13&-10&-82& 71&16\end {array} \right]$$

$$\mathcal F(z)= 235881245121207804587411960701306719265323208859893332240317579038186926384437756105994990174601216 \,{z}^{20}$$ $$- 56667961698421275486477928948110812282902570919602124623489768763827567365679693726348244280967523467264 \,{z}^{19}$$ $$+ 6048947968186588490594584689489725737578714891594212995091650916245701949031766886064062805972149015561961472 \,{z}^{18}$$ $$- 377846885183975583018386700581791195059292064678828743900545237450167549586752063338765565334932349331275516657664 \,{z}^{17}$$ $$+ 15286850506168579971756938126493069729059123282255899177827971904944168221090159136653844248599717397210960888275828736 \,{z}^{16}$$ $$- 417778481982662474312279606319090124688409068830305537046465890506362388398352557327824923750457942847495002170554035355648 \,{z}^{15}$$ $$+ 7771030494185043574901939047538577140107755224327689387162776470154970054916184600792172705003804851456970532720266437742346240 \,{z}^{14}$$ $$- 95723060584238257571962517686503179626848322434408646737956692360054430941710192087699740852430069266176468226297536637618852397056 \,{z}^{13}$$ $$+ 708848924372468800898519873127110051532600831509081114626986332082176811740478349544280452148059286753152955061144561456219665960767488 \,{z}^{12}$$ $$- 2043378171930182925337311458342409372301433234257010109971918209609357207449850626953912476949076005233203533860972454556115183305808965632 \,{z}^{11}$$ $$- 11402275142671714779716932126758879897399008088430445468521879294945884897382502437400727889337926446300220521732139927718701340312799010601984 \,{z}^{10}$$ $$+ 114547622826244056445377277328726763462125950690154774270498630756181669849829279852500024396180260851506438974097123467014226799542400478233227264 \,{z}^{9}$$ $$- 226882524882714075781189714842951904988378043445577736693405391142917079631254218198180833548015166891773226586370991876068828944401170124704735596544 \,{z}^{8}$$ $$- 956373512743708614746647943511020089038420977167743886172764935661538161553324995780032975313900218227981607494970325050593643148735626009258287064068096 \,{z}^{7}$$ $$+ 5542233682542999893883237036168488607515405396377041844693889359289144680499474934211002754312796375943036857444213620168466659176371071097731321741874657280 \,{z}^{6}$$ $$- 10586092799337785416313924920387789726772096904765847431159207057756630233680245153909031381200052809060158821837039236252089893656689067119120680446885844140032 \,{z}^{5}$$ $$+ 8729241954998042991392459081202729565539061146277450486234197126751905575208763534634073245423141684407654983636190798779185452713830508413891812078474848593379328 \,{z}^{4}$$ $$- 2695714340736444872519311053138517046748267655687061595967320585065833019918638554861959830960949865596302727998158395052764145017551335332239821467105180789947990016 \,{z}^{3}$$ $$+ 272854975071077558940512062846322763760529508078582655718446561883679727480774599707490877995942898609482542800158558547167580078337395888898334340567451536649328918528 \,{z}^{2}$$ $$- 819590447025455800354957083985360821048794051645631077492195944601546412621504902339989973876899587137911346190378352522309206835649058548086613368689046604206721466368 \,z$$ $$+ 267266700866144790892111787893870708546596521152099598436632746926826444612743274611243383773303823862413809727652171550190718398587419663467731524597520490917281660928$$

$$z_1 \approx .3719987247368903391870203284002055168574, z_2 \approx 2.715202607618295499484522748713631060636,$$ $$z_3 \approx 210.7048414470474825803269101056168000188, z_4 \approx 283.1738185635839372976835460776575063732,$$ $$z_5 \approx 1429.235894159243498766643804086413948534, z_6 \approx 3200.570392651843624648712803258742164925,$$ $$z_7 \approx 17747.79981517305330845550490649869114386, z_8 \approx 27666.32544879062636968173566583087856247,$$ $$z_9 \approx 28484.42036315244958295941176402335325842, z_{10} \approx 43064.57767671530617486424930438145025088$$

Ex

$n=10$

$$\left[ \begin {array}{cccccccccc} -28&5&13&-10&-82&71&16&83&9&-60 \\ -83&98&-48&-19&62&37&5&96&-17&25 \\ 91&0&98&-64&64&-90&-60&-34&-13&44 \\ -2&71&-47&-39&-53&-72&-97&33&10&7 \\ -89&65&12&-25&-96&50&-60&-42&7&-89 \\ -70&34&-68&-60&16&52&-20&-4&-89&-77 \\ 69&80&28&-42&-33&21&-35&97&30&-64 \\ 89&-16&59&-69&-46&-33&87&-34&40&77 \\ 1&-10&-65&-85&54&0&18&52&36&91 \\ -22&51&-27&50&60&-91&-47&-97&-2&-31\end {array} \right]$$

The polynomial $\mathcal F(z)$ has $\deg =90$, the size of the free term $784$ digits; $28$ real zeros:

$$z_1 \approx 48.48680193535528191493532550815062302298,\ z_2 \approx 77.22721779907582816190427322776061976381,$$ $$z_3\approx 484.7788085635894659564558854189060401953,\dots, z_{28} \approx 1.062378989254227785954990790462024339410\times 10^5$$ $$U_{\ast}\approx \left[ \begin {array}{c} 0.2904187589025009032717235936885024826321 \\ - 0.05355704223074589348324034990899446299459 \\ - 0.2446704866234813747523715185905934045293 \\ - 0.1726643102836018245161326609445307658248 \\ 0.6256007137064552887313822469845455414991 \\ - 0.1892552928427663856774562533242360446312 \\ - 0.2180603992369239957126127513965015518048 \\ 0.4771281919456922203617116517353098553394 \\ - 0.1966526677106362776460268780377429556048 \\ 0.2864336516425271961314272442616988284738 \end {array} \right]$$ $B_{\ast}$ has the double eiganvalue $\lambda_{\ast} \approx -163.64724202$.

Ex.

$n=20$.

$$\left[ \begin{array}{cccccccccccccccccccc} 16&-85&-44&-31&45&49&-58& 49&1&-95&86&-97&-14&83&-96&-8&-54&62&96&-51\\ 89&14& -79&-58&-95&61&-2&86&57&-35&57&28&63&21&-71&-66&-34&72&-40&-68 \\ -15&-32&17&87&-60&7&-61&45&-50&-22&48&-82&46&-49& -66&18&16&-22&38&-48\\ 0&-87&-13&-87&-63&-31&90&66& 88&74&-39&-20&31&-52&37&-59&-54&-22&17&63\\ 21&-12&- 21&-74&55&-17&83&-65&-28&-5&27&0&26&13&83&-19&-62&-12&48&-84 \\ -83&96&6&76&95&-14&61&83&-83&53&-6&52&-40&45&-86& -88&-21&28&92&35\\ -31&-17&17&31&-52&44&-16&-37&-59& -47&58&-56&-45&-15&-67&-62&70&93&74&95\\ 65&-10&11&- 56&-1&49&-27&88&-63&61&46&-36&-76&-22&-55&-34&52&1&87&11 \\ -11&88&94&6&21&18&75&10&18&60&-90&11&-91&-53&37& 76&-44&67&-72&2\\ 4&-84&-60&-83&0&84&77&-46&63&27&- 43&72&-64&8&43&58&-84&78&-12&-2\\ -45&-85&58&-62&-77 &-45&-34&94&67&95&-38&45&-99&-80&42&86&7&-11&2&-4\\ 5&-40&5&4&-89&-54&-81&22&-38&-50&-75&11&80&14&-56&8&80&-91&-48&53 \\ -68&27&-5&33&-87&24&-18&-8&37&-96&-82&50&88&-73&- 88&-52&3&-86&-80&-96\\ -72&-16&-11&94&5&65&-45&-59& 83&-3&-21&-80&98&77&-60&-28&-47&34&-4&31\\ -81&98&- 15&42&-37&67&39&-27&11&72&-51&-33&81&39&76&57&78&49&-10&61 \\ -57&26&-8&-4&-76&92&-2&64&-21&-81&86&71&-80&2&7& 24&-91&-16&-98&44\\ 41&-65&-20&-55&-64&-62&-86&67&11 &78&38&-63&89&30&-42&-97&-78&17&-43&89\\ -3&80&-37&- 89&58&32&-61&83&-9&-52&51&-75&23&-27&-38&-14&-77&24&4&-39 \\ 44&-49&75&14&61&-80&-17&-90&13&68&-84&-81&-60&68& 11&-29&-25&-95&-22&57\\ 18&51&49&34&-64&70&97&21&-27 &-9&1&11&72&-56&53&-63&-63&94&89&-56\end {array} \right]$$

The polynomial $\mathcal F(z)$ has $\deg =380$, the size of the leading coefficient $1920$ digits, the size of the free term $3565$ digits; $36$ real zeros: $$z_1 \approx 220.4058888376608357274124059953815687582,$$ $$z_2 \approx 549.5846573718424009963480745659209497822,$$ $$z_3 \approx 618.5153640869838724279597656518642986433,$$ $$\dots, z_{36}\approx 2.287148964519324976173272434243232377772 \times 10^5$$ $$U_{\ast}\approx \left[ \begin {array}{c} 0.4432676500764833043168173830624969645507 \\ - 0.2302679872742856509008234665269898899889 \\- 0.1880192506368522665970386123270432124132 \\ - 0.1127684597398090445506062176696772699348 \\ - 0.2555841828430564796708408431913744423962 \\ 0.01660895984443809618105841342701105870481 \\ - 0.1093113278361756595437525160483786997786 \\ - 0.06701827307991977638896158497748322437307 \\ - 0.0004475120304850622114343020012568203388628 \\ - 0.01501851121120248344987778803242404258524 \\ - 0.05456782281964901851171859958182039576241 \\ - 0.3312583323172087867528158486405174119479 \\ 0.3489576885512267209983213347347701941314 \\ - 0.4052678342089582821267918400843898666019 \\ 0.2439465154844411160991821723506891596871 \\ 0.01803342835328527587660014655607403188404 \\ 0.3165811793725314022632711738722090435489 \\ - 0.2336011604423947347801579624641497830239 \\ 0.04054967288139043599123242426031884680540 \\ 0.07184311308952182460994748683970921996812 \end {array} \right]$$ $B_{\ast}$ has the double eiganvalue $\lambda_{\ast} \approx -32.89067941838$.

matricese/optimize/distancee/casc2022ex.txt · Последние изменения: 2022/08/16 19:12 — au