CONTRIBUTION TO THE THEORY OF NUMERICAL
INTEGRATION OF NON−LINEAR DIFFERENTIAL
EQUATIONS（IV） BYTATsuJIRo SHIMIZU
16・Theorems on numerical integration We consider the differential equation （52）dy
＝f（x，y） dの of which the solution locally exists and is unique i110≦x＜oo，閣く◎◎46），（for example，∫（￠， y）is colltinuous with respect to x， y， and satisfies locally Lipschitz condition with respect to y）． We can not㎞ow theoretica11y how far the solution y（x）with y（0）＝Yo exists， unless we prolong the solution theoretically or numerically． For examples：To obtain the numerical solution of y’＝y2 with Yo＝1，0r …’一…’一芸w・th・y・一・，・・一・，・f we ca1・u1…by・…t・p…verg… method with step sizeん（su伍ciently small but丘xed）， we shall obtam the numerical values Yn，π＝1，2，…beyond x＝1， which is a stngular po口1t of ・h…u…1・・・…fthe eq・・…n・’一・・，・・輌’一・，・’一芸・ As to the numerical calculation there are two cases：the case in conside・ ration of round o仕errors and the case without consideration of them47）． In the following we．consider some numerical solutions of di丘erential equation（52）by the numerical methods， called“discrete variable methods” or“finite di避erence methods．” Let．Yn be the values obtained by the numerical method at￠＝xn， where xo＝0， xl＝ん1，… xn二九1十… 十んn，… and y（xn）be the values of the true solution of the equation（52）． We shall discuss some theoretically important problems on the numerical solution of di任erential equatiOns． 46） 47） We consider the lst order equation， but the following theorems are perfectly similar for the system of the lst order equations． The Iatter being theoretical calculation with an in丘nite number of decimal Places． 血herent instability（Fox：Numerical solutions of ordinary and partial differential equations）is a result of round off errors． ［47］48． CONT’
qIBUTION TO T且E THEORY OF NVMERICAL INTEGRATION
With・Ut・・n・id・・ing ；・und・丘errρrs it i・kn？wn th・t if th…1岨・叩⑰）吊 of（52）．6Xists in O≦x≦Xo，（Xo being a．丘nite nUmber）， then for． an．arbitrary ε＞0．vb have［yπ一ダ（Xh．）1＜εfor O≦xn≦Xe by the numeriCal mCmQdll（Conver・ 9。nt in D。hlqui、t・s sen・e），．・h…ing．th・・t・p．・i・も九・u組・i・ntly・m・11．48’ When we consider round． 庶d「’?窒窒盾狽刀@it is not so simple as above， since round off errors are not so smal． 、 In the．following we con§ider．「lthe calcUlaΦn of且oating decimal point system（sihce it is usually used）． ． There． were very few papers in which round off errors were discussed rigOrously， before Wilkinson，s papers…Ψpeared．．．．ジ．、．、；．．∵．．．．：．・、 Ro皿d o丘errors depend on the oエder．、：9￡．三姐∼斑a垣Q恥ey煕・。￡gr、：．Ω9 Same formulae， alld depend on the magnitudes of the terms calculated．49） へ tt ． メ ご tt ロ In g・neral・・und・丘er・・rs睡滅∼・σξ単・ll・・th°se discussed ln． tm姐yi瓢鵠㌫躍1；culi͡『elS∵∫「，三∵∴．−．．
T・ti”0・kM・A・・h C・n・id・頑b・・f『ri・u輌彗‘err・rs if th・β91吻興（52）：e垣・t・ in g≦廷X・（X・．b・ing．・．・，・麺t・n叫・ti）・l th・h f・・an・・bit・町ε．≧°．We have the htmeticql s91utioh Yn・n．＝：o・1・．2・．∵ 琴uch．that IYn÷y（Xn）1ζε、foで．q≦xn≦x・ by th・麺eti・al牟gth・d唾1er’・m・th・d g・．T・yl・t’・α麺ig垣・thOd）・ Ch66sihg the』’step’ size hh，．n＝1，2，．…P・・Perly．s血all’apd’tbe加励e・・f dechh：aゴdigi七s being increased properly．・Vhe ’nUmber．of decinial digits Used is r deteぬined by「．ε，㎡Xl，．the㎜merical method；．the《Mfferential ’equation and the、initial value．50） ． ．・ ・ ．． ． ・． ， ：．． ちごちゼコ：． tN On the other、hand we．havE given’the’ follow．血g．・． ㍉ 『L’．一． ’ ・“囀c鋼M‘
a．「Without c㎝siderihg round o茸． errors we、can choose the step size為，’n−＝1，2，・・−and−the numerical solution Yn，％＝0，1，2，…Such that IYn−y（砺）1＜ε，．fOr an arbitraryε＞0， for． all碗in（0， Xn）which．．tendS’t6．the ”6xistence inte押al of・the true． solution by one step加merical method．；．5P 48）∵Dahlquist． has gj恒e真anecessary and su伍cient．condition in order 曲at a 伽lti−step linear me廿10d may be conVerge叫輿D、ahlquist’芦．琴ense）㌧We have 元『giVen『’a neCessary and su伍cient condition「fOr a’general multi−5tep』method． （T．Sh↓mizu：ContributiOll to the theory of mUme虹cal integratio血of．ordinary non−ljnear．differen廿al equations（II）， T．RU Math．e】〔阜atics VoL 3タ、1968． @’S9ア蕊’鵠謬瓢。㌫器鵠瀦譜㌍禦1；≡9C≡y〔gh
50）loe． cit． T． Shimizu：Contribution to… （II），．TRU． Mathematics． Vol．3．（1⑳8）． ・ The． meΦ・d be無g・．EUIeゴs 6ne． in this paper we・di・cussed rePlaCin9．∫（x・y） by a polynomial p（x，・暫）Unifomlly convergent to，・∫（＄t．び）Wh印We ConSidered round：ρ岱e∫rorS． ；C．the method工s Taylor’s，eXPans工On． qne，．：instead gf』Euler，s， we must replace f＠，紗）by a polynomiai p（x， y）not only unifoyrp，ty：conver− ．gent to∫（x， y）， but a1Sg plo，． pv，・．．ρエが．．q， p習劉＿び Unifqr担1y，cg頁yergent to ・fx，∫“，…∫虚虚＿￠，． fry，．1，Y respective》y・、．．．．・、、∴： 、 51）T．Shimizu：Cont曲ution to the theory of numerical integration of ordinary pon．撫ear differential equatio耶（写）， TRU mathematics Vo1．2．（1967）．、 」T．SHIMIZU 49 REMARK． By the existence interval we mean：th6 interval on which：the solution exists and can not be prolonged beyond the end point of it， the end point bei血9’a l singulalr point，（the solutioi of the equation（52）beco血es十◎◎ at the singular−point by the assUmptions on the equation（52））． In theoremB we do not assume that the solution exists in O1≦x≦Xo． If the solutioll eXiSts in O≦x＜oo， then we have Iyh−y（xn）1＜εfOr al1￠π“in （0，Xn）tend血g・to O≦x＜◎◎， and if there eXists a s臼1gular point X such that y（x）becomes◎o（for the system of the differential equations some of the compohents of the l @solution Vdctor y（x）beco㎡e◎o at X）， then we have IYn−y（x。）1＜εfor’all’・X。 in（0， X。）ten《Hng t・0≦x＜X． W吐・ve・h・wn・血・th・d t・r s・1ving the equ・ti・n numeri・aily by・n・ step method−Without knowing，the singular point．（If the existence hiterval is㎞Owh缶st，’we・ will have other wayS easier than七he proof of theorem B． In addition to it if the true solution is㎞own， we proceed in the fdllowidg way・L・t O≦x〈X（X b・ing面t… infulite）b・the exi・t・nce interv・いh・n
we choose X1＜X2＜…＜Xm＜…＜X and ε1＜ll2＜…＜εm…＜ε such that
Xm→4 and・m→・respectively・The s・1uti・n exists ’in ’X・．i≦x≦Xi， i＝1，2， ・… hence We．can choose the step size hε， i＝1，2，… and Yn， n・』0，1，2，・∴ such that IYn−y（xn）1＜εieεi−1 in Xi−1≦x≦Xi， the numerical value at X山eing y（xi）． Thus Yn， n＝」0，1，2，… satisfies l Yn−y（xh）1＜εfor all xn血0≦念元ぐX）． In the followmg we shall give the pr血cipal theorem． ‘ 1一 The principal theorem In consideration of round off errors We can choose the step size九ゴ，ゴ＝1，2，…and the・numerical solutiOn yヵ， n＝o，1，2，…Sudh that【Yn−y（xn）1〈εfor all x in（0， xゴ）such that（0， xj）tends to the existence interval of l@the true solutiOn， for an『arbitraryε＞0， by one stel｝nu血erical method． In this case hn，’n＝1，2：…may become indefinitely small， alld the number of decimal digits may be increased mde丘nitely so that the round off errors may become諏aller than a required number． Proof of the p亘ncipa1’theorem i ： ： Let the initial value be y＝O qt：x＝0．．． Now we draw two nets， the丘rst net with parallel血1es”x＝’10，20，30，’・’・・and y’三十10，十20，十30，…， the ・ec・nd n・t・with・・＝10・20・30…ρ四斗9，＋19，i＋29，…． Suppose that y’（0）≧0， lthen we eonsider the dOmain 1）1：0≦x≦10，0≦y≦10， and M・r購r⑭（’f y’（°）≦O・ i’thett− D・…°≦x≦i°・−1°≦y≦・・姐・M噺託｝ ∫（x，y））・D・aw the st・aight lin・y一晦and 1・t P、：’ i訟、，10）b。 th。 p。int。f 紅1tersection of y＝」胚lx ahd yニ10．’Let QI be the point（Xl，0）． Then the true solution囁Of（52）exists m OPIQ1． By theorem（A）：we choose the step size九1 and the ntimerical sqlution y・・y・・…・y・・…such that ly・−y（x・）1＜・for an arbitraryε＞0血・（0， x、）．1・ 50
CONTRIBUTION TO THE THEORY OF NUMERICAL INTEGRATION
Drawing the numerical solution with step size九，，1et（xl， Yl＊）be． the point of intersection of the numerical solution and the line x＝xl． Draw the straight line y−2J1＊ニM1（x−xl）and let P2：（x2，10）be the point of ihtersection of this line and y＝10 if 2〆（x1）＞0（if y’（xl）〈Othen P2：（x2，0））． By theorem（A）we choose九2 and the numerical solution y1，．Y2，…such that I yn−y（xn）k2εforε＞O inほ1， x2）． By drawing the numerical solutioll with step size砺，1et（x2， Y2＊）be the point of intersection of the numerical solution and the lme x＝x2． Since it． hqs been proved that the solution exists in O≦x≦碗， we can choose九3 and the numerical solution such that IYn−y（xn）1＜εm（0， x2）by theorem（A）， renewing the numerical solution obtained before．（Of course we must choose p＠， y）， convergillg to∫（x， y）， newly for（0， x2））． Drdwing the numerical solutioh with step size九3 in（0， x2），1etほ2， Ys＊）be the point of intersection of it apd the line x＝x2・ Draw the straight．1ine y−Y3＊ニ」胚1（x−x2）， and let（x3，10）・be the point of intersectioh（if ．y’＠1）＜O then（x3，0））of it and the line y＝10． By theorem （A）we choose h4 such that lYn−y（xn）｝＜2εforε＞O in（0， x3）． By drawing the numerical solution With step sizeん4，1et（x3， Y4＊）be the point of inter・ section・of the㎜merical solutiommd the line炉ω3．− Since it has been proved that the solution eXists in O≦x≦x3， we choose ん5suCh that［Yn−y（xn）1〈εin（0， x3）by theorem（A）renewing the solution obtaine（l before again． Contn leing the similar process the numerical solution attains a point lX’， yり，． by a丘nite number of．steps， which satis丘es one of the following conditions （1）10＞y’＞9，10＞x’＞0，（ll）10＞y’＞9，〆≧10，（皿）9＞y’≧0，〆≧10 1f（〆， y’）．satis且es（1）， then we consider． 1）2：0≦x≦．10，9≦y．≦20 if y’（〆）≧0 、 30 29 一 20 19 》 （カり （㎡ッり 10 （1） ⑪ 、9 4⑱
⋮‘ （ ’ クx， Pt） ．川 ll ll 0 苫’ 10 20 52）Since YI＊may be different from y（x1）by±ε， we can not have l 1 ，i 一“‘ IJ（．Tn）Kξ． bqt［yn−y（Xn）．1＜29・T．SHIMIZU
satisfies（H）， then we consider satisfies（皿）， then we consider According as y’（x’）≧O or y，（x’）≦O we respectively．consider
51 10≦x≦20，9≦2ノ≦20 〃 10≦‘t≦20， 0≦1ノ≦10 〃 Max∫（x， y）or Mini f（x， y） 妨汐∈D2 x，y∈D2 Continuing the similar process from the point（x’， y’）as the starting value of the numerical solution， we proceed as before． Obtaillillg as aboveω1’，．T2’， …for each case，1et（x”， y”）be the point at which the numerical solution goes out ofヱ）2 for each case． There are three cases for each case． For the case（1）we have （i）20＞3ノ”＞19，10＞x”＞x’，（ii）20＞2ノ”＞19， x”≧10，（iii） 19＞2ノ”＞9， x”≧10 1f（xtt， y”）satisfies（i）then we consider 1）3：x’≦x≦10，19≦y≦30 satisfies（ii）then we consider 10≦x≦20，19≦y≦30 satis丘es（iii）then we collsider 10≦x≦20，9≦y≦20 For the cese（ll）we have （i）20＞1ノ”＞19， 20＞x”＞10，（ii）20＞y”＞19， x”≧20，（iii） 19≧1ノ”≧9， x”≧20 1f（x’1， y”）satisfies（i）then we consider ．03：ゴ≦x≦20，19≦y≦30 satis丘es（ii）then we consider 20≦x≦30，19≦y≦30 satis丘es（iii）then we consider 20≦x≦30，9≦y≦20 For the case（皿）we have （i） 10＞2ノ”＞9， 20＞x”＞10， （ii） 10＞1ノ”＞9， x”≧20， （iii） 9＞2ノ”＞0， x”≧20 1f＠”， y”）satis丘es（i）then we consider 1）3：xt≦x≦20，9≦y≦20 satisfies（ii）thell we consider 20≦x≦30，9≦y≦20 satis丘es（iii）then we consider 20≦x≦30，’0≦y≦10 1n the above when it has been proved that the numerical soution exists in（Xk’， Xk＋、’）， where the illequality｜Yn−y（xn）1＜2εholds， we choose九ノk’and the numerical solution Yn such that f Yn−y（xn）1＜εil1（0， xk＋1ノ）by．theorem （A），renewing the solution obtained in（0，㊨，）and（xk’， x’ic＋1）． Continuing the similar process indefinitely we colne． to one of the follow．ing two cases： The first case：The solution Ym increases that is，暫n→±o。（for the system of the丘rst order equations at least one of the components of the numerical solution vector increases indefinitely）and the point xm， which corresponds to Ym， apProaches to X（a finite liumber）． The second case：The numerical solution may be・prolonged indefinitely， that is， X＝◎o． In the latter case Ym converges to a finite number， oscilla土es infinitely or tellds to oo as x肌→oo． For the丘rst case the chosen sequence of step． sizes h’， h”，… corresponding to the intervals（0，め（0， xlt）…（where（0， xtt）→（0， X））converges to zero， so we can not choose hm， m＝1，2…and the numerical’solution Ym， m＝0， 1，2，… such that［Ym−y（xm）1＜ε for a11 xm in O≦xm＜X by this process．52 CONTRIBUTION TO THE T且EORY OF NUMERICAL INTEGRATION
、 In general it is inlpossible that we choosg h！，九2，…九ゴ，…alld O＝Xe ＜X，＜X2＜… ＜Xj＜… ＜X， and Xゴ→X（or oO）such that l Yn−y（xn）1〈ε （for an arbitraryε＞0）for all’x血0≦x＜X（or oo）， the step sizeんゴ bemg used for the numerical method in （Xブ＿1， Xゴ），ゴ＝1，2，… ． This is evident from the following method by which we prove the latter part of the theorem in another way． It’has been proved that53）the numerical solution exists in O≦ω＜X and it can not be prolonged beyond X （finite or infinite）， and that for every ゴ，ゴニ1，2，…we can choose the step size九（」），」＝1，2，…of the numerica］ method and the numerical solution仇， n＝0，1，2，…such that［yn−y（xn）1〈εj for everyεゴ＞O where nh（ゴ）＝xn， n＝0，1，2，…in O≦の≦Xゴ（XゴーtX）（if we obtain the numerical solution Yn at xn with the step size九（ゴ））54）Now we take an infinite sequence Xl＜X2＜…＜Xn＜…＜X such that
Xn−X・・n・d・n…ﾓ．㌃・y P・an鴨げ一）（・・1・y M・resp・r・・v・ly・
where Dゴ：Xゴ≦x≦X」＋、， Y」−L≦y≦Yゴ＋L．55） ． First we consider（0， X1）． We choose the step size九（o）of the皿mericaI method， and the numerical solution Yn， n＝0，1，2，…such that ［yn−y（xn）kε （by taking Yo＝y（0））for all xn in O≦xn≦XI by theorem且， where X1＝Noh（o）56）． Next we consider（Xl， X2）． In this case the truncation error is estimated byθ10（1十h（1）Pl）N1十hPG（Xl， P1）， whereθ10 is the error、 of initial value of the 皿merical solution at Xl， that is，θ10＝Ylo−Y（X1）， Ylo being the value of the initial value of the numerical solution at Xl57）． RI being丘xed in（Xl， X2）θ10 must be． so small thatθ10（1十九（1）P1）N1〈εand 九Pσα1；Pl）＜εhold． By the latter we choose九（1）， alld then N1， and thenθ10． Now we chooseん（1・1）and the・numerical solutioll Yn， n＝0，1，2，…such that IYn−y伝π）kθ10 in．’（0， X2）by theorem A． ・ 53）It is impossible to obtain the exact value of X practically by caleUlation since it takes an infinity of time to attain to X． But theoretically we can ．㌃ρb垣in．X． by．the al）oVe process． ’．’ ー ガ ．． 54）The existence interval of the solution may be obtained in another way（by prolongation of Taylor’s expansion along the real axis， calculating the tadii ’吟．@ of co】nvergence）．． ． 、∵． ・ ．ヒ ． ． ． ． 『 ・55） アゴ denotes the numerical solution obtained above for Xj≦x≦Xソ＋1， and L．a SUMCient］y large nUmber． 』 ． 56）If．the”true solution is known， we will、proceed in the following way． Let ε1＜ε2＜επ＜…＜ε，and we choose hm． properly．by theore血（A）and obtaining 仇enumerial solution． Ym sUch that lym−y（輪）｝＜εz−ei．｛in Xi．1≦x≦Xi． for． aU 諺in’（Xi 1”i；’苦）ジi＝1ジ2，・・一・， where・εo＝Xo＝0；．ni’h1十… 十nihi ＝Xi， and y（Xz．1） ．a等．th．e・in．itlal Va1Ue：at．の＝Xi−1てof the nume℃ical．solution）．．・．Then we have lym−y＠肌）1＜εi一ε乞＿1十（εi＿1一εi＿2）十… 十（ξ，一εo）＜ε． By continueing the nume−． rical solutions above obtained in（Xi．1， Xi）， i＝1，2，…， We have the nume− rical solution輪for which iYm−y（xm）1＜εfor aIいv in O≦x＜X． But the true solution is not known in genera1， and only the numerical solUtion which may be different from the true one at most bv±εis known． 57・・（X・・e・）一・・誓・・・・・・・…nera1・・X・・…一・・砦〆声・1T． SHI］班IZU 53 Thus、ve have the加merical solution‘Yl，ηdi登erent from the true one at most by±θ10 in absolute value． Now we choose九（1・1・1）and the numerical solution Yn，％＝0，1，’2，…sUch th・t lyn−Y（xa）1＜2・．f・・all㊨in l（Xi・X・）by th・・rem A・t・king th・V・lu・・f y、，n ・at・Xi， as the’initial v前ue， where N。九…＋N、h（D＝X、．｝ Draw the segment passing through（（」Vo−1）ん（o）， YNo−1）and（X元， Yl，o）， alld consider the curve passing through the points Yo， Yl，・・㍉YNo＿’i， Ylo， Yll’，°”， YIπ1． ・Let the last curve be denoted．by ynt， m＝o・1・2・・・… then wg与ave lYm−y（Xm）1＜2εfor allωm in（0， X2）． Assum・th・t th・・um・・i・ai・・1・ti…u・vr a・b・v・i・・bt・i・輌（O，．．Xゴ）・ then we consider（Xj， XJ＋1）． In this caSe the truncation error is estimated by eゴo（1十ん（〆）P∫）Nj十hPG（Xj，矛ゴ）， whereθ∫。 is the error of the in託ial value of the numerical solution at xゴ， that is，θゴo＝Yio−y（xゴ）， yje being the initial value of the numeric…d solution at，Xみ P」・being．fixed・in（．X」， X」．、），θ允must be s・sma耳l thatθゴ・（1＋h（ゴ）P’）町くε and九幻G（Xj， Fゴ）＜ε． From the latter we choose九（5）and then珊， NO九（o） 十1V‘九（1）十… 十Njん（ゴ）＝Xゴ≒1 and thenθ」0．・ Now we must shooseん（ゴrゴ）and the numerical solution Yn，％＝0，1，2，… such that lyn−y（xn）1＜θゴo in（0， Xゴ＋1）by theorem A． Thgs we haye the nUmerical solution YJn， n＝0，1，2，…different from the true one at most by±θJo． We choose海（碗）a耳d the numerical solution yπ， n r Q，1，2，…号uch that ly。−y㈲1＜2・f・r aU砺in（Xゴ， X」・・）by the・rem A・by taki㎎the value・f Ydn at Xj as the血itial value． D・aw th・・egment p・・扇㎎th・・ugh（（Nゴー「1）九（」rl｝・ y」一・・N・一・一・）叫（X」・y’・・）・ and consider the curve pass迦through the points（0， Yo），（九（o），Yl），… 〈X1，Ylo）， （X・＋ん（1），、ω・…、（X舞、⑭・（X・＋瓦（2’・ Yz・・）・∵・（Xi・．⑭・．（X・＋h（？・yi・・）i．：°’ （Xj，駒，o），（X・＋九（・），・y」，・），…（Xン…鞠）・P・tth・1・・t・u・⑪・ρ・n・t・ぱby Ym， inニ0，1，2，…， then We．have l協一y（Xm）］＜2・’fgr all疏m．（0， Xお・）・ P・・ceedidg in・the same lwaY we can・btaih th6加磯r颯．日01Uゆ．侮l m＝°・1・2・’”・whi麺ti□es｝he．lli°nd’ti°nr；l t͡e°耳∴．．．・ll 17、．Nume，ical inte卿6・n fOr馳①皿一1im滅i亘tervd8・一、『1： ．’ ” −The theOfem∫above・ptioved is a、fundε血enta10nefor・hum．erical．垣tegratibn of ordinary differential equations． 『 ．．・．．・． B・輌P・a・tice・we can h・・但y「di輌・晒inde丘nitely・「sincr．Ch‥umbe「 of decim垣、digits Can not be』so easily麺cfeased indelinitely． ’、 ． Thus we must discuss the case wh6re hn is limited below by a positive
number九。＞0． 『
t54
CONTRIBUTION TO THE THEORY OF NUMERICAL INTEGRATION
Of course if the solution becomes oo at x＝X， thell we can not have IYn−y（xn）1＜εfor O≦x＜X for an arbitraryε．＞0， unless we diminish九π indefinitely． Even for the case where the solutioll exists in O≦x〈oo， the followillg example shows that we can not have［Yn−y（xn）1＜εfor O≦x＜oo， unless we have hn→0． L・・．・he eq・・…n・・農一・， an・we c・n・・der・h…1・…n・（・）w・th y（0）＝Yo． The true s・luti・i’being y＝y・exp 2 hi， the numerical s・luti・n by Euler’s ヵる＝1 method Yn＋1＝Yn十hnYn is Yn＋1＝yoII（1十んi）． i＝1Now
n n ・xpΣ九・−1（1十hz）＝θΣ々1−1（1＋九D／θΣゐり i＝1 る＝1 …h…her h姐・曇1・9（・＋・・）＜Σ・・一ご署・∼＋丁曇屍・・nce・＜・K・Therefore
喜1・g（・＋・・）＜書・・一丁書・∼＜義・・−22・…亘（・＋・∂／・xp書＜θ一￥ ん1beillg a ftxed number． H・・ce θΣ・・（・¶（・＋・∂／・Σ・り〉・…（・一θ一牢）一・・。、。＿．。． Thus iYn−y（Xn）l becomes。。 as n→。。58）． We shall discuss some class of di丘erential equations of which the solution with some initial value call be obtailled by some ntimerical method so that IYn−y伝π）kεfor an arbitraryε＞O for all x m O≦x＜oo with a丘xedん． Babuska， Prager alld Vit亘sek in their book59）gave on the same stand point the folloWi㎎：T…REM・6°）G…n・h・d・…e・…1・q・・…n篇一∫（鋤…h・…erv・1
0≦x＜oo， let y（x）be its solutiol1， which exists il10≦x＜oo， and let the solution be stable under persistent disturbances． Consider a given completely 58） ︶︶90
56
This does mt contradict to theorems in §’ユ6． From ［exhi−1“1（1十九の｜ ＞eh1（1一θ一h1216）we can chooseん1 so that ehi（1一θ一h12／6）＜εand for a properly leΣhi−∬（1十hi）1＜ε if hn→0．chosen hn Babuska， Prager， Vitasek：Numerical proceSses in differential equations． Babuska discussed round off errors ln the above theorem， but the conditioll on the round off errors is too strong（the assumption lδnl〈hδbeing not practica1）we describe here the theorem in the form without consideration of round off errors． Complete regularity onφ（x，y， h）restricts∫（x， y）， that is， f（x，y）should be bounded in O≦x＜。。， lyk。o and uniformly cont血uous蒜霊㌃》・際，P灘φ認認r「ぽ「蒜瓢撫，：1ρ還
bounded， it is necessary and sufficient that∫，∫ω， … ∫｛P−1）are bounded．T．SHIMIZU
55 regular general one step（convergent）method of’degree p＞0．』Thenl：’?盾 every ε＞O there exist ん＊＞O and δ＞0． such that every solution of the difference equatioll Yn＋、＝Yn＋九φ（Xn’s Yn， h） for which l yry（0）1〈δand九くん＊satis丘es the illequality ly・−y（Xn）1＜ε， n＝0，1，2，…in O＜x。＜。。． In Babuska，s theorem the conditions：the boundedness ofφand Lipschitz condition in O≦x＜oo，．｝y l＜∞are too strong， and these can be teplaced in the form（without modifying the．proof）：1et y（x）be the solution bourided in O≦x〈。。andψis bo皿ded for an arbitrary but fixed K ih O≦x〈◎。， i y lくK， Lipschitz constant L」ζdepending on K in O≦x＜◎o， l y l＜K． We shaU give here some similar theorems61） THEoREM C． Without considering round off errors let there be given the differential equation（52）， alld let y（x）be．the solution of（52）with y（0）， and y・＋・＝y・＋九φ（Xn， Yn， h）the di丘erence gquati・n（9・ne・al・ne step c・nverg・填 method）of degree p＞0． Suppose that g（x）exists and is bounded in O≦x＜◎o， ・ndφ（x，y， h）・nd∫（x， y）a・e analytic with resllect t・x，・y，んin．D：0≦x＜。。， ［y］＜∞，0≦九≦九。，andφ（㌶y，h），φ“伝， y，九）and∫（P）（ちy）are』「bounded for an arbitrary but丘xed K＞O inρ≦x＜oo， l y l〈K，0≦九≦んo，and Ov（x， y，九）≦一η＜O in l≦x＜oo， lyl＜K，0≦ん≦んo for a large Z＞0．62） Then there exist九＊＞O andδ＞O for everyε＞O such that every so1Ution Yn， n＝1，2，…for which IYo−y（0）kδandんく九＊satisfies the inequality ［Yn−y（Xm）1＜ε， nニ0， 1， 2， 一・ in O＜Xn＜oo． 九P判 Proof：and
Then
Denotingφv（Xn， θ叫・＝θ・（1＋九9・（0））（1＋hρ、（1））…（1＋九9、（n））＋九ρ＋1｛ρ2（％）＋P2（n−1）（1＋九9、（n）） ＋…＋9・（0）（1＋卿、（n））（1＋卿、（n−1））…（1＋卿、（1））｝ Since．Iy（x）1＜KI by the assumption we haveゆ1×K2，1ψ21＜K3 and we can、 choose h so that lんgij＜1． Thus O＜1十九g）1（n）＜1一η九． Let y（X。＋1）＝y（ωn）＋九ψ（X。， y（ωn），ん）＋ ∫（P）（θn’， θπ”）68） （P十1）！ Yn＋、＝Yn＋九φ（Xn、 Yn， h） hP＋1 θ・＋・＝θ・＋ん｛φ（x。，y（x。），ん）一φ（x。，y。，ん）｝＋ ∫（卯）（θn，， θπグ）・、 （P十1）！ 一・・｛1十hφv（Xn， θn）｝＋（帯，∫…（・・’…〃） ． θπ，九），∫（P）（θn’，θn”）by g1（n），ψ2（n）respectively we have ︶︶ 噌⊥り臼66
63） In ordinary case stability is considered only for the solution which is bounded． We can not determine whether the solution is stable or not unless the solution is known． In general the numerical method is applied when the solution is not known． In this theorem there is not any conditon about stability． Many suMcient conditions in order that the solution may exist in O≦x＜。o or it may be bounded are known． di y（xn，θn， h）， f《P［（θn’，θn”）are the values obtained by mean value thborem．．与6
CONTRIBUTION TO THE THEORY，10F NUMERICAL INTEGRATION
．Hence．’we have、 …・・＋・1＜・（・一・・）叫K…＋11葺器1＜・（・一・・）・＋・＋竿3．・ Takil19δ，んsmall we haVe l ep＋；1＜ε． ’㌧ ・・ REMARK：The．conditions of．the last theorem and of Babuska，S one are equivalent in eSsential parts at、1eaSt for the 丘rst order linear differential equations． ． ． ． ．．・ ”“ ：Complete regularity， that is， thatφis bounded and eontinubus With icespect to x，．y，・．h，、 and’uniformly continuous in x， and satisfies Lipschitz condition With・espect．t・yin O≦X＜。。， lyl〈。。，0≦九≦九。 is equiv・lent・ in eSs6ntial parts to thatφ，φ禦．aPd工（1）．are continuous and bounded for an arbitrary but fixed K＞O in O≦x〈oo， Iyl〈五，二〇≦九≦．海o， uniform continuity ihωcorfe磯）onding t豆．th・P・un血C・S−．gf．1．Ei）・．ロPschitz c・nditi㊤9・rre・pqnding t・the cOnti頭tygf．・多6a＞：∵ ．’1 ． 『
−The cOhditionφy≦≡一η＜O in Z≦x＜oO， iyl＜K，0≦九≦九〇is equivalent to the stability under persistent disturbances for the first order linear differential equations． y’ニーg1（x）y十g2（x）， where￠1ノ（x）has not an infinite humber of z’c｝rO9’for諺．〉彦，’1・b6ing．alarge nu血ber65）． Lef the SolutiOn wrth Yo at x＝O be 」『1． w．’@．㌔ 三 』 ’y一θ一∫言Pl（x）di｛fo Xgフ2（x）・∫：・・（・・d・dx＋・・｝ On the other hand 2’＝−g1（x）2十g2（x）十δ（x）gives the solution −』一’・〒・一∫：・・（・）dx｛∬（9・（・晒））r∫：・1（・・d・d・＋y・｝．H飽ce．．
Q． P二y−・一∫お・1，d・∬δ（∂・∫φ！1・・d・d・ B・一・〃一・・≧・〉・P・・t…B−・∫：・蝿五＝∬δ（x）・∫：・・（・・d・d・ ・ w・h・V・・?齒ｨ書・8’・・・・・・…t・
H…e
@ ㌔÷B二頑1三λ・弓（B−Bパ・…
了∴∵．ノ ・．㌧ ’ジ’∬・≡「 馳’．1’ ． ・・．A δ ．． → By．みで゜°as．ひO・．We have・万く万f°「鋤・・x・bei”g a large．nu頑e「・ that is， y（x）is stable under、 persistent disturbances． ．．一 ．．Conver日e1）ζif the so1Utio．P、 y（x）．of the eqUation is stable under．Pe斑istent disturbances， then 一φ〃〒・gi（x）≧η＞O in Z≦x＜oo for．a chosenηand．a sU伍ci− e耳墾yJarge’乙 旨． ’−．rli五66 g）1’（x）’has‘n6tξ1n infinite hi niber Of zeros，．乎）1（lil）has’ohly a．finite Ilu血ber of血axima． Hence there exists a． humbe士Zsuch that gi（x）＞0， g1（x） 64）Uniform continuity’in九follows from the continuity in九in O≦九≦ho． ．65）FoT」example， it・is、so when、 pl＠）is・a rational function of x． ’s
T．SHIMIZU
研． ＜O or ￠1（の）≡≡O for コ7＞1． 、． ．If． g1《x）．…≡0． for x＞．1， then B→c＞O as・・x→◎◎and 4→◎◎for any small positiye ngmber 6 as x→。。・Hence〃（x）can n・t be stab1・in this case．．1．．．・∴．・ If gi（x）＜O for x＞1， tthen・．8→c whe】陀c≧0．二〇n the other．hand∠4＝・∬e∫：・’（x竺…C・一・’〉…CC・由㎎…〉・…一α．‘・．−
Th・・㍗’蓋・・b・urid・d，・na・ωtS・・t・tdb1・．．・． ．＾
If 9・ω＞Ol・tp「．．q＞P・，and・・瓢．eP・ω＝0・then孕一・＞10・β→・。・・xp・。・B being：t虹gnotoniC… Let，．β一tc，・then・4→◎O「』ξas x：→oo’．ひ憤）f〕ing ＞1， for any Positive small numberδ． Hence y．（￠）・lcan． not be stablC in this case also；L・・ぷ・・h・遣一参、も’・頸音一・・…nee・・《舳e・n・・一・sc・n…
in血1ite number of times， there exists a large positive numbeF液・i’sti6h that多＞K・・r≧・』興lln亘be−・nce・A−A・〉（β二la・趣撒
＞K，．輿号e一β→。r・喚i・h幽≒b。頑h・t y（X）i・nOt・t・b1・・ ． ．t He竺r・興・）jS…1・・U・d・・p・ll・i・t・・t di・tU麺・r…there exi・』p・・ifi・・ numberη、 such that、 gi（x）≧η＞O for￠＞1， l being a large111umber． N・wwe・1 ail give・th・・rem．ρ？n・id・・ing・・und・仕．err・rs・” T・EO・亘頑：−’ ．IO ’b餌・lderati・n・f輌d・仕er・Orr』let th←re ’b・gi随th・ differg・ti迫⑭ti・h、：（52）麺’1・t姻b・th・・61・tigP・fl｛r2）・Wit四（旬‘疸d、 、r．Y・＋、・FYn＋んφ侮。， Yn，九） 、 ．．． th・．’ р奄???窒?氏E6∈呼ign．’・品e9・e・P＞1・SupPQ・∈． th・t yω・Xi・tS頑l i・ b・unded in O≦x＜。6；ahdφ（x，’y，h）．and∫（‘i’， y）are analytic with respect t°x・鱒in°≦・＜°°・・［yl＜◎o・、0≦九≦九〇・．甲｝dφ（x，2ノ，九）・φ・（・・yl’『ん）・nd∫ω（x・ y）征e坤・ded i紅’1’D’ ’L⑩≦x＜。。・コyl＜。。1，ρ≦九≦”…血d．●砲’y，’九）≦一η＜O inl≦x≦°9・］yl細・、9≦lv≦タ・伽担逗9⇒α・ ．．．．
FU・th・r願a・日・血・that the・・Upd・ff ・r・・rs，・tt，be・XPrer・edざuφaS．、．． （53） Φ（2n， On， h）＝｛φ（xh，ずn， h）十βn●n｝（1＋．γ。） whereΦ，9πand疹n denote the values rounded off，1γπklo−z after66）wilk蝕1soh，s paper，βn are67）composed of the errors sUch asγヵ，…． and Tn are uniformly bounded in D． Then if we take a su伍ciently large number of decimal digits there exist 九＊＞Oandδ＞O for everyε＞O such that every solution On， nニ1，2，… for which I豆o−y（0）Kδand九く九＊6atisfies the inequality ︶︶ 67・ nOρU 68） We assum that the computer has“a t・digit mantissa”． βπdepends on the number of additions， multiplications， etc． inφ（x，y， h） and is a complicated error which comes from the errors of calculation ofφ． The magnitude ofβntrn depends on the magnitude ofφ．φ（砺y，h）being bounded in D， we ean express the round off errors i皿the fo㎜（53）in many cases． 、 ’ Remark that in the assumption of theorem D，φ，φ〃…are bounded in O≦x＜oo， lyk◎o，0≦h≦九〇， while in that of theorem C they are bounded in O≦x＜．oo， lyl＜瓦，0≦九≦九〇． It is not easy to prove directly that gn is bounded as n→。。 since n is not limited and Y，． can not be expressed such as Yn＝Yn（1十rn’）．駒
