Maloň, Stanislav

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On nonlinear numerical iteration processes

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeStanislav MaloňOn nonlinear numerical iteration processesCommentationes Mathematicae Universitatis Carolinae, Vol. 3 (1962), No. 3, 14--22Persistent URL: http://dml.cz/dmlcz/104914Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1962Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCommentatibnes Mathematicae Un ive r s i t a t i s Carolina® 3 f 2 (1962) Otí HCmLINKAH NIMBRICA1* 1TSBATICB HIOCBSSES S tan i s l a* • MALOft f Praha 1 Let y to; a Banach space and F it® closed stibset* Let K be a !»±pschltz operátor mapping F i n t o Y s 1*6*^ there e x i s t s a pos i t ive constant fi such t h a t (1) . , i! K t t - K ^ í U / í « ^ - v - l hoids for any <A*f w-Q F • -Let us consider the i t e r a t i o n procese M0 e F , This process i s convergent i f the wall Imom condi t ions given in the Banach the ořem (se e e»g*[2]) are 8a t i s f i ed f i#e» '(3a)-' :'.&<.* » (3bl v* 6 F =*> V* m K ^ e F" ' C3c) SCVn > f )c F , Tibere S i a a closed aphere whose centre i s yA and r ád iu s * - ^ « u , - v. "• We shall' supposs that the secpenee (2) conirerges to the li-ait % « In practice, if the digital. couputation techniqae be u~ sed, there often will be neceasary to transfer the problém oř realising the sequance {^} » defined bj (2), into a space different from the .'originál onet so that the elements ^t might be numerlceúLlý interpreted [l] • T"0 this effect it •" • - 1 4 . - •••'• i s necessary to replscwi the originál proces® (2) by another subsidiary proces© •rtiiqhis easy to be realised as to using the - numrleal techniquej moreotěr» we must naturally desire the originál process (2> to be approximated sufficiently ac-curately by the subsidiary process* In agreement with KTantorotiS [l] f le t us transfer the problém into ápaee Y iaomorphe with Y 9 the isomorphism being realised by the linear bounded operát ion fi j i t is matural to asaume tha t the element® f e F can be.numeri-cally inter$reted* The construction of the subsidiary i tera-t ion proces® wHl be doně by a suitable operátor K appro-ximating the operátor K • Let us assign the analogical pro-cess _ t o the procesa (^)* In t h i s páper m study the approximatite solution of the e~ quation AfmKq'r'[iÍw&» i terat ioň proces® (2) is replaced by the proces® {4) • P i r s t s m shall deal with theduastion mhst aufficient conditiona are to be required from the operátor K to get the l imit i |* of'the procesa (4) sufficiently near to the l imi t <y* of the sequence (2) in terms of the definition and £ i s a giteiá posit i te number* Next, m consider the question of the influence of the roianding-off errors, m ^ _ . ' II We aaaume that the approximating operátor K i* Lipsch-i t s bounded, i . e . for ítŤ iř being tmo arbitrary element© be-longing to F , (6) IKiL-Kfrly * J l * - * ^ holda. — # The proceas (4) will converge to a certaln limit y t if analogical conditiona, the aame as for the proceas (2), l*e« (7a) il < 1 , (7b) i * € F « > § , * K-y* € F, hold. What other conditiona aře to be aatisfted by the appro-ximation K , so that the limita y and y may be 8Uffi~ ciently near in terma of definition (5)7 The answer to thia queation ia given by the following theoreni: Theorea 1: Let the following aeaumptiona be fulfilled: 1) The conditiona (1), (3a,b,c), (7afb,c) and (6) are satiafied* 2) Approximating operátor K ia auch that for any ele-roent ti e F the inequality Ia} _ JyKu-Ry^tt^ á^l^«y x) x) vhe condition of L*V.iCantoroviěf [ l ] , p.107 - 16 -holde. holds, where ^ , /£ are deřined by the proces® (2) and by the íormula (3c), 4) flyflá 4 • Thea (9) Wll9V*-*«Jý * -f^" holds, where c * >*afi. II ^4 v * Froof. As both sequencea {i^n} l í í n . } a*10 convergent according to our assumptions and the operátor g> is continu-oua, the limit in (9) existaj i t remains only to prove the inequallty* 1) First, m shall prove that for any positive integer 7L the inequality holds, where .. u Evidently, for any it € S llptt - fy< II á Il9li il<t -%H4IU - ^ Al ú K holds, i.e« J t t e S c S1 . Then it foliová from our assumptions that and l|cp^+ , - V » w il? - 19ř KV* ~ * V* l ly * •-17 -Ad ty», € S C F , (8) can be applied on the las t by one term further, 9%*, & &9 U* € S^ % conse~ quently ?Viv «A Vn. belong to F and (6) can b© applied on the last temni coase quently ácc2^H^ay< -irr" * 2) I t is evident that _ •y ž i i ^ - v ^ i i - T n holds, Aa íl^^^-yJI-^O and Cu á C f the inequality (9) follows immediately* Note» If the operátor K and i t s approximation K a3?e line ar bounded operátora mapping complete spaces Y resp* Y into themselves, then we can put F*V, F « Y»/3* II Kli,/S « II Kil and the assumptions (3b), Í3c), (7b), (Tc) are to be dropped* i n In practice hoymvůr, as the actual computation i s r e a l i -sed by digital nmabers, rounding-ořf errors in the process (4) aris«* Consequently, the computation proceduře i s not de~ fined by the i teration fomnula (4) but in generál by the fo l -lowing one: £«.+< - K-y«. » n.'m 0,4,1,.... - 1 8 • - • '. where ÍI eY and the operátor K is defined on the samé sub~ space F a® K and appraxiiaate® K in teras of definition (12) ' • • I I K Í - - K 5 ^ ^ ? , í e y , wfaer© | ie a nonnegative number (the upper bound of error eaused by accinmilation of rounding-off errors when computing the valme K u )• I t is the procese (11) only which allways can be realis~ ed. x) The ořem 2» Let the following conditiona be fulfilled: 1) The inequality (12) holde for any ti e Y p 2) (7a),(7b),(7c) hold, 3) % € F , 4) U . f& i % + T )c F , Twhere U (%^%K + f^ i s the closed sphere, y i t s centre, % + f it© radiu©, ' 1-/3 _ í - 4 Then l ) <& € U f or al l i * </, 2 , . . -2) The estimatlon (13) l l ^ - ^ « á ^ « ^ l l + ^ hold©* Proof: 1) We shall pro ve that the sphere S^ defined by the formula (7c) is contalned in the sphere . x) In the following we omit to designate spaces when wrlting noriBS* - 19 -i . e . that h e ŠH implles H tv ~y< U é Z + Reálly, for any V« e ^ From these ineqtxalities we get aimply 2) Evidently ff -neighborhood.cst &, ie contain®d i a f -neighborbood o£ 5L and coD*®qp®ntly in l i • We sbal í prove by induction that i f % £ U 9 then ala# S i ^ c Zi- •» For 1*4 i t i s proved; for i ž 4 wt haf® I * -V i l - I IK^-KVi - . l l» 1*®* the estimation (13) hold®* From i t follows that A® Vi ^ S-i $ -th® f i r s t part ot the aaserlion ia .•!-®o proved* Hote* The influence of truncation and of rounding e r r -ors was studied by M» Tlrábe* Theorem 2 i s alightly geaeri*-lis&ed reaxilt of his páper [3] r vfao&® řoramla (2.5") p# 481 i s - 2 0 -a spec i á l čase of ourformula (13) váien ym « y& * I ? Conclusion* Summing up the results of items 2 and 3, we get the following the ořem: Theorem 3« Let the conditions of the lat and 2nd theo-rem be satisfied, Then the following assertions hold: 1) All elemente of the sequence (11) belong to F . 2) For the distance of the U-th approximaťion ^y^ from the element <f% the estimation (14) )l^*-^«?<^*-^«y*^llíjl^ ^ff^ holds , where C^ i s defined in the the ořem 1 • Proof: Evidently | | ^ * - V « . II á l\<?y%- 9>V«.II * | 9 V * - V* [[ + II Í * - V » // • The f i r s t term of the r igh t i s at most equal t o the e r ror of the TV-th approximation in the process (2 ) . For the s e -cond term we use the estimation (10) and f o r the t h i r d the es t imat ion (13) • Notau The influence of er rors 1 1 ^ * - V** " a n a " *í " diminishes with n -»oo t o zero. But as £ i s a fixed pos i t i ve number, i t doea not f ollow from (14)» t h a t the p ro -cess (11) should converge i n current sense . We can a s s e r t only, t h a t a number V being given, - 21 -such an integer n4 exists that y ^ belongs to the sphere % Í^PH* V^ foT ^ u ^ ^ o * In Pra<^tice, as a rule, Y will be an Euclidean m~dimensional space Km' * In every fi­nite part of RT* the re is a finite number of vectors whose components are digital numbers with given number of figures; let us assume that the sphere S^ contains just Jf eleinents* Then evidently* if the sequence { ^ l does not converge, it will be periodic beginning from % certain *i<, Á ^ t 0 f with the period at most* In other words, the sequence {Vy^} will reach the statě of numerical convergence in the senseof M# Urabe [3]. An arbitr ary element \§^*-£ 9 it m 0 f A $ ... • can be accepted as an approximation of ty* the error of ifhich does not exceed the number V « The statě of numerical convergence need not také pláce when especially from the conditions of the ořem 2 it is the '2nd condition only which is not fulfilled. However, in the čase mentioned in the Notě of item II the 2nd condition is to be dropped. L i t e r a t u ř e : [ i ] L»V* KAHTOROfVTČ: Funkcional 'nyj anal iz i pr ik ladnaja matematika. UMN, 2 , vyp* 6(1948) s 89 - 1.85. JX| L. COXLATZ* Einige Anwendungen funkt ionalanaly t i schér Methoden in der praktischen Analya is , 2AMP, i (1953), 3 2 7 - 357. [3] M. URABB: Convergence of numerical i t e r a t i o n in s o l u t i o n of equat ions . J .of Sci.Hiroshima Univ. , Ser . A, 12 No. 3 (1956), 479 - 489. '- - 22 - ' •' 

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On nonlinear numerical iteration processes

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Institute of Mathematics AS CR, v. v. i.