AN ANALYSIS OF APPROXIMATE NONLINEAR ELIMINATION

AN ANALYSIS OF APPROXIMATE NONLINEAR ELIMINATION - CS{ { A...

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Unformatted text preview: CS{ { A nA na ly s iso f A pp rox im ateN on linearE lim inat ion P au lJ .Lanzk ron ,D ona ldJ .R ose D epartm en to fC om pu terSc ience D ukeU n ivers ity D u rham ,N orthC aro lina { Sep tem ber, ANANALY S ISOFAPPROX IMATENONL INEAREL IM INAT ION PAULJ .LANZKRON y AND DONALDJ .RO SE z A b stract . W ep resen tam ethod forso lv ingsy stem so fnon linearequat ion ssu itab letop rob lem swhere convergenceo fanapp rox im ateN ew tonm ethod isin it ia llys low .Them ethod ,non lineare lim inat ion(N lEm ) , e lim inatesthenon linearequat ion sandapp rop r iatevar iab lesdeem edtobecau s ingthep rob lem .W eg ivean ana ly s iso fthem ethod .Theana ly s islead stoadeta ileda lgor ithmwh ichw eshow reducesau tom at ica llyto app rox im ateN ew tonm ethodneartherooto fthesy stem o fequat ion s .W econc ludew ithsevera lexam p les dem on strat ingtheecacyo fthem ethod . K eyw ord s . non linearequat ion s ,g loba lm ethod s ,app rox im ateN ew tonm ethod AM S (MO S )sub jectc lass icat ion . H,H .In troduct ion . W econ s iderthenum er ica lso lu t iono fanon linearsy stem o fequa- t ion s g ( w )= (.) where g = ( g;g;:::g n ) T and w = ( w ;w ;:::;w n ) T .G ener ica llyw econ s iderthem ethod- o logyo fandtheana ly t icfram ew orkp resen tedthere .Th issett ingin su resthatforany w insom eset S ,asequenceo fiterates w k w illconvergeto wS and g ( w )= .The iteratesaredenedas w k + = w k + t k x k ; (.) where x k approx im a tes z k in g k g ( w k ) z k = ? g k ? g ( w k ) (.) and t k ( ; ]ischosentoforce k g k + k < k g k k ; (.) where << .Thesen seinwh ich x k app rox im ates z k o fequat ion(.)ism easu redby k = k g k x k + g k k = k g k k (.) anda ll k < sucesforconvergence(forsom esequenceo f t k ) .W eca llth isa lgor ithm ic app roachGAN ,forg loba lapp rox im ateN ew ton ,/g loba l"re ferr ingtothesetSwh ichcanbe IR n . Suchvar iat ion so fN ew ton 'sm ethodhavebeenu sedsuccessfu lly ins ign icanttechno logy areasinc lud ingc ircu itanddev ices im u lat ion [,].Thetr ick inanyo ftheseapp licat ion sis Theau thorsareintheD epartm en to fC om pu terSc ience ,D ukeU n ivers ity ,D u rham ,N orthC aro lina- y Supported inpartbytheN at iona lSc ienceF oundat ionundergran tN SF-CCR-- z Supported inpartbytheOceo fN ava lR esearchundergran tN--J- inp ick ing ,ifnecessary ,an/ inner"so lverfor(.)to in su re k < and inchoos ing t k ! toen su re(.)andsuper linearconvergence .Th iscanbepart icu lar lycha lleng ing inp ract ice s inceo ftenw ecannotbesu reap r ior ithatthesuc ientconvergencecond it ion saresat ised , norcanw ew a itfor k ! . Theu seo fnon lineare lim inat ion(N lEm ,en-lem )ism ot ivatedbyp rob lem sinwh ichcon- vergenceseem stobein term inab lyted iou sandyetthereisnoev idenceo fill-cond it ion ing(or s ingu lar ity ) .Th islead su stobe lievethat g hassom ebad lym issca ledcom ponen tfunct ion s , thatis ,som e/sub funct ion" g ( u ;v )regardedasafunct iono f u g iven v cau ses t k tobesm a ll. W ep roposetoe lim inate g asan inneriterat...
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This note was uploaded on 01/16/2011 for the course MATH 224 taught by Professor Layton,a during the Fall '08 term at Duke.

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AN ANALYSIS OF APPROXIMATE NONLINEAR ELIMINATION - CS{ { A...

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