4 Round-Off and Truncation Errors
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1 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 4 Roud-O ad Trucato Errors Errors Roud-o Errors Trucato Errors Total Numercal Errors Bluders, Model Errors, ad Data Ucertaty Recallg, dv dt Δv v t Δt t v t t appromatg te dervatve o velocty ot perect resultat soluto avg error! dgtal computer s also a mperect tool yeldg error! How do we deal wt suc ucertaty? detcato, quatcato, ad mmzato o errors DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr
2 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Errors umercal error bluders model errors data ucertaty - roud-o error due to computer appromato - trucato error due to matematcal appromatos - uma mperecto, computer malucto - complete matematcal models e.g., Newto's d law wtout relatvstc eects eglgble ar resstace ~ v or v - measuremet errors accuracy: ow closely does a computed or measured value agree wt te true value? accuracy or bas systematc devato rom te trut error precso: ow closely do dvdual computed or measured values agree wt eac oter? mprecso or ucertaty magtude o te scatter error DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr
3 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Error Detos We kowg true values, true value appromato error or true error, E t true value appromato true value appromato true ractoal relatve error true value true value appromato or ε t % true value We ot kowg true values, appromato error ε a %, but ow ca we calculate te appromato error wtout true value? appromato preset appromato - prevous appromato ε a % or umercal metods based o terato preset appromato repeat computato utl we we meet ε s ε < ε, a prespeced percet tolerace "stoppg crtero" a.5 s %, te result s correct to at least sgcat gures Eample How may terms sould be added te Maclaur seres epaso o e uder ε s coormg to 3 sgcat gures? Sol. DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr 3
4 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Coee Break Every aalytc ucto ca be represeted by power seres!! For stace, ow ca we epress to power seres? Let us cosder te ollowg geometrc seres. 3 L L Taylor seres or a were a! L! were R remader!! R R R! ξ, Lagrage's orm ξ ξ, Cacy's orm! Taylor's teorem states tat ay smoot ucto ca be appromated as a polyomal. A Maclaur seres s a Taylor seres wt ceter e 3! 3! L L! 3 s 3! 5 5! 7 7! L 4 6 cos! 4! 6! L 4 DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr
5 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Roud-o Errors arse because dgtal computers caot represet some quattes eactly computers' sze ad precso lmts o ter ablty to represet umbers gly sestve umercal mapulatos to roud-o errors Computer umber represetato bts: bary dgts byte 8 bts word: udametal ut wereby ormato s represeted; a strg o bts e.g., 6-bt or -byte word sze decmal or base- system 3 e.g., bary or base- system e.g., Does ca te computer represet rratoal umbers suc as π,e, 7 π or 6-bt word sze sgle-precso loatg-pot umbers π or 3-bt word sze double-precso loatg-pot umbers How about.? Artmetc mapulatos o computer umbers assumg a 4-dgt matssa ad a -dgt epoet e.g., subtractve cacellato 3. 3 ormalzato. large computatos.4 4. addg a large ad a small umber dgt matssa.4 4 smearg - occurrg weever dvdual terms > te summato; proe to roud-o error e.g., seres med sgs e 3! 3! L ad - wat appes? er products y y y L y proe to roud-o error DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr 5
6 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Trucato Errors result rom usg a appromato place o a eact matematcal procedure e.g., te-dvded-derece equato; dv dt Δv v t Δt t t v t I umercal metods, ow ca we epress ucto a appromate aso? Taylor seres were ξ R, ξ Lagrage's orm t-order appromato! predctg a ucto value at oe pot terms o te ucto value ad ts dervatves at aoter pot zero-order appromato rst-order appromato secod-order appromato! 4 3 <Appromatos o at > Taylor teorem: ay smoot ucto ca be appromated as a polyomal e.g., Recogzg tat all te values subscrpted are costats te secod-order appromato, a a a polyomal! 6 DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr
7 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 How may terms are requred to get close eoug to te true value or practcal purposes? based o te remader term o te epaso oly a ew terms most cases cocept o order o, R O - te error s proportoal to te step sze rased to te t p ower - useul judgg te comparatv e error o umercal metods; O vs. O - estmatg trucato errors Eample Use Taylor seres epasos wt to 6 to appromate ad ts dervatves at π/4. Note tat ts meas tat π/3 π/4 π/. cos at π/3 o te bass o te value o Sol. DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr 7
8 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Recall te eample o te bugee jumper, dv dt cd g m v Epressg umercal orm by usg te Euler's metod, Also, vt ca be epeded a Taylor seres, v t v t v t v t t t t t L R! Trucatg te seres, v t v t v t t t R, were R ξ ξ!! Rearragg, R v ξ were, t t O t t t t! Tus, te estmate o v t as a trucato error o order t t or te step sze. 8 DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr
9 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr 9 Numercal deretato orward derece appromato o te rst dervatve O or O backward derece appromato o te rst dervatve - epadg backward, L! cetered derece appromato o te rst dervatve - orward backward Taylor epasos L 3 3 3! or L 3 6
10 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Eample Use orward ad backward derece appromatos o O ad a cetered derece appromato o O to estmate te rst dervatve o at.5 usg a step sze.5. Repeat te computato usg.5. Note tat te dervatve ca be calculated drectly as a d ca be used to compute te true value Sol. DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr
11 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Fte derece appromatos o ger dervatves Epadg a Taylor seres or terms o ;! L Multplyg te Taylor seres or by ad subtractg t rom te above, L Te, we ave, secod orward te derece Smlarly, secod backward te derece secod cetered te derece or DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr
12 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Total Numercal Error trucato error roud-o error Decreasg te step sze trucato error creasg te umber o computatos roud-o error DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr
13 HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5 Bluders, Model Errors, ad Data Ucertaty bluders - maluctos o te computer tsel - uma mperecto - ca occur at ay stage o te matematcal modelg process - ca cotrbute to all te oter compoets o error model errors - bas ascrbed to complete matematcal models - e.g. Newto's secod law wt or wtout relatvstc eects Bugee jumper's problem wt a rst-order or a secod-order drag coecet data ucertaty DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr 3
14 4 DM869/Computatoal Numercal Aalyss/4_Error.doc Avalable at ttp://bml.pusa.ac.kr HK Km Slgtly moded 3//9, /8/6 Frstly wrtte at Marc 5
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