We note that since we are starting with three points, our interpolating polynomial is second order, and therefore the second derivative is constant everywhere. In general, to appro ime derivative of order m we will require at least m +1 Fornberg's algorithm The o, can be computed very efficiently using a recursive algorithm developed by Fornberg(generation of Finite Difference Formulas on Arbitrarily space ics of ce ion,51,184). 1.3 Undetermined coefficients SLIDE 7 Insert Taylor expansions for v abou Uy=20+(x;-x1)+5(x-x1)2 determine coefficients om to maximize accuracy. 1. 3.1 Example sLide 8 m=2, l=r=1, i=0,(uniform spacing A r 62 62(+△x6+分2++ Equating coefficients of k=0 621+6+6 k=1 Δr(6-621) (62+621)õ☎ö✱÷☞ø❨ù❴öqù❪ú✼û❨ù✐ü♦ý❪÷✐þ✉öqÿ✱ö✔û✁￾❲ö❊ü✉ù❴û✁￾♦ù❛ý❪÷✄✂➨ÿ✧ý❪ù✜ú❭ù❪ú☎￾tö✉ö✝✆✏ø✢ý❪÷✼ù✖ü✟✞❚ø✡✠☎￾✣ý❪÷✼ù❴ö☛￾☞✆✼ø✁✌➉û❨ù❛ý❪÷✄✂✍✆✼ø✁✌✏✎❨÷☞ø✁✑⑩ý❧û✁✌ ý❡ü❘ü❍ö❲þ✉ø✢÷✓✒✙ø✡￾✔✒rö✕￾✖✞qû✢÷✓✒➋ù✜ú✏ö✕￾tö✘✗✍ø✁￾tö☛✞✱ù✜ú✏ö❀ü❍ö❲þ✉ø✢÷✓✒✙✒✆ö☛￾♦ý✛✚❨û❨ù❛ý✛✚❨ö❭ý❡ü❭þ✉ø❨÷✼ü♦ùsû✢÷✼ùqö✕✚❨ö☛￾✜✎❨ÿ✗ú✼ö☛￾tö✣✢✥✤♦÷ ✂❩ö✍÷☞ö☛￾tû✡✌✦✞✑ù❴ø û✟✆✧✆✓￾tø✩★rý✛✑➋û❨ù❴ö✚û✪✒rö✕￾♦ý✛✚☛û✢ù✖ý✛✚❨ö✚ø✫✗Òø✁￾✟✒rö✕￾✭✬ ÿ✱ö✥ÿ✧ý✛✌✛✌✮￾tö✖✯✕✠✷ý✛￾tö✙û✢ù✰✌✹ö✉û☛ü♦ù✍✬✲✱✴✳ ✆✼ø❨ý❪÷✼ù✖ü✕✢ ✵✭✶ ✷✹✸✻✺✜✼✾✽ ✿❀✸☎❁✣❂❀❃❀✼✁❁✣❄✰❅❆❈❇☎❉❊❄✝✸☎❁●❋❍✺❍■❑❏ ▲◆▼P❖❘◗✁❙❚✲❯✕❱❳❲❩❨ ❖ ❯☛❬❳❭❫❪P❴P❵ ❖✩❛✙❜❳❖✩❝✔❞❈❖✕❡❯✕❢❖❲✄❵✔❣❞ ❴✐❤✔❢❥❲P❦❫❱ ❝✔❖ ❯☛❴❝ ❤❍❢❜❳❖ ❱✡❣❧❦❳❬❝ ❢❥❵▼❭ ❛P❖✕❜❳❖❣❥❬✧❪❖✣❛ ❨ ❞✹♠❬ ❝❲♥❨❖✩❝❦♣♦❍qsr✜t✓r✖✉✜✈☎✇✘①✛②✁t③②●④✥⑤❑①❥t✐①✘✇⑥r❈⑦⑧①❥⑨⑩r✖✉☛r✖t✻❶☛r❫⑤✓②✁✉✔❷❩❸✡❹✛✈✡❺❻②✁t❽❼✰✉●❾✁①✘✇✘✉✖✈✡✉✔①❧❹➀❿♣➁✡➂⑩✈☎❶✜r●➃ q◆✉✟①❊➃✧❺✩➄P➅✪❱✡❵▼P❖❭❫❱✡❵✔❢➆❯✕❤◆❬❳➇✮➈✍❬✧❭➉❪✐❴☎❵✟❱✡❵✔❢❧❬❳❲⑩➄✐➊ ✳ ➄ ✳✩➋✡➌♥➍✜➎ ➏❀➐✘➑ ➒➔➓⑧→✰➣✻↔☎➣⑩↕✐➙➜➛✔➓✰➣✝→➞➝➠➟◆➣✻➡➢➝✻➛✫➣✝➓✍↔P➤ ➥✓➦✄➧✛➨✝➩♣➫ ➭ ❵✖❱❝ ❵➯➇❝❬❳❭ ➲✄❙s➳ ➲❳➵❙ ➸ ➸ ➸ ➸ ➺✣➻⑩➺✩➼❀➽ ➚➾ ❚ ➻❀➪➹➶ ◗❙ ❚ ➳ ❚❫➘ ➴❲✐❤ ❖✕❝❵ ▲ ❱❞ ❣❧❬❝◆❖☛➷❪✐❱❳❲✐❤❍❢❧❬❳❲➹❤➬➇➮❬❝ ➳ ❚ ❱❳❨➹❬✧❴☎❵ ➵❩➱➔➵➹✃ ➳ ❚ ➱➔➳✁❐ ✱ ➳♥❒❐ ♦➵ ❚✥❮ ➵➹✃ ➍✝✱ ✳ ✶ ➳♥❒❐❒ ♦➵ ❚✰❮ ➵✓✃ ➍❍❰✍✱ ➘✕➘✕➘✩Ï ❛☎❖❵ ❖✕❝❭❫❢❥❲❖ ❯✕❬ ❖☛❡❯☛❢❖❲✄❵✟❤ ◗✁❙❚ ❵✔❬❈❭❈❱➷ ❢❧❭➉❢❧Ð❖ ❱✧❯✕❯☛❴❝❱✧❯❞❳➎ Ñ✓Ò❧Ó✝Ò❊Ñ Ô⑧Õ⑩ÖP×♣ØÚÙ❊Û ➥✓➦✄➧✛➨✝➩✹Ü ✬ ➱ ✶ ➄✐Ý ➱➔Þ❘➱ ✳ ➄☎ß ➱áà ➄⑩♦❊❴P❲P❢❥➇➮❬❝❭â❤✔❪✐❱❳❯✕❢❥❲✐❦❫ã➵➍ ➳❐❒ ❒ ➱ ◗❰➪⑩ä ♦➳❐ ❮ ã➵✐➳❐❒ ✱æå➺✣ç ❰ ➳❐❒ ❒ ❮ å➺✩è é ➳❐❒ ❒ ❒ ✱êå➺✕ë ❰❍ì ➳✐í ì✖î ❐ ✱ ➘✩➘✕➘ ➍ ✱ ◗❐❰ ➳❐ ✱ ◗❰ ä ♦ ➳❐ ✱ ã➵➹➳❐❒ ✱ å➺✩ç ❰ ➳❐❒ ❒ ✱ å➺✩è é ➳❐❒ ❒ ❒ ✱ å➺✩ë ❰❍ì ➳ í ì✟î ❐ ✱ ➘✕➘✩➘ ➍ ➥✓➦✄➧✛➨✝➩✹ï ð➬ñ❴✐❱✡❵✔❢❧❲P❦❫❯☛❬ ❖☛❡❯☛❢❖❲✄❵✖❤✥❬✡➇ ➳ í❧ò î ❐ ó❈➱áà ô ◗❰➪✻ä ✱õ◗❐❰ ✱❽◗❰ ä ➱áà ó❈➱ ✳ ô ã➵ ♦ ◗❰ ä ❮ ◗❰➪✻ä ➍ ➱áà ó❈➱ ✶ ô å➺✣ç ❰ ♦ ◗❰ ä ✱❽◗❰➪✻ä ➍ ➱ ✳ ö