The Lagrange interpolant is the polynomial of lowest degree that passes through A dml 1. 2.1 Example Second order Lagrange interpola =列儿=出-1+=出=吗+ (21)(=2)v+1 (x;+1-2j-1)(a+1-2j) Assuming a uniform grid m= 1(First derivative) 6}-1050 -2△a2-2k2 For ward 4 2= Backward For example a second order backward difference appro imation can be written v=(3 )/(2△x) 6◆☞❖✏P❘◗❚❙❑❯✢❱❲❙❨❳❩❯❩P❭❬❪❳✼❫❴P✍❱❛❵✼❜❨❝✹❙✢❳✼❫❊❬❡❞❢❫✜❖✏P✔❵✏❜✢❝❤❣❨❳✐❜✢❥❦❬❧❙✢❝✦❜✬♠❘❝✹❜❨♥✱P✍❞♦❫q♣rPs❯r❱tP✉P❀❫✜❖✏❙✢❫●❵✼❙☛❞✉❞❍P♦❞✈❫✜❖✞❱t❜✢✇r❯①❖ ❙❨❝❪❝✗❫❪❖✼Pq❵✼❜❨❬❪❳✠❫❧❞❀②✜③✏④✆⑤✬⑥①④①⑦✍⑤❛⑧⑩⑨❷❶✑❸✬⑤❍❹❍❹❑❹❅❺✏❻ ❼✐❽❩❾❪❿❚➀❇➁ ➂✑➃✼➃✏➄t➅☛➆✷➇➉➈➋➊❨➌t➍ ➎❩➏ ⑥ ➎③➏ ➐ ➐ ➐ ➐ ➑①➒✩➑❑➓✑➔ ➎❩➏✙→⑥ ➎③➏ ➐ ➐ ➐ ➐ ➑①➒✩➑❑➓ ⑨ ↔➣ ④ ➒❂↕✠➙ ➎❩➏❘➛④ ➎③➏ ➐ ➐ ➐ ➐ ➑❑➒✩➑①➓ ⑥①④ ➜➞➝➍❑➄t➍❍➟✜➅r➄❲➍r➠ ➡④➏ ⑨ ➎❩➏❘➛④ ➎③➏ ➐ ➐ ➐ ➐ ➑❑➒✩➑ ➓ ❹ ➢✐➤❡➥✩➤❧➢ ➦➨➧✩➩✏➫➯➭❂➲❧➳ ❼✐❽❩❾❪❿❚➀✁➵ ➸➍❍➌ ❸●⑨➺❺❀⑨➼➻ ➠ ②✜③④ ↕✩➽ ⑤t③④ ⑤t③④t➾ ➽ ⑦ ➸ ➍❑➚✍➅✆➪✼➶✚➅r➄✉➶✷➍❍➄✔➹✗➊✢➘r➄✉➊✢➪✼➘r➍❃➇➉➪❩➌t➍❑➄t➃✐➅r➴❡➊✢➪❩➌ ⑥☞②✜③☞⑦➷⑨ → ➬ ➑r↕✐➑❑➮❅➱ ➬ ➑r↕✐➑❑➮✖✃✞❐❴➱ ➬ ➑ ➮❴❒❩❐ ↕✐➑ ➮ ➱ ➬ ➑ ➮❴❒❩❐ ↕✐➑ ➮✖✃✞❐ ➱ ⑥④ ↕●➽❰❮ ➬ ➑✢↕☞➑❍➮❴❒❩❐✬➱ ➬ ➑r↕✐➑❑➮✖✃✞❐❴➱ ➬ ➑ ➮ ↕☞➑ ➮❴❒❩❐ ➱ ➬ ➑ ➮ ↕✐➑ ➮✖✃✞❐ ➱ ⑥④ ❮ ➬ ➑r↕✐➑❑➮❴❒❩❐❴➱ ➬ ➑✢↕☞➑❍➮❅➱ ➬ ➑❍➮✖✃✞❐❲↕☞➑❍➮❅❒❩❐❅➱ ➬ ➑❍➮✖✃✞❐♦↕✐➑❑➮❅➱ ⑥①④t➾ ➽ ❼✐❽❩❾❪❿❚➀❇Ï ➂❃Ð❲Ð✬Ñ✼➈⑩➇➉➪✏➘Ò➊⑩Ñ✏➪✏➇✹➟✜➅r➄❲➈Ó➘✆➄t➇❡➶ Ô ⑨❷➻ ②❧Õ➇✹➄✉Ð✬➌❃➶✷➍❍➄❲➇✹Ö❨➊✢➌t➇➉Ör➍ ⑦ ➡ ➽④ ↕✩➽ ➡ ➽④ ➡ ➽④t➾ ➽ × ⑨Ø⑧✈❶Ù➻ ❶ÛÚ Ü❲Ý➑ Ü Ý➑ ❶ ➽ ÜtÝ➑ Õ➅r➄❲Þ➞➊✢➄✉➶ × ⑨Ø⑧ ❶ ➽ Ü❲Ý➑ ß ➽ Ü❲Ý➑ à➍❍➪❩➌t➍❑➄t➍①➶ × ⑨Ø⑧ ❮ ➻ ➽ Ü❲Ý➑ ❶ Ü Ý➑ Ú Ü❲Ý➑ á➊✆➚✉â✞Þq➊r➄❲➶ ã✗❜❨❱✙P❲ä❩❙✢❥✑❵☞❝✹På❙å❞❍P✉æ✉❜❨❳✐♣ç❜❨❱t♣✆P✍❱✚è❲❙✆æ❲é✢♥✱❙❨❱❲♣❇♣✢❬ê❃P❍❱tP✍❳☞æ✉Pë❙✉❵r❵✐❱❲❜❍ä✆❬❪❥✥❙✢❫✖❬❧❜✢❳➺æ❲❙✢❳Ùè✉P✙♥✧❱♦❬❪❫✖❫❴P✍❳ ❙☛❞✔⑥✞ì④ ⑨í②✖î✢⑥①④✔❶çï✆⑥①④ ↕●➽ ❮ ⑥①④ ↕ Ü ⑦❲ð✷②✖ñ✆ò❦③✐⑦❍❻ ❼✐❽❩❾❪❿❚➀❇ó Ô ⑨✳ñ ② ➸➍①➚✍➅r➪✠➶ë➶✷➍❑➄t➇➉Ö❨➊❨➌❲➇✹Ö✆➍ ⑦ ➡Ü ④ ↕✩➽ ➡Ü ④ ➡Ü ④t➾ ➽ ➽ Ý➑①ô ❶ Ü Ý➑❑ô ➽ Ý➑❑ô à➍❑➪✆➌❲➍❍➄❲➍❑➶ ñ