Question

Math: Numerical Analysis

Write a summary of what is learn on this topic with their subtopics: Topic: Interpolation and Approximation: Subtopic: Polynomial interpolation: 1. Linear 2. Quadratic 3. Higher Order 4. Divided Differences homii 5. Properties of Divided Differences 6. Interpolation of Divided Differences

I want a summary of the Theme with their subtopics

Answer 1

Aerpolatin Apnimatim IntoPALANm The ndin thoug girtn dala caled intrpo latn is weiers tras App.ouimaha Theortm Suppose that t duhned CA, blFr any t0, anhnuos m Pln, dehined there enish ly no mial a Property t ta,63 wih he in ,b) 14(N)-P)) E, 1. Liner 9ntupolahn 9t involwes hneg atjacar a nen valine b annechng 2 tnoun valweus Sraigit ne let (Mo,y) A (*I,41) Then the valuu alory the to Dedinades

Straijtt lie given by the es y y, y ) (n,- ) t0 rmale for linear intrp- aterval Thi is he - lah'in n(he Qualrahe erp. laha: The urdralie tayange inle- Polahig polynmid thr the point 3. ishr ordur 9ntpolatin: Mn91)1 Let tn,y) sin tin poin arc nt

then Lagranges Pormuh he foly no mid inkolah an 4 hne pointais PA (x4 Lo ()t y,IlL) 4n Ln(1) Divided Dift evenca 9tls a ricursine Aiviim used hr tomputg logarina Proc em trlynomeh tas Zemth diwded dit firr aidel dif tDi, 1in) Cai1J-t ) itl

th divided tren a: t[1i,Min s. Propertiy t Oividel DHeren n. al tinedi Lineari ty: teib niz kule Cb) C.. +Co, ..M.9[7) us

(c) rtled difuen ces art symmehc permuh n'in hen - 1 - In) lo the Valne Meo rem diniked diffiren it othws hat Gin Whurt du tmind by te he un inKral Larget the 's Smallers 6 nterpolafnt Dinded Diteren cn New An's dinkl ieenu mtpo- 1a tplahn - lahan fmula teeknige med whn he atral ditt as4un u t values

Let tH1, H), 4(1). .. fo be he (nt valvs t the An chm n) Angume a ts w here mval differtus n are Nt intiplahen dinded Nents n's rln om ula is given by. 7 (1-1,)(1-X1) ,1,, (1-M)(-1,) . . (1 -F,, 1 oy -- M ]