Foundations — Polynomial interpolation — Lagrange form, Newton's divided differences

4.8.10 · D1 · Maths › Numerical Methods › Polynomial interpolation — Lagrange form, Newton's divided d

Is page pe assume kiya gaya hai ki aapko sirf itna pata hai: "$x$ times $x$ is $x^2$" aur "ek graph mein ek horizontal axis aur ek vertical axis hoti hai". Parent note ke har letter, subscript, product sign aur baaki sabhi symbols ko yahan, floor se, us order mein build kiya gaya hai jisme aapko unki zaroorat padegi.

0. Woh picture jisme hum hamesha hote hain

Koi bhi symbol aane se pehle: hamare paas ek graph paper ka tukda hai. Kuch dots marked hain. Hum ek rule chahte hain — ek function — jiska graph ek single connected curve ho jo har dot ko touch kare.

Amber dots dekho. Unse hokar infinitely many wiggly curves kheechi ja sakti hain, lekin ek chosen low degree ke liye sirf ek polynomial hoti hai. Yahi uniqueness is poore topic ka kaam karti hai — is baat ko yaad rakho.

1. Ek point aur uske coordinates: $(x_i,y_i)$

Picture: ek dot chuno. Horizontal axis ke saath right chalo jab tak us dot ke neeche na pahunch jaao — woh distance $x$ hai. Phir upar chalo — woh distance $y$ hai.

Topic ko iske liye zaroorat kyun: hamara data hi dots ki ek list hai, toh har dot ko ek naam chahiye.

Subscript $i$

Picture: imagine karo ki har dot se numbered tags latke hain: dot 0, dot 1, dot 2. Hum count 0 se shuru karte hain (ek computing ki aadat) taaki "$n+1$ dots" ko $0,1,\dots ,n$ number mile.

2. Dots ke peeche ka function $f$

Picture: $f$ ko ek sacchi chupi hui curve samjho; hamare dots woh jagahein hain jahan humne uski height pe nazar daali. Interpolation un jhalkiyon se $f$ ke baaki hisse ko guess karne ki koshish karti hai.

Topic ko iske liye zaroorat kyun: error formula aur divided differences $f[\dots ]$ dono is chupi hui $f$ ke baare mein baat karte hain — toh $f$ ko un symbols ke aane se pehle ek named object ke roop mein exist karna chahiye.

3. "Polynomial" aur "degree" ka matlab

Neeche ki picture dikhati hai ki degree kaise control karti hai ki curve mein kitne bends aa sakte hain.

Topic ko iske liye zaroorat kyun: pura subject inhi dials ko is tarah choose karne ke baare mein hai ki curve dots ko hit kare.

4. Greek letters aur "distinct" condition

Picture: ek hi vertical line mein stack ki hui do dots curve se demand karengi ki woh ek $x$ ke upar do alag heights pe ho — ek function ke liye impossible. Isliye nodes horizontally spread hone chahiye.

Topic ko iske liye zaroorat kyun: distinctness exactly wahi cheez hai jo solution ko unique banati hai (agla section) — Lagrange mein denominators $x_i-x_j$ kabhi zero nahi hote, aur Vandermonde determinant nonzero hota hai.

5. Sum $\sum$ aur product $\prod$ — do bade loops

Parent note do symbols pe bahut zyada depend karta hai. Yeh sirf shorthand hain "yeh kaam baar baar karo" ke liye.

Condition $\prod _{\begin{array}{c}j=0\\ j\ne i\end{array}}$ ka bas matlab hai "saare $j$ ke liye multiply karo sivaay $j=i$ ke". Picture: factors ki ek row jisme ek gap hai jahan $j=i$ hota.

Topic ko iske liye zaroorat kyun: Lagrange ka basis ek product hai; Newton ka polynomial products ka ek sum hai. Yeh do loops hi parent formulas ki poori notation hain.

6. Lagrange basis "switch" $L_i(x)$

Ab jab product sign mil gayi, hum $L_i$ symbol se mil sakte hain jise parent page (aur neeche ka map) use karta hai.

Yeh formula kaam kyun karta hai, do moves mein:

• Numerator ${\prod }_{j\ne i}(x-x_j)$ zero ho jaata hai jab bhi $x$ doosre kisi node pe land karta hai — isse "0 elsewhere" milta hai.
• ${\prod }_{j\ne i}(x_i-x_j)$ se divide karna (numerator ki value $x=x_i$ pe) use exactly 1 tak $x_i$ pe scale kar deta hai.

Topic ko iske liye zaroorat kyun: in switches ko stack karna, har ek ko apni height se scale karke, curve ko har dot se guzarne par majboor karta hai — yahi parent page pe Lagrange interpolant hai.

7. Vandermonde matrix aur uska determinant (kyun "unique")

Vandermonde grid ke liye yeh number saare nodes ke ordered pairs ke product ke roop mein nikalta hai: $\det ={\prod }_{i<j}(x_j-x_i).$

Topic ko iske liye zaroorat kyun: yeh non-zero product exactly "distinct nodes ⇒ ek aur sirf ek interpolating polynomial" ka statement hai.

8. Slope, secant slope, aur divided differences $f[\dots ]$

Newton ke divided differences slope ke idea pe built hain, §2 ke hidden function $f$ pe apply kiya hua.

Topic ko iske liye zaroorat kyun: yeh numbers $2,3,-2$ exactly parent page pe Newton ke coefficients hain.

9. Error toolkit: factorial aur derivatives

Error formula do aur symbols use karta hai. Aapko inhe bas yahan pehchanna hai; Taylor series inhe poori tarah develop karta hai.

Topic ko iske liye zaroorat kyun: saath mein $\frac{f^{(n+1)}(\xi )}{(n+1)!}$ product $\prod (x-x_i)$ ko true function $f$ aur $P(x)$ ke beech actual gap mein scale karta hai — Runge phenomenon yahi product hai jo interval ends ke paas badata hai.

10. Sab kuch kaise connect hota hai

Upar se neeche padho: dots ko naam dena, hidden function $f$, aur powers mila ke dials milte hain; distinct nodes Vandermonde ke non-zero determinant ke zariye dial-setting ko unique banate hain; do loop-symbols us unique curve ko Lagrange switches $L_i$ ya Newton brackets $f[\dots ]$ ke roop mein dress karte hain; slopes Newton ko feed karte hain; factorial aur derivatives mystery point $\xi$ pe bachey hue error ko naapte hain.

Equipment checklist

Right side cover karo aur khud test karo.

Connections

• Parent topic
• Vandermonde matrix — woh grid jo uniqueness guarantee karta hai.
• Runge phenomenon — ends ke paas error product kya karta hai.
• Chebyshev nodes — smarter node placement.
• Cubic splines — piecewise low-degree alternative.
• Numerical differentiation — derivatives kahan se aate hain.
• Newton-Cotes quadrature — interpolant ko integrate karna.
• Taylor series — factorials aur derivatives, poori tarah develop kiye.