Foundations — Error in polynomial interpolation

4.8.11 · D1 · Maths › Numerical Methods › Error in polynomial interpolation

Is page pe assume kiya gaya hai ki abhi aap kuch nahi jaante. Error formula padhne se pehle $E(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}{\prod }_{i=0}^n(x-x_i),$ aapko isme har ek symbol ka malik hona chahiye. Hum unhe ek ek karke build karte hain, har ek apni jagah banata hai agle se pehle. Jab aap finish kar lein, Error in polynomial interpolation padh lena.

1. $f(x)$ — woh true function jo hum poori tarah nahi jaante

Picture. Graph paper pe ek smooth curve draw karo. Har horizontal position $x$ ke liye, curve ki ek height hoti hai — woh height $f(x)$ hai.

Topic ko iski zaroorat kyun hai. Interpolation ka matlab hai is curve ko guess karna jab hum sirf kuch heights jaante hain. Agar "true curve" hi nahi hogi toh galat hone ka sawaal hi nahi uthega.

2. Nodes $x_0,x_1,\dots ,x_n$ — woh dots jo hum actually jaante hain

Picture. Usi curve pe kuch dots laga do. Har dot ek jaani-poochhi horizontal position $x_i$ aur jaani-poochhi height $f(x_i)$ pe baitha hai. Dots ke beech ki jagah unknown territory hai.

Topic ko iski zaroorat kyun hai. Yeh dots hi hamare paas ek maatra pakke facts hain. Poori error story unke beech ke gaps ke baare mein hai.

3. Distinct — dots ko stack nahi karna chahiye

Picture. Ek hi horizontal spot pe do dots hona ek dot ka do honge ka natak hoga — aap usme se ek unique curve fit nahi kar sakte. Unhein sideways spread karo.

Topic ko iski zaroorat kyun hai. Distinctness hi yeh guarantee karta hai ki degree $\le n$ wala exactly ek polynomial dots se guzarega. Us polynomial ko banane ke do standard tarike dekhne ke liye Lagrange Interpolation aur Newton Divided Differences dekho.

4. Polynomial $p_n(x)$ aur "degree"

Picture. Degree $0$ ek flat horizontal line hai. Degree $1$ ek seedhi tirhee line hai. Degree $2$ ek single U-shaped bend hai (parabola). Zyada degree $=$ zyada allowed wiggles.

Topic ko iski zaroorat kyun hai. $p_n$ haara guess hai: degree $\le n$ wala unique polynomial jo saare $n+1$ dots se guzarta hai. Error is guess ko sach se compare karta hai.

5. $E(x)$ — error, "sach minus guess"

Picture. Har $x$ pe, guess curve se true curve tak upar (ya neeche) ek vertical stick draw karo. Us stick ki sign-wali length $E(x)$ hai. Har node pe dono curves milte hain, toh stick ki length zero hai: $E(x_i)=0$.

Topic ko iski zaroorat kyun hai. Yahi woh quantity hai jo parent page measure karta hai. Baaki sab kuch in vertical sticks ki shape describe karne ke liye hai.

6. The product ${\prod }_{i=0}^n(x-x_i)$ — dots se doori

$(x-x_i)$ ko ek signed distance ki tarah padho. Agar $x$ node $x_i$ se door hai, toh $(x-x_i)$ size mein bada hai; agar $x$ khud hi node hai, toh $(x-x_i)=0$.

Picture. $\omega$ khud ek aisa polynomial hai jo har dot pe zero tak dip karta hai aur gaps mein bulge karta hai. Har gap mein ek hump banata hai; sabse bade humps interval ke ends ke paas hote hain.

Topic ko iski zaroorat kyun hai. $\omega (x)$ woh "dots se kitna door" knob hai jo aap nodes ki placement choose karke control karte ho — yahi reason hai ki Chebyshev Nodes evenly spaced nodes se behtar hote hain aur Runge Phenomenon se bache rehte hain.

7. Derivatives $f^{\prime }$, $f^{\prime \prime }$, aur $f^{(n+1)}$ — twistiness measure karna

Yahaan achanak derivatives kyun chahiye. Jitna seedha true curve hoga, utna aasaan hoga dots ke beech guess karna; jitna zyada bend karega, utna zyada yeh hamare polynomial se dots ke beech door bhaag sakti hai. "Bending" exactly wahi hai jo higher derivatives measure karte hain — isliye error mein koi aur tool nahi balki ek derivative aata hai.

Specifically $(n+1)$-th derivative kyun? $n+1$ dots ke saath, hamar guess $p_n$ pehle se hi sach se "$n+1$ pieces of information" mein match karta hai. Pehli cheez jise woh control nahi kar sakta woh hai $(n+1)$-th derivative — isliye wahi exactly woh leftover twistiness hai jo error mein leak hoti hai. Yeh Taylor's Theorem with Remainder se mirror karta hai, jahaan leftover term bhi next derivative use karta hai.

8. $\xi$ — mystery sample point

Picture. Sabse left aur sabse right dots ke beech kaheen ek hidden spot $\xi$ hai jiska bending value $f^{(n+1)}(\xi )$ formula ko exactly sahi banata hai. Yeh har test point $x$ ke liye alag jagah chhup jaata hai, toh actually $\xi =\xi (x)$ hai.

Topic ko iski zaroorat kyun hai — aur yeh kahaan se aata hai. Yeh existence guarantee hume Rolle's Theorem deta hai (Mean Value Theorem ka woh special case jahaan endpoints ki height equal hoti hai). Rolle kehta hai: equal height wale do points ke beech, slope kahin zero hoga — lekin exactly kahaan nahi batata. Woh "kahin" hi haara $\xi$ hai.

9. Factorial $(n+1)!$ — softening denominator

Picture. Ek aisi staircase jo jaldi steep ho jaati hai: $1,2,6,24,120,720,\dots$ Yeh kisi bhi fixed power se zyada tezi se badhti hai.

Topic ko iski zaroorat kyun hai. Error formula mein $(n+1)!$ denominator mein baitha hai, error ko chhota karta hai. Yeh derivation mein naturally aata hai: $\omega$ ki sabse unchi power ko exactly $n+1$ baar differentiate karne se $(n+1)!$ nikalta hai. Ise miss karo aur aapka bound wildly galat hoga.

10. $C^{n+1}[a,b]$ — smoothness ticket

Picture. Ek aisi curve jise aap pen uthaye bina draw kar sako, chahe use kaafi baar bend kiya gaya ho, koi sharp corners nahi.

Topic ko iski zaroorat kyun hai. Error formula secretly $n+1$ baar differentiate karta hai. Agar woh derivatives exist hi na karti ya idhar-udhar jump karti, toh poora argument collapse ho jaata. $C^{n+1}[a,b]$ woh fine print hai jo machinery ko chalane deta hai.

Yeh topic ko kaise feed karte hain

Ise upar se neeche padho: true function aur uske jaane-poochhe dots guess aur error banate hain; distances $\omega$ deti hain; derivatives twistiness deti hain; Rolle $\xi$ deta hai; factorial aur smoothness ticket formula finish karte hain.

Equipment checklist

Right side cover karo aur reveal karne se pehle har ek ka jawab do.

Jab har reveal instant ho jaaye, Error in polynomial interpolation padh lena.