Foundations — Newton-Raphson method for root finding

4.1.33 · D1 · Maths › Calculus I — Limits & Derivatives › Newton-Raphson method for root finding

Parent note ke saath chalne se pehle, tumhare paas uske har ek symbol ki poori pakad honi chahiye bina kisi jhijhak ke. Yeh page har ek symbol ko bilkul zero se list karta hai — seedha matlab, woh picture jisme woh rehta hai, aur kyun method uske bina kaam nahi kar sakta.

1. Do axes: $x$ aur $y$

• Plain words: $x$ = "kitna across", $y$ = "kitna upar".
• Picture: graph paper jisme ek horizontal line ($x$) aur ek vertical line ($y$) origin $(0,0)$ par cross karti hain.
• Yeh topic kyun chahiye isse: neeche har ek curve, tangent, aur root is $x$–$y$ plane par rehti hai, isliye hum shuru mein hi dono labels fix kar lete hain, inke aane se pehle.

2. Function $f$ aur equation $f(x)=0$

• Plain words: "$x$ andar jaata hai, ek height $y$ bahar aati hai."
• Picture: $x$-axis ke upar ek tedhi-medhi curve; $x$ ke har value par uski upar-neeche ki position woh $f(x)$ value hai jo $y$-axis se padhi jaati hai.
• Yeh topic kyun chahiye isse: poora game hai "$f(x)=0$ solve karo". Bina function ke zero karne ke liye kuch hai hi nahi.

"$=0$" kyun, "$=5$" kyun nahi? Kisi bhi equation ko rearrange kiya ja sakta hai taaki ek side zero ho. $g(x)=5$ solve karna waisa hi hai jaisa $f(x)=g(x)-5=0$ solve karna. Toh "curve kahan axis se milti hai dhundhna" har equation ki universal shape hai. Isliye method specifically zero ko target karta hai.

3. Guess $x_n$ aur subscript $n$

• Plain words: $x_0,x_1,x_2,\dots$ guesses ki ek list hai, har ek hopefully pehle se behtar.
• Picture: $x$-axis par dots ki ek row root ki taraf kheenchti hui.
• Yeh topic kyun chahiye isse: Newton-Raphson ek repeated process hai. Hume "current guess $x_n$" aur "next guess $x_{n+1}$" ke baare mein baat karne ke liye names chahiye. Counter ek formula ko har step describe karne deta hai ek saath.

4. Height: $f(x_n)$ — zero se kitne door hain hum

• Picture: axis par guess $x_n$ se curve tak ek vertical line upar (ya neeche). Us segment ki signed length hai $f(x_n)$.
• Yeh topic kyun chahiye isse: yahi woh cheez hai jise hum zero tak drive karna chahte hain, aur yeh formula mein $f/f^{\prime }$ fraction ke top par aata hai.

5. Slope, limit, aur derivative $f^{\prime }(x_n)$

Yeh woh tool hai jis par parent note sabse zyada depend karta hai, isliye hum ise poori care se build karte hain.

• Picture: line par sawaar ek right triangle — horizontal leg $\Delta x$, vertical leg $\Delta y$. Ek steep line ka triangle lamba hota hai; ek flat line ka chota.

Lekin ek curve seedha nahi hota — uski steepness jagah-jagah badlati hai. Toh ek single point par the slope kya hai?

• Plain words: "$\Delta x$ ko aur chota karte raho aur dekho kaunsa number answer approach karta hai."
• Picture: curve par doosra point pehle ki taraf slide ho raha hai; unke beech ki chord ek settled tilt ki taraf jhuk rahi hai.

• Plain words: derivative jawaab deta hai "exactly is jagah curve kitni steep hai?"
• Yeh topic kyun chahiye isse: Newton-Raphson curve ko tangent line se replace karta hai, aur ek line tab fix hoti hai jab tum ek point plus uski slope jaante ho. $f^{\prime }(x_n)$ wahi slope hai. Yeh fraction $f/f^{\prime }$ ke bottom par aata hai.

6. Tangent line — method ka star

• Picture: ek ruler curve ke against ek point par chhookar rakhaa gaya, curve ki lean se match karta hua.
• Yeh topic kyun chahiye isse: touch point ke paas tangent aur curve almost identical hote hain (yeh hai Tangent line and linear approximation). Isliye tangent ka apna zero curve ke zero ka ek badhiya stand-in hai — aur ek line ka zero exactly dhundhna aasaan hai.

7. Line zero kahan hit karti hai — aur Newton update

Tangent $y-f(x_n)=f^{\prime }(x_n)(x-x_n)$ lete hue aur $y=0$ rakhte hue: $0-f(x_n)=f^{\prime }(x_n)(x_{n+1}-x_n)\Rightarrow x_{n+1}-x_n=-\frac{f(x_n)}{f^{\prime }(x_n)}.$

Ab guess ko akele paane ke liye dono sides mein $x_n$ add karo — yeh standard Newton-Raphson update hai jo parent note use karta hai: $\boxed{x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}}$

• Plain words: ratio $\frac{f(x_n)}{f^{\prime }(x_n)}$ woh horizontal distance hai jo tangent height $f(x_n)$ se axis tak drop karne ke liye travel karta hai. Height divided by steepness = kitna sideways jaana hai zero tak pahunchne ke liye. Naya guess hai "purana guess minus woh jump".
• Minus sign kyun? Agar curve axis ke upar hai ($f>0$) aur upar ja raha hai ($f^{\prime }>0$), toh zero tak pahunchne ke liye tumhe left (backward) step karna hoga — minus yahi karta hai. Har sign combination isi tarah kaam karta hai; test karo:

8. Tolerance $\epsilon$ aur stop rule

• Picture: width $\epsilon$ ka ek patla band root ko hug karta hua; jab tumhara dot band ke andar aa jaata hai, tum quit karte ho.
• Yeh topic kyun chahiye isse: true root mein usually infinite decimals hote hain, isliye hum kabhi exactly uski par land nahi kar sakte. $\epsilon$ process ko batata hai ki "practically wahan" kab done count hota hai.

9. True root $r$, aur $O((\text{error}{)}^2)$

• Picture: axis par ek fixed green mark (woh hai $r$); tumhare marching dot $x_n$ se us mark tak ki doori error hai.

• Plain words: agar tumhara current error $0.01$ hai, toh agla error lagbhag $(0.01{)}^2=0.0001$ hoga — sahi digits ki sankhya roughly double hoti hai har step mein.
• Yeh topic kyun chahiye isse: yeh poore method ka payoff line hai, Taylor series ke zariye fully explained. Yahan tumhe sirf symbol padhna hai, ab jo $r$ ka naam aa gaya hai.

10. Method kab trustworthy hai? (convergence conditions)

• Yeh topic kyun chahiye isse: yeh har "it converges" claim ke fine print hain. Parent note ka misbehaving Example 3 condition 3 todata hai; uski double-root warning condition 2 hai; uski "guard against tiny $f^{\prime }$" advice condition 2 aur Section 7 ke flat-tangent case ko protect karti hai.

Prerequisite map

Equipment checklist

Khud ko test karo — right side cover karo aur jawab zor se dو.

Connections

• Tangent line and linear approximation — tangent method ka poora engine hai.
• Derivatives — definition and rules — slope $f^{\prime }$ actually compute kaise hota hai.
• Taylor series — jahan se quadratic convergence aur $O(\cdot )$ symbol aata hai.
• Fixed-point iteration — "guess repeat karo" pattern generalized.
• Bisection method — safe fallback jab $f^{\prime }\approx 0$ ho.
• Roots of polynomials — ek common jagah jahan se yeh functions aate hain.