Foundations — Cubic spline interpolation — natural, clamped

4.8.12 · D1 · Maths › Numerical Methods › Cubic spline interpolation — natural, clamped

Parent note padhne se pehle, tumhe page par har ek mark bina jhijhke padhna aana chahiye. Neeche har ek symbol aur concept ka zero-se tour hai, is order mein ki har ek cheez sirf usse pehle waali cheez par rely kare. Agar parent note ne use likha aur maan liya ki tum jaante ho — toh woh yahan build kiya gaya hai.

1. Points, aur subscript trick

Mark $x_i$ ko "x-sub-i" padha jaata hai aur iska seedha matlab hai "point number $i$ ki $x$-value". Toh $x_0$ pehla hai, $x_1$ uske baad, $x_n$ tak (jo last hai). Letter $n$ last point ka number hai, isliye total mein $n+1$ points hain (kyunki hum $1$ se nahi, $0$ se ginne lage the).

Figure dekho: dots $(x_i,y_i)$ hain. Condition $x_0<x_1<\cdots <x_n$ (parent bilkul yahi likhta hai) bas yeh kehti hai ki points strictly left se right ki taraf jaate hain — koi bhi do ek $x$ share nahi karte, aur koi bhi backwards nahi jaata. Yahi ordering hai jo humein "neighbours ke beech ka interval" bolne deti hai.

2. Interval aur uski width $h_i$

Picture: $h_i$ do neighbouring dots ke beech ka horizontal gap hai (figure s01 dekho, neeche orange bracket). Topic ko yeh kyun chahiye: har formula $h_i$ se divide karta hai — kyunki slope ya curvature hamesha "width ke per unit kitna change" hota hai, aur tum width jaane bina per-width change measure nahi kar sakte.

3. Cubic kya hota hai (aur cubic kyun)

Picture: ek single smooth wiggle jo upar phir neeche ja sakti hai (ya ulta) — figure s02 dekho. Ek straight line ($x^1$) bend nahi kar sakti; ek parabola ($x^2$) sirf ek taraf bend hoti hai; ek cubic bend, straighten, aur phir se bend ho sakti hai — itni flexible ki do heights ko smoothly link kare aur joins par slope aur curvature bhi control kare.

4. Function notation $S(x)$ aur piecewise idea

Toh $S$ piecewise hai: alag-alag stretches par alag-alag formula, lekin overall ek continuous curve. Parent likhta hai $S_i(x_i)=y_i$ — ise is tarah padho: "piece $i$, left node $x_i$ par evaluate kiya, data height $y_i$ deta hai." Yahi interpolation condition hai: curve ko dots se exactly guzarna chahiye.

5. Slope: first derivative $S^{\prime }$

Picture: curve par ek point par tangent rakhe gaye chhote straight ruler ki tilt (figure s03, magenta tangent). Humein isse kyun chahiye: agar do pieces ek node par milti hain lekin unki slopes alag hain, toh curve mein ek visible kink hoga — jaise chhath ka corner. $S_{i-1}^{\prime }(x_i)=S_i^{\prime }(x_i)$ demand karna (slopes left aur right dono se agree karein) kink ko mita deta hai. Yahan pieces "ek doosre se baat" karti hain.

6. Curvature: second derivative $S^{\prime \prime }$

Picture: figure s03 do arcs dikhata hai — ek gently curved (chhota $|S^{\prime \prime }|$) aur ek sharply curved (bada $|S^{\prime \prime }|$). Humein isse kyun chahiye: slope match karna tilt mein kinks hataata hai, lekin phir bhi ek subtle "bendiness jump" ho sakta hai. $S^{\prime \prime }$ bhi match karna curve ko fair banata hai — aankhein curvature mein abrupt change nahi dekhti. $S$, $S^{\prime }$, aur $S^{\prime \prime }$ teeno ko continuous demand karna bilkul wahi hai jo parent ==$C^2$== kehta hai.

7. Moment $M_i$ — end-curvature ka nickname

$M_i$ ke saath kaam kyun karein? Kyunki ek single interval par, $S^{\prime \prime }$ ek straight line hai (ek cubic ka second derivative linear hota hai). Ek straight line completely uske do endpoint values $M_i$ aur $M_{i+1}$ se pin hoti hai. Toh chaar coefficients per piece ke peeche bhaagne ki bajaye, hum sirf ek number per node track karte hain — bahut kam unknowns, aur woh ek solvable chain mein neatly link up ho jaate hain.

8. Slopes-of-slopes: right-hand side discrete curvature hai

Parent ki master equation ka right side hai $6\left(\frac{y_{i+1}-y_i}{h_i}-\frac{y_i-y_{i-1}}{h_{i-1}}\right).$ Inner pieces padho: $\frac{y_{i+1}-y_i}{h_i}$ "rise over run" hai = right interval par average slope; doosra term left interval par average slope hai. Unka difference hai node $i$ cross karte waqt slope kitna badla — curvature ka ek raw, data-only estimate. Yeh "difference of differences" idea Finite differences ki duniya hai. $6$ ek constant hai jo cubic ko do baar integrate karne se nikalta hai; tum ise yahan faith par le sakte ho.

9. Boundary conditions — do missing facts

Parent $4n$ coefficient-dials count karta hai lekin interpolation + smoothness se sirf $4n-2$ equations milti hain, system ko 2 short chodke. Do aur facts do free ends par supply karne padte hain:

Dono missing two rows ki tarah plug in hote hain. "Natural" word = No curvature; "clamped" = Controlled slope.

10. Tridiagonal system — equations ki shape

Kyun care karein: ek tridiagonal system Tridiagonal systems & Thomas algorithm se bahut fast aur stably solve hota hai — heavy general matrix methods ki zaroorat nahi. Yahi ek bada practical reason hai ki splines compute karna sasta hai.

Prerequisite map

Ise neeche ki taraf padho: points se widths aur cubic pieces milti hain; cubics se slope, phir curvature, phir moments; smoothness plus moments plus boundary facts tridiagonal system assemble karte hain; ise solve karne se spline milti hai.

Equipment checklist

Answers cover karo aur check karo ki parent padhne se pehle tum memory se har ek produce kar sakte ho.