Adaptive step-size — RK45, error control: Visual walkthrough

4.8.25 · D2 · Maths › Numerical Methods › Adaptive step-size — RK45, error control

Step 1 — Solution ek curve hai, aur stepping matlab agla dot guess karna

KYA. Hum ek equation solve kar rahe hain jiska form hai $y^{'} = f (t, y)$ . Ise zor se padho: "curve ki slope kisi bhi point par ek rule $f$ se milti hai." Toh $f$ ek slope machine hai — usse ek time $t$ aur height $y$ do, woh tumhe bata dega ki wahan curve kitni steep hai.

YE KYUN. Baad mein har symbol ( $k_{i}$ , $ε$ , $h$ ) ek cheez ke baare mein hai: true curve par ek jaane-pehchane dot ko lena aur guess karna ki curve aage kahan jaati hai. Agar tum us dot ko picture mein nahi dekhte, toh baki kuch bhi samajh nahi aayega.

PICTURE. Figure dekho. Safed curve true solution hai (jo hum jaante nahi). $t_{n}$ par amber dot wahan hai jahan hum abhi hain, height $y_{n}$ par. Letter $h$ horizontal step hai — $t$ mein hum kitna aage jump karte hain. Hamara kaam: $t_{n} + h$ par height estimate karna.

Figure — Adaptive step-size — RK45, error control

Step 2 — Ek method answer deta hai lekin honesty check nahi

KYA. Ek Runge–Kutta method step ke andar kuch clever points par slope machine ko sample karta hai, phir un slopes ka weighted average lekar aage leap karta hai: $y_{n + 1} = y_{n} + h \sum_{i = 1}^{s} b_{i} k_{i} .$

Term by term:

$k_{i}$ — $i$ -wa slope sample, ek number jo slope machine $f$ ne step mein kisi point par diya.
$b_{i}$ — ek fixed weight jo batata hai ki slope sample $k_{i}$ par kitna trust karna hai.
$h \sum b_{i} k_{i}$ — total rise: "step width times average slope."

YE KYUN KAAFI NAHI. Figure dikhata hai estimated dot $y_{n + 1}$ true dot ke paas; dono mein gap hai — true curve aur hamara guess alag hain. Lekin method kabhi batata nahi gap kitna hai. Humne ek number compute kiya aur nahi pata kitna galat hai. Woh silent gap hi poora problem hai.

PICTURE. Cyan dashed leap $y_{n + 1}$ par land karti hai; amber true dot thoda upar baitha hai. Dono ke beech vertical amber bar local error hai — real, lekin method ko invisible.

Step 3 — Ek hi slopes se do answers gap reveal karte hain

KYA. Yeh hai "embedded pair" trick. Wahi slope samples $k_{i}$ rakho, lekin unhe do alag weight rows se average karo: $y^{(5)} = y_{n} + h \sum_{i} b_{i} k_{i} (order 5), y^{(4)} = y_{n} + h \sum_{i} b_{i}^{*} k_{i} (order 4) .$

$b_{i}$ — weights jo sharper (order-5) estimate dete hain.
$b_{i}^{*}$ — weights jo rougher (order-4) estimate dete hain.
Dono ek hi $k_{i}$ use karte hain — slope machine ke koi naye calls nahi.

Unka difference hamara free error estimate hai: $ε = y^{(5)} - y^{(4)} = h \sum_{i} (b_{i} - b_{i}^{*}) k_{i} .$

YE KYUN KAAM KARTA HAI. "Order" measure karta hai ki jab $h$ shrink hota hai toh error kitni tezi se shrink hoti hai. Order-5 dot true curve ke bahut zyada paas rehta hai order-4 dot se. Toh rough dot se sharp dot tak ki doori basically rough dot se truth tak ki doori hai — yaani order-4 error. Humne do visible dots se invisible gap measure kiya.

PICTURE. Do cyan dots paas paas land karte hain; amber true dot lagbhag order-5 dot ke upar hai. Dono cyan dots ke beech bracket $ε$ hai — chhota, sasta, aur honest.

Recall "Free" kyun?

Costly kaam hai $f$ ko call karna har $k_{i}$ ke liye. ::: Dono answers saare $k_{i}$ reuse karte hain; doosra answer sirf ek aur weighted sum hai — ek sasta dot product, koi extra $f$ calls nahi.

Step 4 — Gap kaise step size ke saath badhta hai: $h^{p + 1}$ law

KYA. Ek order- $p$ method ke liye, local error obey karta hai $ε (h) \approx C h^{p + 1}, p = 4 controlled RK45 result ke liye.$

$p$ — method ka order (yahan 4).
$p + 1 = 5$ — $h$ par power; local error shrink hoti hai ek power tezi se order se.
$C$ — ek unknown constant jo is point par curve ki shape mein baka hua hai. Hum kabhi iski value nahi jaanenge — aur jaanne ki zaroorat bhi nahi padegi.

YE EXACT POWER KYUN. Aata hai Taylor Series Expansion se: method true curve ki Taylor series ko $h^{p}$ term tak match karta hai, toh pehla surviving mismatch $h^{p + 1}$ term hai. Har lower term design se cancel ho jaata hai.

PICTURE. Error versus $h$ ka log–log plot. Relationship $ε = C h^{5}$ ek slope 5 ki straight line ban jaati hai. $h$ aadha karne se error $2^{5} = 32$ se girta hai. Woh steepness woh lever hai jo hum abhi pull karne wale hain.

Step 5 — Ratio lekar unknown constant cancel karo

KYA. Hamare paas ek hi curve ke baare mein do facts hain: $ε = C h^{p + 1} (hamara current step), tol = C h_{new}^{p + 1} (woh step jo hum CHAHTE hain) .$

Doosri line ek wish hai: "woh step dhundho jiska error exactly tolerance $tol$ par land kare." Wish ko reality se divide karo: $\frac{tol}{ε} = \frac{C h _{new}^{p + 1}}{C h ^{p + 1}} = (\frac{h _{new}}{h})^{p + 1} .$

DIVIDE KYUN. Hum $C$ nahi jaante aur ise measure nahi kar sakte. Dono equations divide karne se $C$ cancel ho jaata hai — wahi unknown upar aur neeche dono mein hai. Jo bachta hai woh sirf un quantities par depend karta hai jo hamare paas hain: $tol$ , $ε$ , aur $h$ .

PICTURE. Step 4 ki usi log–log line par, do points mark karo: hamara current $(h, ε)$ aur target $(h_{new}, tol)$ . Dono ek straight line par hain — same $C$ , same slope. Ratio bas us line par slide karna hai.

Step 6 — $h_{new}$ unlock karne ke liye fifth root lo

KYA. Power $p + 1$ undo karo dono sides ko $1/ (p + 1)$ se raise karke: $(\frac{h _{new}}{h})^{p + 1} = \frac{tol}{ε} ⟹ \frac{h _{new}}{h} = (\frac{tol}{ε})^{1/ (p + 1)} .$

$h$ se multiply karo: $h_{new} = h (\frac{tol}{ε})^{1/ (p + 1)}, p + 1 = 5.$

$tol / ε > 1$ → hamare paas error ki gunjaaish thi → $h$ grow karo.
$tol / ε < 1$ → error bahut bada → $h$ shrink karo.
Fifth root change ko control karta hai: 32 ka ratio $h$ ko sirf $3 2^{1/5} = 2$ se grow karta hai, Step 4 ki slope-5 line ke saath match karta hua.

ROOT KYUN, RAW RATIO NAHI. Kyunki error $h$ par $h^{5}$ jaisa respond karta hai, $h$ mein chhoti si nudge error mein badi nudge hai. Error ko target par land karne ke liye hume $h$ ko sirf error ratio ke fifth root se move karna hoga — warna hum bahut zyada overshoot kar denge.

PICTURE. Ek dial: input "error ratio" needle $0.03$ se $32$ tak bahut zyada swing karta hai, lekin output " $h$ multiplier" needle muskil se hilta hai ( $0.5$ se $2$ ). Fifth root woh gearbox hai jo response ko gentle karta hai.

Step 7 — Safety factor aur clamps (overconfidence se bachav)

KYA. Raw formula assume karta hai ki $C h^{5}$ exact hai. Nahi hai — $ε$ khud ek estimate hai. Toh real controller padded hota hai: $h_{new} = h \cdot min (f_{m a x}, max (f_{m i n}, S (tol / ε)^{1/5})) .$

$S \approx 0.9$ — safety factor: tol se thoda neeche aim karo taaki retried step usually pehli baar pass ho.
$f_{m i n} \approx 0.2$ — ek baar mein kabhi bhi $5 \times$ se zyada mat shrink karo.
$f_{m a x} \approx 5$ — ek baar mein kabhi bhi $5 \times$ se zyada mat grow karo.

CLAMP KYUN. $h^{5}$ law sirf current $h$ ke paas trustworthy hai. Predicted $14 \times$ jump hume aisi jagah phenk deta jahan local model ne kabhi nahi dekha. Clamping har jump ko model ke valid zone ke andar rakhta hai.

PICTURE. Raw multiplier curve (cyan) vs clamped one (amber): $f_{m a x} = 5$ par flat ceiling, $f_{m i n} = 0.2$ par flat floor, aur har jagah $S$ se thoda neeche shift.

Step 8 — Poora loop: accept, grow, reject, shrink

KYA. Sab pieces ko us decision mein assemble karo jo algorithm har step par leta hai:

$k_{i}$ ek baar banao; $y^{(5)}$ , $y^{(4)}$ , aur $ε = ∥ y^{(5)} - y^{(4)} ∥$ form karo.
$tol = atol + rtol ∣ y_{n} ∣$ compute karo.
Agar $ε \leq tol$ : accept karo, $t \to t_{n} + h$ advance karo, phir Step 7 se agla $h$ set karo (usually bada).
Agar $ε > tol$ : reject karo, $t = t_{n}$ rakho, Step 7 se $h$ shrink karo, retry karo.

YE ROAD-DRIVING IDEA SE KYUN MATCH KARTA HAI. Do cameras (orders 4 aur 5) $ε$ se disagree karte hain; agar woh barely differ karte hain toh road seedha hai (grow $h$ , floor it); agar agree nahi karte toh turn sharp hai (shrink $h$ , brake).

PICTURE. Loop ka ek flowchart do branches colour-coded ke saath: cyan "accept → advance & grow," amber "reject → stay & shrink."

Worked check — dial par teen parent examples

Ek-picture summary

Upar sab kuch ek diagram mein collapse hota hai: log–log error line (slope $p + 1 = 5$ ) tumhe tumhare measured point $(h, ε)$ se target point $(h_{new}, tol)$ tak le jaati hai, fifth-root gearbox move ko gentle karta hai, aur accept/reject branch decide karta hai ki tum aage badhte ho ya retry karte ho.

Recall Feynman: poora walkthrough simple shabdon mein

Tum ek chhupe winding road par ek dot steer kar rahe ho. Har baar tum agle spot ke do guesses lete ho — ek rough aur ek sharp — road ki ek hi jhaanki use karke, toh doosra guess basically free hai. Dono guesses kitni door hain woh tumhara error $ε$ hai. Tum jaante ho ki agar tum step aadha karo toh error $32 \times$ shrink hoti hai, kyunki error ek slope-5 line follow karti hai. Toh apni error ko allowed budget $tol$ par exactly hit karne ke liye, tum us line par slide karte ho — aur kyunki line itni steep hai, tum apna step sirf ratio $tol / ε$ ke fifth root se nudge karte ho. Phir tum safe khelte ho: $0.9$ se multiply karo, aur step ko kabhi bhi kisi bhi direction mein $5 \times$ se zyada jump mat karne do. Agar dono guesses kaafi agree karte hain, tum accept karte ho aur speed up karte ho; agar nahi, tum ruk jaate ho aur slow down karte ho. Woh ek rule — "tol over error, fifth root mein, clamped" — hi poora RK45 step controller hai.

Connections

Runge–Kutta Methods (RK4) — fixed-step method jiske stages $k_{i}$ hum reuse karte hain.
Order of a Numerical Method — woh $p$ define karta hai jo fifth-root exponent banta hai.
Local vs Global Truncation Error — kyun local error extra $h^{p + 1}$ power carry karta hai.
Taylor Series Expansion — leading $C h^{p + 1}$ error term ki origin.
Dormand–Prince (RK45 used by scipy solve_ivp) — actual $b_{i}, b_{i}^{*}$ weight rows.
Stiff ODEs and Stability — jahan yeh clever controller bhi struggle karta hai.
← Parent topic par wapas