Adaptive step-size — RK45, error control: Foundations

4.8.25 · D1 · Maths › Numerical Methods › Adaptive step-size — RK45, error control

Yeh page ek toolbox hai. Parent note padhne se pehle, tumhe har wo symbol khud se samajhna hoga jo wahan aata hai. Hum har ek ko zero se banate hain — seedhe words mein, phir ek picture, phir topic ko yeh kyun chahiye — aur is order mein jahan har tool pichle par tikha ho.

1. Differential equation $y^{'} = f (t, y)$ asal mein kehti kya hai

Toh $y^{'} = f (t, y)$ ek sentence hai: "Har point $(t, y)$ par, yeh hai jaane ki disha." Yeh tumhe curve nahi deta; yeh tumhe har point par ek arrow deta hai, aur tumhe curve trace karne ke liye arrows ke saath chalna padta hai.

Figure — Adaptive step-size — RK45, error control

Topic ko yeh kyun chahiye. RK45 ek machine hai jo un arrows ke saath sahi tarike se chalne ke liye hai. Baad ke har symbol ( $k_{i}$ , $h$ , $ε$ ) arrow field mein ek achha step lene ke baare mein hai. Agar field nahi dikhti, toh kuch bhi samajh nahi aata.

Recall Check:

$f$ kya deta hai? $f (t, y)$ return karta hai... ::: point $(t, y)$ par solution ki slope (steepness) — solution khud nahi.

2. Step size $h$ — ek hop ki length

"Chhota" kyun? Arrow field bend kar sakta hai. Agar tum ek seedhe arrow ke saath bahut door tak chalo toh tum true curve se bhatak jaate ho. Ek chhota hop sach ke kareeb rehta hai — lekin chhote hops slow hote hain. Yeh tug-of-war (accuracy vs speed) hi poora reason hai ki adaptive methods exist karte hain.

3. Slope samples $k_{i}$ — jump se pehle jhaankna

Ek crude method poore hop ke liye ek slope (start par ka arrow) use karta hai. Ek Runge–Kutta method zyada smart hai: yeh interval ke andar kaafi arrows jhaankta hai aur unhe blend karta hai.

Greek $Σ$ (sigma) ka matlab sirf sab add karo hai: $\sum_{j < i} a_{ij} k_{j} = a_{i 1} k_{1} + a_{i 2} k_{2} + \dots$ $i$ se pehle wale sample tak.

Topic ko $k_{i}$ kyun chahiye. Yeh samples hi mahenga hissa hain — har $k_{i}$ ek call hai $f$ ko. Poora "free error estimate" wala trick theek isliye kaam karta hai kyunki RK45 do alag answers ke liye same $k_{i}$ reuse karta hai. Tum "free" appreciate nahi kar sakte jab tak pata na ho ki $k_{i}$ hi paisa lagata hai.

Recall Check:

$\sum_{j < i}$ kyun hai aur $\sum_{j}$ kyun nahi? Kyunki har slope sample pehle se compute hue walon se banta hai — ek explicit chain, toh ek sample kabhi khud par depend nahi karta.

4. Weighted sums $\sum b_{i} k_{i}$ — peeks ko ek answer mein blend karna

Jab tumhare paas saare peeked slopes $k_{1}, \dots, k_{s}$ ho jaate hain (inki $s$ hoti hain), tum unhe weights $b_{i}$ use karke ek "effective slope" mein combine karte ho, phir hop lete ho:

$y_{n + 1} = y_{n} + h \sum_{i = 1}^{s} b_{i} k_{i}$

Sirf pehle arrow ki jagah weighted average kyun? Step ke paar kaafi arrows ko average karna curving error ko bahut better cancel karta hai — yahi cheez RK ko crude ki jagah "high order" banati hai. Exact $b_{i}, c_{i}, a_{ij}$ Dormand–Prince (RK45 used by scipy solve_ivp) se aate hain.

5. Doosri weight row $b_{i}^{*}$ — EK set of peeks se do answers

Yeh woh piece hai jis par poora topic tika hai, isliye hum ise dheere dheere banate hain.

Tumhare paas peeked slopes $k_{1}, \dots, k_{s}$ hain. Step khatam karne ke liye tum unhe weight row $b_{i}$ se blend karte ho. Lekin koi majburi nahi ki sirf ek weight row use karo. Wahi $k_{i}$ ek doosri baar ek alag set of weights se blend kiye ja sakte hain, jise hum $b_{i}^{*}$ likhte hain (padho "b-i-star" — blend numbers ki ek doosri, alag list).

Topic ko $b_{i}^{*}$ kyun chahiye. Bina doosri weight row ke, error estimate karne ke liye ek poora alag method chalana padta (taaza, mahenga $k_{i}$ ). Star-row woh trick hai jo RK45 ko sasta banati hai: same peeks, doosra blend, instant error ruler.

Recall Check:

$y^{(5)}$ aur $y^{(4)}$ mein kya alag hai? Sirf weight row: $b_{i}$ vs $b_{i}^{*}$ — slope samples $k_{i}$ identical hain, isliye estimate almost free hai.

6. Order $p$ — error kitni tezi se shrink hoti hai

$h^{p + 1}$ padhna. RK45 step ko order-4 answer $y^{(4)}$ se control karta hai, toh $p = 4$ aur exponent $p + 1 = 5$ hai. $h$ aadha karne se error $2^{5} = 32$ guna shrink ho jaati hai. Yeh steep power hi wajah hai ki chhote steps itni accuracy khareedte hain, aur wajah hai ki update formula mein fifth root use hoti hai.

Yeh power law seedha Taylor Series Expansion se aata hai: woh pehla term jo method cancel karne mein fail hota hai woh $h^{p + 1}$ ke proportional hai. Order $p$ kahan tay hota hai yeh Order of a Numerical Method aur Local vs Global Truncation Error mein dekho.

7. Error estimate $ε$ — ek number jo batata hai "kitna galat"

Kyunki $y^{(5)}$ (order 5) $y^{(4)}$ (order 4) se kahin zyada accurate hai, unka gap essentially kamzor wale ki poori error hai — ek free ruler, exactly jaisa Section 5 mein picture mein dikhaya.

8. Tolerance $tol$ — woh error budget jo tum allow karte ho

Dono kyun? Agar $y$ bahut bada ho jaata hai, toh ek tiny fixed $atol$ ek impossible demand ban jaata hai; $rtol ∣ y_{n} ∣$ term tumhe bachata hai budget ko scale karke. Agar $y$ zero se guzre, $atol$ budget ko nothing par collapse hone se rokta hai. accept/reject decision simply comparison $ε \leq tol$ hai.

Recall Check: sirf

$atol$ kyun nahi use karte? Kyunki jab $∣ y ∣$ bada ho jaata hai toh ek fixed absolute floor impossibly strict ho jaata hai — relative term budget ko solution ke saath scale karne deta hai.

9. Symbols ko saath mein rakhna — woh vocabulary jo topic assume karta hai

Symbol	Seedha matlab	Picture
$y^{'} = f (t, y)$	har point par slope-recipe	arrow field
$h$	$t$ mein ek hop ki length	horizontal gap
$t_{n}, y_{n}$	jahan hum abhi khade hain	curve par ek dot
$k_{i}$	$i$ -wa peeked slope	step ke andar chhote arrows
$b_{i}$	higher-order (order 5) blend weights	accurate average ki recipe
$b_{i}^{*}$	lower-order (order 4) blend weights, same $k_{i}$	doosre average ki recipe
$y^{(5)}, y^{(4)}$	dono weight rows se do answers	do landing dots
$p$	order (error shrink rate)	error curve ki steepness
$ε$	estimated step error	dono answers ke beech gap
$tol$	error budget	woh line jiske neeche rehna hai

Parent ke boxed formula $h_{new} = h (tol / ε)^{1/ (p + 1)}$ mein har symbol ab tumhara banaya hua hai: ek hop length, ek error budget, ek estimate (do weight rows se), aur ek order.

Prerequisite map

Equipment checklist

I can state what

f (t, y)

returns

Point

(t, y)

par solution ki slope — ek arrow, curve nahi.

I can say what

h

is and the trade-off in choosing it

t

mein ek hop ki length; chhota

h

= accurate lekin slow, bada

h

= fast lekin bends se uda jaane ka risk.

I know what

t_{n}

and

y_{n}

mean

Woh time jahan hum abhi khade hain aur wahan solution ki height (

n

steps ginata hai).

I can read

\sum_{j < i} a_{ij} k_{j}

"Pehle ke slope samples ko add karo, har ek

a_{ij}

se scaled" — ek chain jahan har peek pehle ki peeks use karta hai.

I know what a slope sample

k_{i}

is and why it's expensive

Step ke andar

f

ka ek evaluation; har ek ek costly function call hai, isliye RK45 unhe reuse karta hai.

I can explain the difference between

b_{i}

and

b_{i}^{*}

SAME

k_{i}

par do weight rows:

b_{i}

order-5 answer deta hai,

b_{i}^{*}

order-4 answer deta hai.

I can say what

y^{(5)}

and

y^{(4)}

are

Do weight rows se banaye do hops; superscripts order label karte hain, powers nahi.

I can explain why two answers give a free error estimate

Accurate

y^{(5)}

truth ki jagah kaam karta hai, toh

y^{(5)} - y^{(4)}

essentially

y^{(4)}

ki error hai, aur costly

k_{i}

reuse hote hain.

I can explain what "order

p

" measures

Local error

h

ke saath kitni tezi se shrink hoti hai: error

\approx C h^{p + 1}

I know why the local exponent is

p + 1

, not

p

Local (per-step) error

h^{p + 1}

ki tarah scale hoti hai; global error

h^{p}

ki tarah scale hoti hai.

I can state what

ε

estimates and where it comes from

Step ki error ka size,

∥ y^{(5)} - y^{(4)} ∥ = ∥ h \sum_{i} (b_{i} - b_{i}^{*}) k_{i} ∥

se.

I can write and read

tol = atol + rtol ∣ y_{n} ∣

Ek per-step error budget jo ek fixed floor aur solution-scaled part ko milata hai.

I can decide accept vs reject

Hop accept karo agar

ε \leq tol

, warna reject karo aur chhote

h

se dobara try karo.

Connections

4.8.25 Adaptive step-size — RK45, error control (Hinglish) — woh parent topic jise yeh foundations feed karte hain.
Runge–Kutta Methods (RK4) — jahan stages $k_{i}$ aur weights $b_{i}$ aate hain.
Taylor Series Expansion — $h^{p + 1}$ error law ka origin.
Order of a Numerical Method — exponent mein $p$ kahan tay hota hai.
Local vs Global Truncation Error — kyun per-step error $h^{p + 1}$ hai.
Dormand–Prince (RK45 used by scipy solve_ivp) — actual $a_{ij}, b_{i}, c_{i}$ table.
Stiff ODEs and Stability — jab adaptive explicit steps bhi struggle karte hain.