4.8.4
Ek condition number measure karta hai ki kisi problem ka output, input mein ek choti si wiggle ko kitna amplify karta hai . Yeh problem ki khud ki property hai, kisi bhi algorithm ki nahi. Agar data mein ek tiny relative error answer mein ek huge relative error ban jaaye, toh problem ill-conditioned hai — aur koi bhi clever code tumhe bacha nahi sakta.
Intuition Yeh kyun matter karta haiJo bhi real number tum computer mein daalte ho, usme error hoti hai: rounding (≈ 1 0 − 16 (error in answer) ≤ (condition number) × (error in input) . gain knob hai input aur output error ke beech. Isse pata chalta hai ki tumhara answer 10 digits tak trust kiya ja sakta hai ya sirf 2 tak.
Ek problem ko function y = f ( x ) diya gaya x f ( x ) . Hum input ko perturb karte hain x → x + Δ x Δ y = f ( x + Δ x ) − f ( x )
Definition Absolute condition numberAbsolute condition number measure karta hai ki ek absolute input change kaise ek absolute output change mein map hoti hai:
κ abs = lim Δ x → 0 ∣Δ x ∣ ∣Δ y ∣ = ∣ f ′ ( x ) ∣.
Definition Relative condition numberRelative condition number measure karta hai ki ek relative input change kaise ek relative output change mein map hoti hai:
κ rel = lim Δ x → 0 ∣Δ x ∣/∣ x ∣ ∣Δ y ∣/∣ y ∣ = ∣ f ( x ) ∣ ∣ x ∣ ∣ f ′ ( x ) ∣ . dimensionless hai — yahi woh hota hai jise log usually "the condition number" kehte hain, kyunki data error naturally relative hoti hai (har number mein 1 0 − 16
Definition Ill- vs well-conditionedEk problem well-conditioned hai agar κ rel O ( 1 ) 1 0 3 ill-conditioned agar κ rel ≫ 1 tum log 10 κ rel
Hum memorise nahi karte; hum build karte hain.
Intuition "Digits lost" rule — yeh kahan se aata haiAgar input relative error ε in ε out ≈ κ rel ε in log 10 log 10 ε out ≈ log 10 κ rel + log 10 ε in . ≈ − log 10 ε lagbhag log 10 κ rel .
f ( x ) = x f ′ ( x ) = 2 x 1 κ rel = x x ⋅ 2 x 1 = 2 x x = 2 1 . Yeh step kyun? Hum f f ′ half ho jaati hai — square-rooting beautifully stable hai. (Generally x p κ rel = ∣ p ∣
f ( x ) = tan ( x ) x = π /2 f ′ ( x ) = sec 2 x x = π /2 tan x → ∞ sec 2 x → ∞ κ rel = t a n x x s e c 2 x = s i n x c o s x x = s i n 2 x 2 x . x → π /2 sin 2 x → 0 κ rel → ∞ Yeh step kyun? Humne sec 2 / tan = 1/ ( sin cos ) sin 2 x
Worked example 3. Nearly-equal numbers ka subtraction — catastrophic cancellationMaano f ( x , y ) = x − y y x f ′ = 1 κ rel = x − y x . x = 1.00001 y = 1.00000 x − y = 1 0 − 5 ∣ x ∣ ≈ 1 κ rel ≈ 1 0 5 Yeh step kyun? Denominator x − y x cancellation problem hai; problem ill-conditioned hai, toh subtraction se bachne ke liye algebra rearrange karo.
Worked example 4. Linear systemsA x = b A x = b matrix condition number
κ ( A ) = ∥ A ∥ ∥ A − 1 ∥ = σ m i n σ m a x (in 2-norm) , σ b → b + Δ b A Δ x = Δ b Δ x = A − 1 Δ b ∥Δ x ∥ ≤ ∥ A − 1 ∥∥Δ b ∥ ∥ b ∥ ≤ ∥ A ∥∥ x ∥ ∥ x ∥ ∥Δ x ∥ ≤ κ ( A ) ∥ A ∥∥ A − 1 ∥ ∥ b ∥ ∥Δ b ∥ . Yeh step kyun? Humne numerator aur denominator ko alag-alag bound kiya, phir combine kiya — product ∥ A ∥∥ A − 1 ∥ A σ m i n ≈ 0
Common mistake "Mera answer bura hai, toh mera algorithm bura hai."Kyun sahi lagta hai: jab results garbage hote hain, code ko blame karna natural lagta hai.
Fix: Conditioning problem ke baare mein hai, stability algorithm ke baare mein. Ek ill-conditioned problem bure answers deta hai chahe perfect algorithm use karo. Pehle κ backward stable hai ya nahi.
Common mistake "Ek bada absolute condition number matlab ill-conditioned."Kyun sahi lagta hai: "bada number = bura" ek tempting shortcut hai.
Fix: κ abs dimensionless relative κ rel f ( x ) = 1000 x κ abs = 1000 κ rel = 1
det ( A ) ≈ 0 A Kyun sahi lagta hai: singular matrices ka determinant zero hota hai.
Fix: Determinant size aur units ke saath scale karta hai; 0.1 I 100 1 0 − 100 κ = 1 κ ( A ) = σ m a x / σ m i n nahi .
Recall Feynman: ek 12-saal ke bacche ko samjhaoEk magnifying glass imagine karo. Input error page par ek tiny sa smudge hai. Condition number yeh hai ki magnifying glass kitni strong hai. Ek weak glass (chhota condition number) — smudge chhota rehta hai, tumhara answer clear hai. Ek super-strong glass (huge condition number) — woh tiny smudge ek giant blurry blob ban jaata hai aur tum answer padh nahi sakte. Magnifying glass ki strength problem ki property hai, tumhari aankhon (algorithm) ki nahi. Kuch sawaal naturally "blurry" hote hain — jaise yeh measure karna ki ek see-saw kaise tip karta hai jab woh bilkul edge par perfectly balanced ho: ek tiniest push usse crash kar deta hai.
"CRAB" — C ondition = R elative A mplification of B lunders.
Aur formula ke liye: "x f-prime over f" → κ rel = f x f ′ × ÷
Condition number kya measure karta hai? Input mein relative (ya absolute) change output mein kitna amplify hota hai — problem ki property hai, algorithm ki nahi.
y = f ( x ) κ abs = ∣ f ′ ( x ) ∣ ∣Δ y ∣/∣Δ x ∣
Relative condition number ka formula? κ rel = f ( x ) x f ′ ( x ) , dimensionless.
Ill-conditioning decide karne ke liye kaun sa condition number use hota hai, aur kyun? Relative wala — yeh dimensionless hai, toh scaling/units se dhoka nahi khaata.
Condition number κ Lagbhag log 10 κ
f ( x ) = x p κ rel = ∣ p ∣ (jaise
x ka
1/2 hota hai).
Nearly-equal numbers ko subtract karna dangerous kyun hai? κ rel = ∣ x / ( x − y ) ∣ x ≈ y
A x = b κ ( A ) = ∥ A ∥∥ A − 1 ∥ = σ m a x / σ m i n
Conditioning aur stability mein kya fark hai? Conditioning = problem ki sensitivity; stability = algorithm ki quality. Accurate result ke liye dono chahiye.
Kya det ( A ) ≈ 0 Nahi — determinant size/units ke saath scale karta hai; σ m a x / σ m i n
dimensionless, usual meaning
large, much bigger than 1
property of problem, not algorithm
Absolute condition number
Relative condition number
