Numerical differentiation — forward, backward, central differences: Question bank

4.8.13 · D5 · Maths › Numerical Methods › Numerical differentiation — forward, backward, central diffe

True or false — justify karo

Central difference hamesha forward difference se zyada accurate hota hai same

h

ke liye.

False in general. Iski truncation error ek order chhoti hoti hai (

O (h^{2})

O (h)

), toh smooth

f

aur moderate

h

ke liye ye jeet jaata hai; lekin agar

f^{'''}

bahut bada ho ya

h

roundoff-dominated regime mein ho, toh ordering toot sakti hai.

Central formula

f (x)

ki value khud use karta hai.

False. Central first-derivative

\frac{f ( x + h ) - f ( x - h )}{2 h}

sirf do neighbours use karta hai; midpoint ki value cancel ho jaati hai. Central second derivative zaroor

f (x)

use karta hai.

h

ko half karne se central first-derivative formula ki error roughly quarter ho jaati hai.

True. Iski leading error

\frac{h ^{2}}{6} f^{'''}

hai, aur

(h /2)^{2} = h^{2} /4

hota hai, toh truncation error lagbhag

4 \times

kam ho jaati hai — bas tab tak jab tak roundoff floor nahi aa jaata.

Forward aur backward differences ki accuracy ka order same hota hai.

True. Dono

O (h)

hain; inki leading error terms hain

- \frac{h}{2} f^{''}

aur

+ \frac{h}{2} f^{''}

— same magnitude, opposite sign. Yahi opposite sign reason hai ki inhe average karne par central milta hai.

Second-derivative formula

\frac{f ( x + h ) - 2 f ( x ) + f ( x - h )}{h ^{2}}

O (h)

hai kyunki ye

h^{2}

se divide karta hai.

False. Order surviving Taylor term se aata hai, denominator se nahi. Expansions ko add karne par

f^{'}

aur

f^{'''}

cancel ho jaate hain, aur

\frac{h ^{2}}{12} f^{''''}

error bachti hai →

O (h^{2})

Agar

f

ek straight line hai, toh teeno first-derivative formulas exact slope dete hain.

True. Linear

f

ke liye

f^{''} = f^{'''} = 0

, toh har error term zero ho jaata hai; har formula

h

ki parwah kiye bina exact constant slope return karta hai.

h

ko chhota karna answer ko worse bana sakta hai.

True. Ek certain

h

ke neeche, numerator mein almost equal numbers subtract hote hain → catastrophic cancellation, aur roundoff error

\sim ε / h

badhta jaata hai. Total error ka minimum finite

h

par hota hai.

Central difference data table ki pehli row par theek kaam karta hai.

False. Ise dono

x - h

aur

x + h

chahiye. Pehli row par left neighbour missing hota hai, toh aapko forward difference par fall back karna padta hai (ya ek one-sided higher-order formula par).

Error dhundho

Ek student central difference ko

\frac{f ( x + h ) - f ( x - h )}{h}

likhta hai. Kya galat hai?

x - h

x + h

tak ki span ki width

== 2 h ==

hai,

h

nahi.

2 h

ki jagah

h

se divide karne par answer double ho jaata hai.

Ek student forward difference ko peeche wale point se compute karta hai:

\frac{f ( x ) - f ( x - h )}{h}

aur ise forward kehta hai.

Ye backward difference hai. Forward aage dekhta hai

f (x + h) - f (x)

use karke; yeh wala peeche dekhta hai.

f^{''}

nikalne ke liye, ek student do Taylor expansions ko add karne ki jagah subtract karta hai. Kya toot jaata hai?

Subtract karne se even terms cancel ho jaate hain (

f^{''}

mar jaata hai),

f^{'}

bachta hai — isse aapko first derivative milta hai.

f^{''}

term bachane ke liye aapko add karna hoga.

Ek student claim karta hai ki forward-difference error

- \frac{h ^{2}}{2} f^{''}

hai. Sahi karo.

Error

- \frac{h}{2} f^{''}

hai,

h

ki pehli power (isliye ye

O (h)

hai).

h^{2}

central first-derivative error se belong karta hai, jisme

f^{'''}

hota hai,

f^{''}

nahi.

Koi "

O (h^{2})

" ko "error

h^{2}

ke barabar hai" padhta hai. Ye galat kyun hai?

O (h^{2})

ka matlab hai error proportional hai

h^{2}

ke, ek constant times

f

ka derivative tak; actual size

\approx \frac{h ^{2}}{6} ∣ f^{'''} ∣

hoti hai. Dekho Truncation error and Big-O.

Ek student double precision mein

h = 1 0^{- 14}

use karta hai "safe rehne ke liye" aur garbage paata hai. Diagnose karo.

h

par,

f (x + h)

aur

f (x - h)

~15 digits tak agree karte hain, toh unka difference almost poora roundoff hai.

ε / h

term explode ho jaata hai. Dekho Roundoff error and floating point.

Why questions

Do Taylor expansions ko subtract karne par

f^{''}

term kyun cancel hota hai?

f (x - h)

mein even-power terms ka sign

f (x + h)

ke jaisa hi rehta hai, toh subtract karne par wo remove ho jaate hain, jabki odd-power terms (

f^{'}

f^{'''}

) sign flip karte hain aur reinforce karte hain. Even ↔ symmetric, odd ↔ antisymmetric.

Central difference "ek extra order of accuracy for free" kyun hai?

Symmetry hi kaam karti hai: dono sides par equally sample karne se, dono sides ki leading

O (h)

curvature error equal aur opposite hoti hai aur cancel ho jaati hai, toh pehla surviving term

O (h^{2})

hota hai — koi extra function evaluations nahi chahiye.

Computer

h \to 0

limit definition ki tarah kyun nahi le sakta?

Machine sirf finite floating-point numbers store karti hai jinka ek smallest gap

ε

hota hai; infinitesimals nahi hote. Kisi bahut chhote

h

ke baad, subtraction ke baad koi bhi significant digits nahi bachte.

Central difference ka optimal

h

ε^{1/3}

ke scale par kyun hota hai?

Total error

\approx \frac{h ^{2}}{6} ∣ f^{'''} ∣ + \frac{ε}{h}

; sum minimize karne par (rising

ε / h

aur falling

h^{2}

ko balance karke) derivative zero set hoti hai, jisse

h_{opt} \propto ε^{1/3}

milta hai.

Interior table points par central difference prefer kyon karte hain lekin edges par nahi?

Interior points ke dono neighbours available hote hain, toh central free mein

O (h^{2})

deta hai; edges par ek neighbour missing hota hai, jo ek one-sided (

O (h)

) formula force karta hai jab tak aap higher-order stencil ke liye aur aage nahi jaate.

Expansions add karne par

f^{''}

ke liye denominator

h^{2}

kyun hota hai lekin

f^{'}

ke liye

2 h

kyun?

Add karne ke baad surviving term

h^{2} f^{''}

hai, toh

f^{''}

isolate karne ke liye

h^{2}

se divide karte hain; subtract karne ke baad surviving term

2 h f^{'}

hai, toh

2 h

se divide karte hain. Denominator bas wo power of

h

match karta hai jo target derivative ko multiply kar rahi hai.

Richardson extrapolation central differences ke baad ek natural next step kyun hai?

Kyunki error

h^{2}

mein ek clean power series hai,

h

aur

h /2

par results combine karne se leading

h^{2}

term algebraically cancel ho sakta hai, accuracy

O (h^{4})

tak boost ho jaati hai bina naye derivatives ke.

Edge cases

Agar

f

exactly ek parabola ho toh central second-derivative formula kya deta hai?

Kisi bhi

h

ke liye exact

f^{''}

. Quadratic ke liye

f^{''''}

se aage ke saare terms zero hote hain, toh koi truncation error nahi bachti.

Aapke paas sirf ek data table hai (koi formula nahi) — kya aap phir bhi accuracy ke liye chhota

h

choose kar sakte ho?

Nahi.

h

table ki spacing se fix hota hai; aap rows ke beech sample nahi kar sakte. Aap sirf which stencil (forward/central) choose kar sakte ho aur yeh ki pehle interpolate karein ya nahi.

Aapki table mein node spacing uneven hai (

x - h_{1}, x, x + h_{2}

jahan

h_{1} \neq = h_{2}

). Kya standard central formula abhi bhi hold karta hai?

Nahi — clean

O (h^{2})

cancellation ne symmetric spacing assume ki thi. Unequal spacing ke liye ek general finite-difference weight derivation chahiye, aur naive formula ek order kho deta hai.

Kink par (jahan

f^{'}

discontinuous hai) forward difference ka kya hota hai?

Taylor's theorem ko

f

ki smoothness chahiye; kink par expansion invalid hai, toh error estimates toot jaate hain aur formula quietly ek slope return karta hai jo corner par average karta hai.

Agar

f (x)

constant ho, toh har formula kya return karta hai?

Exactly

0

first derivative ke liye aur

0

second derivative ke liye — equal values ke saare differences vanish ho jaate hain, jo kisi bhi

h

ke liye true zero derivative se match karta hai.

Itne bade

h

par ki

x + h

us region se bahar chala jaaye jahan

f

smooth hai, kya central abhi bhi "best" hai?

Zaruri nahi. "

O (h^{2})

" label sirf

h \to 0

ke saath leading term describe karta hai; bade

h

ke liye higher-order terms dominate karte hain aur accuracy comparison guaranteed nahi rehta.

Kya second-derivative formula kabhi negative value de sakta hai jahan true

f^{''}

positive hai?

Haan — roundoff floor ke paas,

f (x + h) - 2 f (x) + f (x - h)

mein cancellation noise tiny result ka sign poora flip kar sakta hai, jo

h

bahut chhota hone ki ek classic symptom hai.

Connections

Taylor series — upar ke har cancellation argument ka source.
Truncation error and Big-O — $O (h)$ vs $O (h^{2})$ actually kya claim karta hai.
Roundoff error and floating point — "chhota better nahi" wale traps ke peeche $ε / h$ floor.
Richardson extrapolation — upar puche gaye $h^{2}$ error structure ko exploit karta hai.
Finite difference methods for PDEs — jahan uneven grids aur edge stencils sach mein takleef dete hain.