Sources of error — truncation error, round-off error

4.8.1 · Maths › Numerical Methods

1. The two error sources

2. Derive truncation error from scratch (forward difference)

Humein $f^{\prime }(x)$ chahiye. Taylor's theorem se shuru karte hain (yeh ek exact tool hai):

$f(x+h)=f(x)+h{f}^{\prime }(x)+\frac{h^2}{2}{f}^{\prime \prime }(x)+\frac{h^3}{6}{f}^{\prime \prime \prime }(x)+\cdots$

Yeh step kyun? Taylor expansion exact hai (remainder ke saath), isliye jo bhi error milegi woh honest hai, assumed nahi.

$f^{\prime }(x)$ ke liye solve karo:

$f^{\prime }(x)=\frac{f(x+h)-f(x)}{h}-\underset{\text{jo hum throw away karte hain}}{\underbrace{\frac{h}{2}{f}^{\prime \prime }(x)-\frac{h^2}{6}{f}^{\prime \prime \prime }(x)-\cdots }}$

Yeh step kyun? Rearrange karne se finite-difference formula isolate hota hai aur precisely woh discarded tail dikhayi deti hai.

$f^{\prime \prime }(\xi )$ kyun aur $f^{\prime \prime }(x)$ kyun nahi? Taylor's remainder form leftover ko kisi unknown point $\xi \in (x,x+h)$ par rakhta hai — exact, koi infinite sum nahi.

3. Quantify round-off error

Jab machine $f(x)$ store karti hai toh actually $\tilde{f}(x)=f(x)(1+\delta )$ store karti hai jahan $|\delta |\le {\epsilon }_{\text{mach}}$.

Forward difference ke liye, numerator $f(x+h)-f(x)$ mein absolute error $\sim {\epsilon }_{\text{mach}}|f(x)|$ hoti hai. $h$ se divide karne par:

$E_R\approx \frac{{\epsilon }_{\text{mach}}|f(x)|}{h}$

$h$ se divide kyun? Numerator mein error roughly fixed hoti hai (order ${\epsilon }_{\text{mach}}|f|$), lekin hum tiny $h$ se divide karte hain, isliye $h\to 0$ ke saath relative damage badh jaata hai.

4. Worked numerical example

5. Common mistakes

6. Active recall flashcards

Connections

• Taylor's Theorem — truncation error formulas ka source
• Finite Difference Methods — forward/central/backward differences
• Floating Point Representation — machine epsilon ka origin
• Catastrophic Cancellation — round-off amplification
• Order of Accuracy — $O(h^n)$ ka matlab
• Newton-Raphson Method — jahan derivative errors propagate karti hain

Concept Map