The backward difference approximation to the gradient of a function
𝑓
at the point
𝑥
𝑖
, which is uniformly sampled with spacing
Δ
𝑥
is:

Question

The backward difference approximation to the gradient of a function 
𝑓
 at the point 
𝑥
𝑖
, which is uniformly sampled with spacing 
Δ
𝑥
 is:

BlackTom AI · Accepted Answer

To tackle this question, let's compare the given finite-difference formulas for estimating the gradient.
Option 1: fi − fi−1 Δx. Interpreting this as (fi − fi−1)/Δx, this is the standard backward difference formula, which uses the current and the previous sample to approximate the derivative at xi with spacing Δx. This aligns with the definition of a backward difference.
Option 2: fi+1 − fi Δx. If read as (fi+1 − fi)/Δx, this is the forward difference formula, not the backward one, since it uses the current and the next sample rather than the current and the previous one.
Option 3: fi+1 − fi−1 2Δx. This expression corresponds to the central difference formula: (fi+1 − fi−1)/(2Δx). It typically provides a more accurate estimate than backward or forward differences for smooth functions, but it is not the backward difference.
In summary, the backward difference at xi with spacing Δx uses the current and previous samples in the ratio (fi − fi−1)/Δx, which matches Option 1 if we interpret the formatting as a division by Δx. The remaining options represent the forward difference and the central difference, respectively, not the backward one.

AMME2000 BMET2960 BMET9960 (ND) Week 5 Pre-work

The backward difference approximation to the gradient of a function 𝑓 at the point 𝑥 𝑖 , which is uniformly sampled with spacing Δ 𝑥 is:

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Similar Questions

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