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.

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:单项选择题

类似问题

In the context of discretising space and time for numerical calculations, 𝐴 𝑖 − 1 𝑛 + 1 and 𝐴 𝑖 + 1 𝑛 + 1 refer to estimates of parameter A at the same time, but at spatial locations differing by 2 Δ 𝑥 .

In the context of discretising space and time for numerical calculations, and refer to estimates of parameter A at the same time, but at spatial locations differing by .

In the context of discretising space and time for numerical calculations, 𝐴 𝑖 − 1 𝑛 refers to the estimate of parameter A immediately adjacent to location index i, at the current time.

The central difference approximation to the gradient of a function fat the point xi, which is uniformly sampled with spacing Δxis:

What was the world's first widely adopted biodiversity policy?

The aim of the Paris Agreement is to ensure that the earth’s preindustrial temperature is not exceeded by more than º Celsius.

The IPCC's mission is to Blank ______.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单