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Flashcards in Differentiation Deck (32):
1

What is the main idea of numerical differentiation?

Approximate f by an interpolating polynomial and differentiate that

2

If we use a linear Lagrange polynomial, differentiate and evaluate at x = xwhat formual do we get?

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3

What is the forward difference equation?

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4

What is the truncation error for our forward difference approximation?

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5

What is another way to estiamte the truncation error of finite difference formula?

Use Taylors theorem and 

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6

What do we say if the truncation error if it is proportional to h?

We say the approximation is linear or first order 

7

What three nodes do we use for central difference?

x0, x1 = x0 + h, x2 = x0 + 2h

8

What are the two types of finite differences?

Forward and Backward

9

What do you have to set x equal to get the forward difference?

x0

10

What do you have to set x equal to to get the backward difference?

x1

11

Prove that the central difference approximation of f'(x1) is the following

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12

What is another way to write the central difference in the following?

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f(x2) - f(x1) / 2h

13

What is a proboem with numerical differentiation?

It involes subtraction of nearly equal numbers - which leads to rounding errors.

14

Suppse for th central difference methof we have the following rounded versions of f(x1 ± h), show what the upper bound of the followinf is.

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15

What are the two errors called in the following upper bound?

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  1. Truncation error
  2. Rounding error

16

In central differences how does the truncation and error change as h ➝ 0?

  • Truncation ➝ 0
  • Rounding ➝ ∞

17

Differentiating Lagrange polynomialsis tedious, what other formula to we use instead to find higher-order formulae?

Richardson extrapolation.

18

What is the equation for Dh using the central-difference formula?

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19

What is the equation if we use Taylor's theorem to expand the truncation erros in f(x ± h).

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20

If we substiute the following equation into the following equation for Dh what equation do we get.

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21

Once we have found Dh what is the main idea of Richardson extrapolation.

To repreat with step size h/2.

22

What is the final formula for Dh/2?

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23

What equation do you get if you do the following?

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24

Why do we the following calcualtion?

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To eliminate the h2 term.

25

What is the formula for Dh(1)?

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26

How accurate is the following?

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4th order accurate 

27

What is the general n-order approximation for equation for Dh?

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28

What do we get by evaluating the folloiwng at h/2?

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29

Eliminate the hn term in the following two equations, what do you get?

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30

What is the formula for Richardson extrapolation?

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31

Why did you choose h/2 for richardson extrapolation?

It is convenient, there is nothing special about it. Could have picked h/3 or even 2h

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