Differentiation

Question 1

What is the main idea of numerical differentiation?

Answer 1

Approximate f by an interpolating polynomial and differentiate that

Question 2

If we use a linear Lagrange polynomial, differentiate and evaluate at x = x₀ what formual do we get?

Answer 2

Question 3

What is the forward difference equation?

Answer 3

Question 4

What is the truncation error for our forward difference approximation?

Answer 4

Question 5

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

Answer 5

Use Taylors theorem and

Question 6

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

Answer 6

We say the approximation is linear or first order

Question 7

What three nodes do we use for central difference?

Answer 7

x₀, x₁ = x₀ + h, x₂ = x₀ + 2h

Question 8

What are the two types of finite differences?

Answer 8

Forward and Backward

Question 9

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

Answer 9

x₀

Question 10

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

Answer 10

x₁

Question 11

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

Answer 11

Question 12

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

Answer 12

f(x₂) - f(x₁) / 2h

Question 13

What is a proboem with numerical differentiation?

Answer 13

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

Question 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.

Answer 14

Question 15

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

Answer 15

1. Truncation error
2. Rounding error

Question 16

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

Answer 16

• Truncation ➝ 0
• Rounding ➝ ∞

Question 17

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

Answer 17

Richardson extrapolation.

Question 18

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

Answer 18

Question 19

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

Answer 19

Question 20

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

Answer 20

Question 21

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

Answer 21

To repreat with step size h/2.

Question 22

What is the final formula for D_h/2?

Answer 22

Question 23

What equation do you get if you do the following?

Answer 23

Question 24

Why do we the following calcualtion?

Answer 24

To eliminate the h² term.

Question 25

What is the formula for D_h⁽¹⁾?

Answer 25

Question 26

How accurate is the following?

Answer 26

4^th order accurate

Question 27

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

Answer 27

Question 28

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

Answer 28

Question 29

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

Answer 29

Question 30

What is the formula for Richardson extrapolation?

Answer 30

Question 31

Why did you choose h/2 for richardson extrapolation?

Answer 31

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