L1: Basic concepts in numerical computations

Question 1

What is a norm?

What is the Euclidean norm exactly?

Answer 1

Also called the standard norm

Question 2

What is a L^p norm?

What are the three interesting cases?

Answer 2

Question 3

What are the relative and absolute error approximate formulas?

Answer 3

Question 4

What are the two broad types of errors?

Answer 4

Question 5

What is a mantissa? And how do computers represent floating-point numbers?

Answer 5

Question 6

What is the standard implementation of a floating-point number?

Answer 6

WHY DOES THIS LEAD TO ERRORS

NEED TO KNOW BASE 2, BASE 10 CONVERSION

e.g. example 0.3 + 0.3 + 0.3 != 0.9

Question 7

Do we need to worry about roundoff errors?

What shouldn’t you do with floating-point numbers?

Answer 7

Question 8

Why does subtracting a floating-point number cause a catastrophic cancellation error?

Answer 8

ADD TO THIS

Question 9

What are we interested in about the calculation of any algorithm?

Answer 9

Question 10

Why do we care about computation cost when looking at error formulas?

Answer 10

Question 11

Generally, what is big-O notation?

Answer 11

• Look at the biggest power for the limit we are looking at (which way the independent variable tends to in the long run)

LOOK INTO THIS MORE

Question 12

What is the definition of Big-O notation?

Answer 12

Question 13

How do I make a Grid from a continuous interval?

Answer 13

Question 14

What is a finite-difference approximation (forward difference)?

Answer 14

Question 15

What are the other finite-difference approximations?

Answer 15

• forward = tangent point and next forward point
• backward = tangent point and the previous backwards point
• centre = previous point and next point (average of the forward and back point) –> interval is twice as big so divide by 2h
  
  • tends to be the best approximation of the slope of the tangent

Question 16

How do we use the Taylor series in the finite-difference approximation? Why do we use it?

Answer 16

Check why we use it!

Question 17

From the Taylor series representation, how do we find the absolute error between the true derivative and the forward-difference approximation?

Answer 17

Question 18

What are the absolute errors of all the finite differences of the approximations?

Answer 18

This is useful for two reasons:
1) Compare which algorithm is better e.g. for smaller errors, O(h²) is better than O(h)
2) use for checking if what Im going is create

(Check the scaling law discussed in the lecture recording)