For the first question, we need to perform this calculation using standard arithmetic, and then well perform this calculation using a calculator that rounds all numbers after calculation to three places to the right of the decimal point
The first step for both of these are the same
We add these two numbers together and get the following
If we were using normal arithmetic, we would then simply add the denominator, receiving this, which is -5,000
However, instead, if we were to use a calculator that rounded to three places to the right of the decimal place after every calculation, we would instead get this in the denominator
Which is equal to -1,000
You can see that this roundoff error, while it may seem insignificant for most calculations, can become very significant in some circumstances
And this is something that you need to be aware of when doing any kind of numerical calculations
This is the answer to Question 1(a) and Question 1(b)
Question 2 asks, if the trajectories that your ODE solver produces with h = 0.1 do not change when you change the time step to h = 0.05, then h = 0.1 is probably a good choice
And this is true
This is equivalent to saying, if you take one step with one step size, versus taking two steps with half the step size
If you get to the same point, theres no point taking two steps, youre just doing extra work
And in this circumstance, h = 0.1 was probably sufficient
So this question is true
Question 3 is almost identical to Question 2, but it goes in the other way of adaptation
That is, if you have a step size of 0.1, and then you try a step size of 0.2, and you dont see any change, then a step size of 0.2 is probably a good choice
This is true by the exact same logic as Question 2
For Question 4, if the trajectories that your ODE solver produces change when you increase the precision of all the variables (that is, going from single-precision to double-precision arithmetic), then the computers arithmetic is introducing dynamical error into the solver results
This is true, this is exactly whats happening
The error being caused by the single-precision arithmetic, for example, is being fed back into the system at every time step
That error would then snowball, causing dynamical error
If you wanted to, you could also call this roundoff or truncation error, but it is absolutely also dynamical error
These types of error do not need to be mutually exclusive, and these different types of error can often compound, causing even greater error
Question 5 asks if the systems derivative affects the error of any ODE solvers solution of that system
This is also true
For example, consider the error for forward Euler
You can see that the form of the system derivative plays into the error
While this error does not hold for any ODE solver, this general statement does hold true
The systems derivative does affect the error of any ODE solvers solution to a system