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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
2
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The first step for both of these are the same
3
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We add these two numbers together and get the following
4
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If we were using normal arithmetic, we would then simply add the denominator, receiving this, which is -5,000
5
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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
6
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Which is equal to -1,000
7
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You can see that this roundoff error, while it may seem insignificant for most calculations, can become very significant in some circumstances
8
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And this is something that you need to be aware of when doing any kind of numerical calculations
9
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This is the answer to Question 1(a) and Question 1(b)
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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
11
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And this is true
12
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This is equivalent to saying, if you take one step with one step size, versus taking two steps with half the step size
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If you get to the same point, theres no point taking two steps, youre just doing extra work
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And in this circumstance, h = 0.1 was probably sufficient
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So this question is true
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Question 3 is almost identical to Question 2, but it goes in the other way of adaptation
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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
18
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This is true by the exact same logic as Question 2
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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
20
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This is true, this is exactly whats happening
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The error being caused by the single-precision arithmetic, for example, is being fed back into the system at every time step
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That error would then snowball, causing dynamical error
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If you wanted to, you could also call this roundoff or truncation error, but it is absolutely also dynamical error
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These types of error do not need to be mutually exclusive, and these different types of error can often compound, causing even greater error
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Question 5 asks if the systems derivative affects the error of any ODE solvers solution of that system
26
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This is also true
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For example, consider the error for forward Euler
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You can see that the form of the system derivative plays into the error
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While this error does not hold for any ODE solver, this general statement does hold true
30
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The systems derivative does affect the error of any ODE solvers solution to a system