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CR3 (Submission phase)
Submission phase
Instructions for submission
Using your preferred code editor (e.g. VSCode), in a Python script called "CR3.py", write code to answer the following problem.
Problem
Consider the following ODE, describing the displacement of an oscillator (e.g. a mass attached to a spring) at time away from its
resting position:
where is the angular frequency of the oscillator, given as a fixed parameter. The initial conditions are given as
, the initial displacement,
, the initial velocity.
We seek to solve this equation numerically, using a finite difference method. We discretise time with a step size , and denote by the
approximation to computed at time step , that is .
We approximate the second derivative in the ODE with , to obtain the difference equation:
Rearranging this equation allows us to compute using values computed at the two previous time steps, and . We can initialise
and by discretising the initial conditions:
,
Then, we can compute , followed by , etc., for as many time steps as we desire.
Note that this method provides a "valid" solution as long as . This is called the stability condition.
Your task
Write a function "oscillator(w0, u0, v0, nmax, dt)" which takes 5 input arguments:
a positive number "w0", representing the parameter ,
two numbers "u0" and "v0", representing the initial conditions and respectively,
a positive integer "nmax" greater than or equal to 2, representing the total number of time steps,
a positive number "dt", representing the step size ,
and returns a Numpy vector "U" with a total of "nmax" elements, where the th element is the value of , the approximated solution for the
oscillator displacement at the th time step, computed using the method described above.
Dashboard / My courses / www.learn.ed__81855_1 / Code reviews / CR3
Computing and Numerics (2020-2021)[SEM2]
Setup phase Submission phase
Current phase 
Submit your work
Open for submissions from
Monday, 8 March 2021,
2:00 PM (5 days ago)
Submissions deadline:
Monday, 15 March 2021,
12:00 PM (2 days left)
Assessment phase
Open for assessment from
Monday, 15 March 2021,
4:00 PM (2 days left)
Assessment deadline:
Monday, 22 March 2021,
12:00 PM (9 days left)
Grading evaluation phase Closed
Additionally, before computing the result, the function should check the value of and display a message if necessary:
if , display a message warning the user that the stability condition has been violated. Your message should include the maximum
value of "dt" allowed for the chosen value of "w0", to help your user choose a better step size next time.
if , display a message warning the user that the chosen step size is at the stability limit.
In either case, you should still compute and return the solution.
Testing
After the function definition, write a few tests to check that your function is working. Your first test should use the following values:
You can compare the computed solution to the exact solution, which is given by:
You could plot the exact and computed solutions on the same graph over time -- for valid step sizes (such that ), the two curves should
be close, but they will never be exactly overlapping; the peak values will often be where you see the largest error. For values of and close
to the stability condition (but still valid), you should see that the computed solution oscillates slightly faster than the exact solution. As we have
seen in many other situations, decreasing the step size should generally improve accuracy.
Try different values for the initial conditions and , and check that the first two values of your computed solution are set correctly. Try different
values for the frequency , increasing it should produce a faster-oscillating solution.
Make sure you check that the warning messages appear for appropriate values. Plot the computed solution for both problematic cases -- this is
what you should see:
when (and only when) , the amplitude of the oscillations in the computed solution should grow exponentially over time, and you
should see the first warning message.
when (and only when) , the amplitude of the oscillations in the computed solution should grow linearly over time, and you should
see the second warning message.
when (and only when) , the computed solution should oscillate without growing linearly or exponentially over time, and you should
not see any message.
You should include your test code in the script CR3.py, after the function definition. Make sure you label any plots you produce appropriately
and clearly.
Scope
This is for the more experienced programmers amongst you: remember that this is peer-assessed, and that Computing & Numerics is a course
designed for beginner programmers. Please try to stick to what we've seen in the course material so far, even if you feel confident in doing
something more advanced -- keep it simple.
You should not need to -- but if you absolutely want to use something we haven't seen in the course, then please keep it to a minimum, and make
sure you explain clearly what you are doing (and why) in the code comments, so that a student with little or no previous programming
experience can assess your work with confidence. If you don't follow these guidelines, and submit code which requires much more advanced
knowledge/experience with Python to fully understand than we have seen in the course so far, then your assessors will be able to penalise you
(see marking scheme below).
Submission
Upload your file "CR3.py" here when you are finished. Make sure you upload your file before Monday 15th March, 12pm (noon), otherwise you
won't be able to participate to the peer-assessment phase, and you will get a zero for the whole CR3 task!
Marking scheme
The marking scheme which you will use when assessing next week is given in the table below. Each criterion counts equally towards the grade,
and is assessed on a scale of 0 to 3.
Criteria 0 1 2 3
Δ𝑡
𝜔0Δ𝑡 > 2
𝜔0Δ𝑡 = 2
𝜔0 = 5, 𝑢0 = 0.2, 𝑣0 = 5, 𝑛m = 500, Δ𝑡 = 0.03. ax
𝑢(𝑡) = 𝑢0 cos(𝜔0 𝑡) + sin( 𝑡).
𝑣0
𝜔0 𝜔0
𝜔0Δ𝑡 < 2
Δ𝑡 𝜔0
Δ𝑡
𝑢0 𝑣0
𝜔0
𝜔0Δ𝑡 > 2
𝜔0Δ𝑡 = 2
𝜔0Δ𝑡 < 2
2021/3/13 CR3 (Submission phase)
https://moodle2.maths.ed.ac.uk/live2018/mod/workshop/view.php?id=3628 3/3
Was the file CR3.py submitted
correctly, and the code written as
instructed?
If you penalise here, indicate what
the problem was in the text box.
The file was in the
wrong format, which
made it difficult or
impossible to open.
The file was in the correct format,
but either didn't have the correct
name, or the code inside did not
follow the instructions (e.g. the
function was named differently, or
performed a completely different
task, or there was no function, or
there were no tests).
Everything was submitted in
the correct format and the
code followed the
specifications, but the code
was much too
complex/advanced compared
to what we have seen in the
course.
Everything was
submitted in the
correct format
and the code
followed the
specifications
(regardless of
whether it was
correct or not).
Is the code correct and without
bugs? Does the function
"oscillator()" produce the correct
results for any appropriate inputs?
If you can find any bugs or issues,
then you can indicate them in the
text box below to help the author fix
them. If you know that there is an
issue but you aren't sure why, then
indicate which test(s) look incorrect,
and give your best guess as to what
could be wrong.
(Use the test code provided in the
script by the student, as well as the
model answer and test code
provided by the instructors, to make
sure the code is fully functional.)
The code does not
work at all, there are a
lot of problems which
would take a long time
to find and fix.
There are a few bugs, so the code
works only for some input values,
but fails for many others (either
giving an error or returning the
wrong result).
There is a small number of
minor bugs or omissions, so
the code perhaps works for
most input values but fails for
a few special cases; or
perhaps the code doesn't work
but only because of a minor
issue which is trivial to fix.
The code works
correctly and
returns the
correct results for
all input values.
Was the function tested with an
appropriate range of different
inputs, to check if it worked?
(Whether or not the tests passed is
not relevant here.)
No tests were included
in the script.
Very few tests were performed,
not nearly enough to fully check
that the function worked.
A number of tests were
performed, which covered a
good number of possible
values, but perhaps some
important cases were missed.
A good number of
tests were
performed, which
covered all (or
almost all) of the
possible values
we could use with
this function.
Are the plots clear, easy to
understand, and appropriately
labelled?
The plots are extremely
messy, very hard to
read, and not clearly
The plots are perhaps too small or
squeezed, overlapping plots may
The plots are generally well
presented although a couple
The plots are very
well presented,
clearly and
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辅导Python编程设计、讲解data程序、Python语言编程辅导 解析Haskell程序|讲解R语言程序

辅导Python编程设计、讲解data程序、Python语言编程辅导解析Haskell程序|讲解R语言程序