IT1005

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IT1005

• Lab session on week 12 (8th meeting)
• 1 more week to go

We will take class photo today! At the end of the
1st hour of this session
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This Weeks Agenda

• Discussion of my Term assignment answers.
• I believe that most of you have seen the same
  tables (for q1),and the same plots (for q2-3),
  but do you curious to know whatare my (Lab TA)
  answers compared with yours?? Note, my answers
  are not 100 correct. This is not official
  answer!
• Review on
• Various ODE IVP solvers,
• Curve Fitting using Matlab, and
• The initial part of Optimization using Matlab
• Overview of Lab 8, our last lab
• Preview of what is in store for the last week
• Class photo D

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Term Assignment Q1.A

• Q1. A. Naïve trapezium rule, definitely
  inaccurate.
• (b-a)(f(a)f(b))/2, f(a) is (f(0)), f(b) is
  f(1000), the result is very imprecise.
• Lets observe the plot of f(t) from 0 1000 ? Parameter c controls the curvature of
  f(t).
• Only if c 0.001 then naïve trapezium rule still
  give a good approximation.

When c is larger, the error around this area
is bigger
Wrong if you draw a triangle here! Note that the
y axis start from 0.5!
We should draw a trapezium ?
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Term Assignment Q1.B

• Q1. B. More precise trapezium rule, larger n
  should be more precise.
• But why all n produces similar values between
  Matlab quad (likely correct) and my traprule2
  (likely not ok)?
• You get no error because the tested integral
  range is from -10 to 10 and the graph is
  special (symmetrical)
• (positive-negative errors cancel each other).
  That is how you should explain why there is no
  error.
• If you change the range of your integral, you
  will see differences!
• If you do this, you can show that n 2 is
  definitely imprecise.
• The error from larger/slower n (where it should
  logically be more precise) is likely because of
• floating point error around t 0. Larger n is
  still better (and slower) for most cases.
• You can also mention that there is no major
  effect of increasing n when t is far from 0 for
  this f(t) with high c.

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Term Assignment Q2.A-B

• Q2. A. IVP, IVs 0.1 0.1
• Q2. B. Interpretation of the plot the population
  change periodically (roughly every 5 years),
• population curve of the predator (lion) is
  steeper and trails the curve of prey (zebra) by a
  bit.
• Bla bla (tell some nice story)
• Tricky question which ODE solver should I use
  for this case?

No major difference between ode45 and ode15s for
10 years Pick the faster solver then ode15s?
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Term Assignment Q2.C-D

• Q2. C. To find steady state solutions, use fsolve
  or solve (symbolic math),
• Steady state means both dx/dt 0 and dy/dt 0
• We have two possible steady states x 0.5, y
  1/3 or x 0, y 0 (both extinct)
• Q2. D. If you supply any of the two steady state
  IVs, lion/zebra population will be unchanged over
  time.
• The x 0, y 0 can be ignored since it is
  very trivial.
• NOTE for x 0.5, y 1/3, if you write 0.33,
  the graph may looks curvy (this is imprecise)!

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Term Assignment Q2.E

• Q2. E. If population of zebra versus lion is
  plotted, what will we see?
• A circle (oval/ellipse/egg-shape to be precise)
  but which ODE solver that we should use this
  time?
• In predator prey relationship, predator depends
  on the prey to survive.
• If both has non zero population, their population
  count will always be on the border of this
  ellipse.
• This one looks growing perhaps because of the
  constants a, b, c, d used in the question.
• Intermezzo
• If prey extinct, predator also extinct soon (no
  food),
• but if predator extinct first, prey can survive
  and even fill the earth (no predator D),
• but this model is inaccurate as we omit the death
  factor of the prey (look at problem description
  again).

ode45 left (steady). ode15s right (growing). Is
it because ode15smore precise or what?I think
ode15s is better
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Term Assignment Q3.A

• Q3. A. IVP again, but now 4 ODEs, create a loop
  to compute the dx/dt.
• Looking at the differential equation, ri is the
  birth rate, a is interaction matrix, we know
  that
• Species 1 has average birth (1), but is preyed
  by/interact with species 2 3 (total -2.61)
• Species 2 has least birth (0.72), but is preyed
  by/interact with species 3 4 (total -1.8),
• Species 3 has most birth (1.43), but is preyed
  by/interact with species 1 4 (total -2.8),
• Species 4 has 2nd most birth (1.27), but is
  preyed by/interact with species 1, 2 4 (total
  -2.07),
• Looking at the interaction matrix and the plot
• Species 1 is likely predator of Species 3 and 4.
• Species 2 is likely predator of Species 1.
• Note that there are other ways to describe
• this plot, e.g. several species competing
• for the same resource, etc

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Term Assignment Q3.B-C

• Q3. B. 3-D plot for 3 species, 3-D plot plus 4th
  axis is color if you want to show 4 species.
• If these 4 species are alive, their population
  count will fall on the surface of this mesh.
• Q3. C. Lotka-Volterra, a study of population
  dynamics!
• Open ended Hard to grade this one

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Application 5 IVP Solvers

• Matlab has several ODE IVP solvers
• Which one to use?
• Generally, ode45, ok for normal cases
• But if the ODEs are stiff, use the more precise
  solver
• ode15s
• Intermezzo, what do you think the plot of number
  of term assignmentsubmitted by students over
  time will looks like?
• A stiff plot p, perhaps something like this.

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Application 6 Curve Fitting

• This is a graph for my/our research paper (due
  Friday, 11 Apr 08, 6pm)
• http//www.cs.mu.oz.au/cp2008
• Look at how I use polyfit/polyval.

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Application 7 Optimization

• Hm, fyi, my research is about optimization too.
• A little bit more complex than what you see in
  your lecture notes,but the idea is the same
  given so many options, choose one optionthat
  minimizes (or maximizes) some objective function.
• e.g. I deal with problems like TSP (Traveling
  Salesman Problem),n cities, n! possible tours,
  one of the tour is shortest.
• How to deal with simple optimization in Matlab
• Code the function that you want to optimize in a
  dedicated function,similar like what we have
  done with fsolve and ODE solver.
• Call x, f fminsearch(_at_function_handle,
  initial_guess)
• Wrong initial guess will bring us to local
  optima (good, but not the best).
• Next fmincon
• Constraints redefine the search space, the
  global optima may be differentthan the
  unconstrained one.

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Lab 8 - Overview

• No ODE (already tested in term assignment), Lab 8
  tests fzero(the leftover from Lab 6), polyfit
  (Curve fitting), and fminsearch (Optimization),
  in fact, fminsearch is used to do curve fitting.
• Q1. A-B. fzero, this is the former Lab 6 question
  3.If you have not tried this before, do this
  now!PS I have taken out my answers from my ppt
  slides 2 weeks ago.
• Q1. C. Use polyfit to the results in Q1.B above,
  and plot the graph.
• Q2. A. Create a function to compute sum of
  squared errors.
• Q2. B. Use fminsearch to find k1, k2, and k3 that
  minimizesthe error in Q2. A and plot the fit.
• Should be straightforward,you do not need to
  wait until week 13 to finish this.

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What is left for next week?

• Fiuh
• 10 Labs almost done (we still have Lab 8, due on
  week 13, the last week)
• I am generous right?, so at least you get
  8..10
• 20 Midterm test done
• Compare with my modal answer, you should
  approximately know your score.Assuming you score
  10..15, your total is now about 18..25
• 30 Term assignment done
• Most likely your score will fall in the range of
  20..25 as long as you submit proper
  stuffs!So your total is now about 38..50
• So, ?? out of 60 of IT1005 grade is roughly in
  your hand
• Most probably your total is already 40, the
  minimum passing grade (D)
• If it is, 99 you will pass this module, you can
  skip exam and still pass (with D or C).
• So, what left is 40 Final exam to decide your
  final score
• Next week, we will discuss PAST YEAR PAPER (you
  only have 1 sample).
• Come, if you want to fight for a good score
  (B/B/A-/A/A) ?.

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Class Photo (1)

• I always do this every semester
• You can download our photos here
• http//www.comp.nus.edu.sg/stevenha/myteaching/re
  cord.html
• Photos from my Wednesday classes

2B 3B 4B
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Class Photo (2)

• Photos from my Friday classes

5A 7A
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Free and Easy Time

• Now, you are free to explore Matlab to
• Do your Lab 8!
• Review the lessons that you have received during
  IT1005 course!