Maple
Seminar provided by:
Mathematics Faculty
Computing Facility
Presented by:
Marcus Shea
Seminar Schedule
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Maple
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AMPL
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Wed. July 23, 5-6pm
Wed. July 30, 5-6pm
Both in MC 3006, pizza provided afterward
Goal of Seminars
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Provide new tools and useful information to
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Help you with homework
Approaching problems during co-op terms
Assist personal research
Give you a competitive edge
You will learn these new tools through real
world examples
Maple in the Real World
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Maple plays a vital role in 3-D biomechanical modeling and stability
analysis of bipedal motion
Maple aids in the development of new aerospace manufacturing
technologies
Maplesoft technology speeds up concept vehicle prototyping over 10,000
times
Maple used for single-cylinder engine design to help EMAK achieve high
efficiency
Maple contributes to revolutionary advancements in medical robotic
technology
Maple helps advance Maglev train technology
Use of Maple optimizes financial modeling at Mitsubishi UFJ Securities
International
Maple helps Ford Motor Company with analytical predictions of chain
drive system resonances
Overview
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Generating a curve
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Prime Functions
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Cryptography Problem
Planning a Road Trip
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Plot Builder
Finding Shortest Paths
Basic Statistical Analysis
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Google Stock Data
Generating a Curve
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There are many different ways to specify a function in Maple
We can define the function explicitly
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Then display your function
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You can then right click the function to perform many operations
including integration, differentiation and solving at a point.
This can be very helpful when you need to check simple
calculations involved in homework
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Generating a Curve
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More interestingly, we can provide Maple initial
conditions for a function, and it will return the
satisfying function
Can be helpful in fitting observed data to a curve
Suppose we want a cubic polynomial that has a local
minimum at x=(3,-6), passes through (0,24), and has
a zero at x=-3
Specify the general formula for a cubic polynomial
Generating a Curve
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Provide Maple the initial conditions
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The equations that our function must satisfy
are then displayed
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We can right-click these equations and click
Solve > Solve
Generating a Curve
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Maple outputs the
coefficients for our
polynomial
We can get the cubic
that we want and store
it to g(x) as follows
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Instead of the ??, press
ctrl+L and enter the
equation number for the
solution of our
coefficients
Plot Builder
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Let’s ensure that we
received the correct function
and get comfortable with
Maple’s plot builder tool
We will define our points
Then right click the points
and choose Plots > Plot
Builder
Right click the plot, select
Symbol > Symbol Size and
set the size to 20
Plot Builder
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We can highlight and
drag our polynomial
g(x) onto the plot
Select the plot, then
select the Axis
Properties button that
has appeared in the
Context Bar above
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Change the y-axis to
display values -50..50
Plot Builder
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By changing the axis scales you are able to
zoom in on particular segments of curves
This can help you to understand the
behaviour of curves you may be working with
during homework or research
The plot builder makes it easy to create
professional graphs for reports
Using Prime Functions
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Maple has many useful functions dealing
with primes
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isprime(n)
ithprime(n)
nextprime(n)
prevprime(n)
Cryptography Problem
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Suppose you work for a software company which
has just designed a fast new encryption algorithm
that relies critically on two secret primes p and q
The value n = p*q is publicly available
The prime choosing algorithm is also publicly
available
We need to determine if this algorithm is secure –
that is, we need to find out if one could determine p
and q given n
Cryptography Problem
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Algorithm to choose p and q
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Set p to a random large prime
Set q = p + 2
While (q is not prime)
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Set q = q + 2
Output (p,q)
The value of n is provided in a text file
Can we use Maple to determine p and q?
Cryptography Problem
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Hints
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Note that p and q are consecutive primes
Also, note that p < sqrt(n) < q
Will probably need the evalf[precision](number)
command
Solution
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This will reveal the p and q
sn = evalf[100](sqrt(n));
z = floor(sn);
p = prevprime(z);
q = nextprime(p);
Cryptography Problem
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As we have seen, the computational power
of Maple allows us to perform mathematical
calculations that would be infeasible by hand
Maple is widely used in the analysis of
algorithms
This could be helpful during co-op terms or
personal research
Planning a Road Trip
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Say we are planning a road trip to Los
Angeles, California
To break up the trip, we want to pass through
some major cities along the way
We list all the cities between Waterloo and
Los Angeles that interest us
We want to get to Los Angeles quickly
Planning a Road Trip
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Here are the major cities along the way that interest us
Planning a Road Trip
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Cities with distances (in hours)
Planning a Road Trip
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We create a graph where cities are nodes
(numbered 1,2,…,12) and edges between
cities are the distances
We then want to find the shortest path from
Waterloo (node 1) to Los Angeles (node 12)
We can use code from the Maple Application
Center to find the shortest path
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Uses Djikstra’s Algorithm (CO351)
Planning a Road Trip
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Algorithm takes an input text file
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Line 1: #START
Line 2: numNodes numEdges startNode
Subsequent lines each define an edge:
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tail head distance [no decimals in distance]
Last line: #END
Planning a Road Trip
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With the input file created and saved in a file
called trip.txt, you can determine the solution
by running the following commands:
Dijkstras_Algorithm("trip.txt", true);
draw(G);
Planning a Road Trip
Tree of Shortest Paths:
Planning a Road Trip
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We can use the tree of shortest paths to
determine the optimal route to any one of the
11 destinations
For Los Angeles, our schedule would be:
Waterloo (1) to Detroit (2) to Chicago (4) to
Kansas City (7) to Denver (8) to Phoenix (9)
to Los Angeles (12)
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Total travel time: 43.1 hours
Planning a Road Trip
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The analysis we have seen can be applied to
a variety of situations
The shortest path problem can be used to
model:
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Web mapping (Mapquest, Google Maps)
Telecommunication networks
Scheduling problems (CO351)
Many other Operations Research problems
Basic Statistical Analysis
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We know that gas prices have been steadily rising
lately
Let’s take a look at the stock prices of an American
independent oil and gas company, Apache
Corporation (APA:NYSE)
We will be working with the closing prices over the
past year
Historical data taken from Yahoo Finance
You are provided with a file that will read in the data
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Variable prices holds daily prices
Variable data holds (day,price) points
Point Plot
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A point plot will show each individual daily stock
price
Output data to ensure that we are plotting points
p1 := pointplot( data, color=blue ):
p1;
Moving Average
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Let’s create a one-week
moving average
A moving average is
less sensitive to daily
fluctuations
Provides a better trend
curve
Moving Average
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We get the y values for the moving average
movingAvg := moving[5](prices):
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Then create points
movingAvgData :=
[seq([i,movingAvg[i], i=1..n-5)]:
Moving Average
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Then we create a point plot using our moving
average data, but connect the points
p2 := pointplot(movingAvgData, connect = true,
color = green, thickness = 2):
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And we display it with our original point plot
display(p1,p2);
Histograms
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Maple can easily create two different
histograms
A histogram with equal area bars
histogram(prices);
– A histogram with equal width bars
histogram(prices, area = count);
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Histograms
histogram(prices);
histogram(prices, area = count);
Box Plot
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We can also create a boxplot, providing
parameters for its center and width
b1 := boxplot( prices, shift=maxTime/2,
width=maxTime, color=red, thickness=2 ):
Box Plot
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Then we display it with
our point plot
display( b1, p1 );
Statistical Analysis
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You can use Maple to generate informative
plots for reports, personal, and business
research
Maple is very useful for Financial Analysis,
and there is a good library available at the
Maplesoft Application Center
Maplesoft Application Center
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Free Maple Applications
http://www.maplesoft.com/applications/