Maple

Seminar provided by:

Mathematics Faculty Computing Facility

Presented by:

Marcus Shea

Seminar Schedule

 Maple – Wed. July 23, 5-6pm  AMPL – Wed. July 30, 5-6pm  Both in MC 3006, pizza provided afterward

Goal of Seminars

 Provide new tools and useful information to – 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

 Generating a curve – Plot Builder  Prime Functions – Cryptography Problem  Planning a Road Trip – Finding Shortest Paths  Basic Statistical Analysis – Google Stock Data

Generating a Curve

  There are many different ways to specify a function in Maple We can define the function explicitly  Then display your function   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

Generating a Curve

    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

 Provide Maple the initial conditions  The equations that our function must satisfy are then displayed  We can right-click these equations and click Solve > Solve

Generating a Curve

  Maple outputs the coefficients for our polynomial We can get the cubic that we want and store it to g(x) as follows – Instead of the ??, press ctrl+L and enter the equation number for the solution of our coefficients

Plot Builder

  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

  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 – Change the y-axis to display values -50..50

Plot Builder

 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

 Maple has many useful functions dealing with primes – isprime(n) – ithprime(n) – nextprime(n) – prevprime(n)

Cryptography Problem

    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

 Algorithm to choose p and q – Set p to a random large prime – – Set q = p + 2 While (q is not prime)  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

 Hints – Note that p and q are consecutive primes – – Also, note that p < sqrt(n) < q Will probably need the evalf[precision](number) command

Solution

 This will reveal the p and q sn = evalf[100](sqrt(n)); z = floor(sn); p = prevprime(z); q = nextprime(p);

Cryptography Problem

 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

 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

 Here are the major cities along the way that interest us

Planning a Road Trip

 Cities with distances (in hours)

Planning a Road Trip

 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 – Uses Djikstra’s Algorithm (CO351)

Planning a Road Trip

 Algorithm takes an input text file – Line 1: #START – – Line 2: numNodes numEdges startNode Subsequent lines each define an edge:  tail head distance [no decimals in distance] – Last line: #END

Planning a Road Trip

 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

 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) – Total travel time: 43.1 hours

Planning a Road Trip

 The analysis we have seen can be applied to a variety of situations  The shortest path problem can be used to model: – Web mapping (Mapquest, Google Maps) – Telecommunication networks – Scheduling problems (CO351) – Many other Operations Research problems

Basic Statistical Analysis

     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 – – Variable prices holds daily prices Variable data holds (day,price) points

Point Plot

  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

   Let’s create a one-week moving average A moving average is less sensitive to daily fluctuations Provides a better trend curve

Moving Average

 We get the y values for the moving average movingAvg := moving[5](prices):  Then create points movingAvgData := [seq([i,movingAvg[i], i=1..n-5)]:

Moving Average

 Then we create a point plot using our moving average data, but connect the points p2 := pointplot(movingAvgData, connect = true, color = green, thickness = 2):  And we display it with our original point plot display(p1,p2);

Histograms

 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);

Histograms

histogram(prices); histogram(prices, area = count);

Box Plot

 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

 Then we display it with our point plot display( b1, p1 );

Statistical Analysis

 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

 Free Maple Applications http://www.maplesoft.com/applications/