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

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 qsn = 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 pointsp1 := 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 averagemovingAvg := moving[5](prices):

Then create pointsmovingAvgData :=

[seq([i,movingAvg[i], i=1..n-5)]:

Moving Average

Then we create a point plot using our moving average data, but connect the pointsp2 := pointplot(movingAvgData, connect = true,

color = green, thickness = 2):

And we display it with our original point plotdisplay(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 widthb1 := boxplot( prices, shift=maxTime/2,

width=maxTime, color=red, thickness=2 ):

Box Plot

Then we display it with our point plotdisplay( 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/