CS 314 Specification 2 - Implementing a Class - Mathematical Matrix

"Linear algebra is a fantastic subject On the one hand it is clean and beautiful. If you have three vectors in 12 dimensional space, you can almost see them."
- Gilbert Strang, Linear Algebra and its Applications

Programming Assignment 2: Individual Assignment. You must complete this assignment on your own. You may not acquire from any source (e.g.  another student or an internet site) a partial or complete solution to a problem or project that has been assigned. You may not show another student your solution to the assignment. You may not have another person (current student, former student, tutor, friend, anyone) “walk you through” how to solve the assignment. You may get help from the instructional staff. You may discuss general ideas and approaches with other students but you may not develop code together. Review the class policy on collaboration from the syllabus.

The purposes of this assignment are

1. To practice implementing a stand alone class.
2. To work with two dimensional arrays.

Provided Files:

 File Responsibility Implementation MathMatrix.java Provided by me and you. (Okay, mostly you.) Documentation MathMatrix.html Provided by me. Implementation Stopwatch.java (For use in experiments) Provided my me Documentation Stopwatch.html Provided by me Testing MathMatrixTester.java Provided by me and you

Description: Implement a class that represents a mathematical matrix. You are implementing a stand alone class that is a new data type.

Mathematical matrices are used  to solve systems of linear equations. Matrices are used in applications such as physics, engineering, probability and statistics, economics, biology, and computer science. (especially in the area of computer graphics. Here is a page on how matrices are used to perform rotations on 3d objects in a graphics system. )  There is a course in the UT Math department that covers matrices, 340L, and many CS students take this course. Dr. Maggie Meyers and CS Professor Robert van de Geijn offer an online linear algebra with a programming component.

Matrices appear in the following form:

These matrices represent this system of linear equations:

x +  5y + 10z + 5w =   4
6x +  4y + 12z + 4w =   5
10x +  5y + 12z + 11w = 12
5x + 11y + 23z + 9w  =  7

The above matrix has 4 rows and 4 columns, but the number of rows and columns do not have to be equal. In other words mathematical matrices do not need to be square, but they must be rectangular. Each entry can be an integer or real number. For this assignment the matrices will only contain java `int`s. You will implement a class, `MathMatrix`, that models a mathematical matrix and supports various operations on matrices. See this page for an explanation of the mathematical operations you are implementing. The Wikipedia article may also be useful.

Requirements: The provided source file MathMatrix.java contains a skeleton implementation of a class to model mathematical matrices.

Implement all of the methods in MathMatrix.java under the constraints of the general requirements.

• You may use other classes and methods from the Java standard library you wish on this assignment.

• Add conditional statements (ifs) that throw ``` IllegalArgumentException```s to methods to check preconditions.

• You must use a "native" two dimensional array of `int`s as your underlying storage container in the matrix class:

private int[][] values;
// or nums or cells or some other appropriate name
// DO NOT USE some variation of mathMatrix or matrix.
// That is much too confusing. The two dimensional array of ints
// is NOT a MathMatrix!

• The first row of a `MathMatrix` is numbered 0. The first column of a `MathMatrix` is numbered 0.

• The provided source file, MathMatrixTester.java contains various tests for the `MathMatrix` class. Some of these tests may be incorrect. You must find and fix any incorrect tests. Your `MathMatrix` class must pass the included tests. Use the class discussion group to identify incorrect tests and share your own tests.

• Add at least 2 new tests for each of the public methods you are completing (24 tests total.) I encourage you to share you tests with others via the class discussion group. Everyone must write their own tests, but you are allowed to share the tests you write to check your code.

• You are encouraged to create `private` helper methods and use other `public` methods in the `MathMatrix` class when completing methods in the ``` MathMatrix``` class if this simplifies the solution.

• In CS312 we emphasized minimize redundant code. That still holds true. In CS314 we also want to minimize redundant instance variables. Do no add unnecessary instance variables.

• Note, once a `MathMatrix` object is created there are no methods to alter its size. So unlike the `IntList` we created in class it does NOT make sense to have extra capacity. The size of the 2d array of ints will be the same size as the Mathematical Matrix it is representing. (This implies it is not necessary to keep track of the number of rows and columns with separate instance variables.)

Experiment: In addition to completing the MathMatrix.java class and adding tests to the MathMatrixTester.java class, perform the following experiments and answer the following questions. Place your results and answers in a comment at the top of MathMatrixTester.java. Recall, you cannot share your experiment code with others. You CAN share tests on Piazza to help each test your solution code with a wide variety of test cases. The tests you turn in must be your own.

Include the code that conducts the experiments in the MathMatrixTester.java class, but comment it out..

Use the Stopwatch class to record the time it takes to perform various operations on `MathMatrix` objects.

```Stopwatch s = new Stopwatch(); s.start(); //code to time s.stop();```

The Stopwatch class has methods that return the elapsed time in seconds or nanoseconds between starting and stopping the Stopwatch. See the Stopwatch class documentation for more details.

Experiment 1: Create two matrices and fill them with random values. Initially try matrices that are 1000 by 1000 in size. You may have to adjust the dimension as described below. Reuse the same matrices for the following experiments. Repeat the each experiment 1000 times and note the average time of the 1000 experiments for each value of N. (3 total, N, 2N, 4N)

• Use the Stopwatch class to record the time it takes to add the 2 `MathMatrix` objects together.

• You must choose a value for the number of rows and columns so the total time of the 1000 tests give a result of at least 1 second of elapsed time. (1 second total for all 1000 tests, NOT 1 second per test.) You should, of course, automate these 1000 repetitions.

• Record the dimension of the matrix and the average time it took for the add operation based on 1000 repetitions.

• Now double the dimension of the matrix, create two matrices with random values, and repeat the experiment. For example if the original `MathMatrix` was 800 by 800. In this step the size would be increased to 1600 by 1600.

• Record the dimension of the matrix and the average time it took for the add operation on the larger matrix based on 1000 repetitions.

• Double the dimension of the matrix one more time, create two matrices with random values, and conduct 1000 additions, and determine the average time.

• If you get an out of heap space error, increase the size of the heap. All of the possible command line flags are on this page. In Eclipse you can set a command line flag for your program. Follow the instructions for enabling assertions and include the flag -Xmxsize where size is the new requested heap size. For example, to increase the heap size to 120 mb include the command line flag -Xmx120m.

Experiment 2:

• Perform the same basic experiment as experiment 1, but use the `multiply` method instead of the `add` method. Repeat the experiment 100 (one hundred not 1000 in this case) times to get the average time.

• You must choose a value for the number of rows and columns so the total time of the 100 tests give a result of at least 1 second of elapsed time. (1 second total for all 100 tests, NOT 1 second per test.) You should, of course, automate these 100 repetitions.

1. Based on the results of experiment 1, how long do you expect the add method to take if you doubled the dimension size of the `MathMatrix` objects again?

2. What is the Big O of the add operation given two N by N matrices? Does your timing data support this?

3. Based on the results of experiment 2, how long do you expect the multiply method to take if you doubled the dimension size of the `MathMatrix` objects again?

4. What is the Big O of the multiply operation given two N by N matrices? Does your timing data support this?

5. How large a matrix can you create before your program runs out of heap memory? (When using the default heap size. No command line flag to increase heap size.)  In other words what size matrix causes a Java `OutOfMemoryError`, Estimate the amount of memory your program is allocated based on the largest possible matrix object it can create successfully. (Recall, an int in Java requires 4 bytes.)

Submission: Fill in the header for MathMatrix.java and MathMatrixTester.java. Replace <NAME> with your name. Note, you are stating, on your honor, that you did the assignment on your own as required. I will use plagiarism detection software on your submissions. If you copy solution code from another source you are cheating. I will submit an academic dishonesty case with a recommended penalty of an F in the course.

Create a zip file name a2.zip with your MathMatrix.java and MathMatrixTester.java files. The zip file must not contain any directory structure, just the two required files.

See this page for instructions on how to create a zip via Eclipse.

Turn in a2.zip via your Canvas account to programming assignment 2.

• Ensure you are turning in the version of MathMatrix.java that has your completed methods and the version of MathMatrixTester.java with your extra tests added and the original tests deleted. You may have more than one version of the files on your system. Do not turn in the .class file; turn in the Java source code. If you turn in the wrong one you will get a zero on the assignment.

• Ensure you files are named MathMatrix.java and MathMatrixTester.java. Failure to do so will result in points off.

• Ensure MathMatrix.java and MathMatrixTester are part of the default package. Do not add a `package` statement to the either file.

• Ensure your zip has no internal directory structure. When the file is unzipped no directories (folders) are created.  (Note the built in unzip feature of some versions of Windows and Apple OS X "help" by adding a directory when you unzip with the same name as the file. The unzip we use on the CS Linux system does not do this. Neither do unzip programs such as 7zip.)

• Ensure you submit the assignment under Programming Assignment 2 in Canvas.

Checklist: Did you remember to:

1. review and follow the general assignment requirements?
2. work on the assignment by yourself and complete all the solution code on you own?
3. fill in the headers in the MathMatrix and MathMatrixTester classes? If you are using Eclipse be sure to expand the header comment.
4. implement the required methods?
5. ensure your program does not suffer a compile error or runtime error?
6. find and fix any incorrect tests in MathMatrixTester?
7. ensure your program passes the tests in MathMatrixTester?
8. add your own tests (at least 2 per public method) to the main method of MathMatrixTester?
9. complete the experiments and place you answers to the questions in a comment at the top of the MathMatrixTester file?
10. turn in your files (MathMatrix.java and MathMatrixTester.java) in a zip named a2.zip with no internal directory structure?
11. turn in your zip named a2.zip to Programming Assignment 2 via Canvas no later than 11 pm on Thursday, February 8?

Tips:

1. Be clear on the difference between `MathMatrix` objects and the 2d array of` int`s that serves as the storage container.

Assume the 2d array of ints instance variable for each MathMatrix object is named myCells.

MathMatrix result = new MathMatrix(numRows(), numCols(), 0);
int valueFromThisMathMatrix = myCells[0][0];
int valueFromRightHandSide = rightHandSide.myCells[0][0];
int valueFromResult = result.myCells[0][0];

// the following line results in syntax error
// valueFromRightHandSide = rightHandSide[0][0];

2. Familiarize yourself with the concept of deep copying. (As opposed to shallow copying.) One of the constructors requires you make a deep copy of a 2d array of ints.

3. If possible use other methods from the `MathMatrix` class instead of repeating code.

4. An explanation of the requirements for the `toString` method.

In the String that is returned from the `toString` method the space for each "cell" is equal to the longest value in the matrix plus 1. (Don't forget to consider a minus sign in on of the values.) All cell entries are right justified with newline characters  between rows. For example, given the following `MathMatrix`.

 10 100 101 -1000 1000 10 55 4 1 -1 4 0

You should return a String that would appear like this. Use newline characters ("\n") to create line breaks.

|    10   100   101 -1000|
|  1000    10    55     4|
|     1    -1     4     0|

In example above it can be hard to tell how many spaces there are between numbers. In this example the spaces have been replaced by periods to the number of "spaces" is more clearly shown.

|....10...100...101.-1000|
|..1000....10....55.....4|
|.....1....-1.....4.....0|

Note, the last line includes a newline character.

One way of finding the length of an int is to convert it to a `String` and find the length of the `String`. Here is an example:

int x;
//code to give x a value.
String s = "" + x;
int lengthOfInt = s.length();

//or more simply given an int x
int lengthOfX = ("" + x).length();

Implementing the `toString` method using just loops and `String`s (and / or `StringBuilder`s) and if statements is a very interesting exercise. Alternatively you can learn how to use the format method from the String class and formatting string syntax. Here is an introduction to formatting String syntax.

5. The `isUpperTriangular` method determines if the `MathMatrix` is an upper triangular matrix. A matrix is upper triangular if it is a square matrix and all values below the main diagonal are 0. The main diagonal consists of the cells whose row and column are equal. (Runs for the top left to the bottom right.) The values of the elements on the main diagonal don't have to be zero, just the ones below it. A 1 by 1 matrix is upper triangular.

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