### C++代写 | MA117 Project 3: Determinants of Matrices

这个作业是用C++完成矩阵的各种操作

MA117 Project 3: Determinants of Matrices

Administrative Details

• This project is the third of the three assignments required for the assessment in this course. It is

to be submitted by Noon, Monday 4th May 2020. Details of the method of the submission

via the Tabula system have been described in the lecture notes and are also available on the

course web page.

• This assignment will count for 40% of your total grade in the course.

• The automated submission system requires that you closely follow instructions about the format

of certain files; failure to do so will result in the severe loss of points in this assessment.

• You may work on the assignment during the lab session, provided you have completed the other

tasks that have been set. You can use the work areas at all times when they are not booked for

teaching, 7 days per week. If you are working on the assignment on your home system you are

advised to make regular back-up copies (for example by transferring the files to the University

systems). You should note that no allowance will be made for domestic disasters involving your

own computer system. You should make sure well ahead of the deadline that you are able to

transfer all necessary files to the University system and that it works there as well.

• The Tabula system will be open for the submission of this assignment starting from 23rd April

2020. You will not be able to test your code for correctness using Tabula but you can resubmit

your work several times, until the deadline, if you find a mistake after your submission. A later

submission always replaces the older one, but you have to re-submit all files.

• Remember that all work you submit should be your own work. Do not be tempted to copy

work; this assignment is not meant to be a team exercise. There are both human and automated

techniques to detect pieces of the code which have been copied from others. If you are stuck

then ask for assistance in the lab sessions. TAs will not complete the exercise for you but they

will help if you do not understand the problem, are confused by an error message, need advice

on how to debug the code, require further explanation of a feature of Java or similar matters.

• If you have more general or administrative problems e-mail me immediately. Always include the

course number (MA117) in the subject of your e-mail.

1 Formulation of the Problem

Matrices are one of the most important mathematical concepts to be modelled by computer, being

used in many problems from solving simple linear systems to modelling complex partial differential

equations.

Whilst a matrix (in our formulation) is simply an element of the vector space R

m×n

, it usually possesses

some structure which we can exploit to gain computational speed. For example, a matrix-matrix

multiplication generally requires of the order of n

3 floating-point operations. If the matrix has some

special structure which we can exploit using a clever method, then we might be able to reduce this to

n operations. For large values of n, this significantly improves the performance of our code.

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MA117 Programming for Scientists: Project 3 Deadline: Noon, Monday 4th May 2020

In this project, you will write two classes representing matrices of the form:

A =

a11 . . . a1n

.

.

.

.

.

.

.

.

.

am1 . . . amn

and B =

b11 b12 0 0 0

b21 b22 b23 0 0

0 b32 b33 b34 0

0 0 b43 b44 b45

0 0 0 b54 b55

A is a dense m × n matrix which, in general, has no special structure and no zero entries. B is a

tri-diagonal matrix, where all entries are zero apart from along the diagonal and upper and lower

diagonals. Note that although B is only a 5 × 5 matrix, your classes should represent a general n × n

tri-diagonal matrix. Also, the tri-diagonal matrices you need to represent will always be square.

In a similar fashion to Fraction, you will then write functions to perform various matrix operations:

1. addition and subtraction;

2. scalar and matrix-matrix multiplication;

3. calculating the determinant of the matrix.

Clearly calculating the determinant is the most tricky task here. Probably you will already have

seen expansion by minors as a possible method. Whilst this is an excellent method for calculating

determinants by hand, you should not use it for this task. The reason is that calculating the

determinant of a n×n matrix requires O(n!) operations, since for each n×n matrix, we must calculate

the values of the n − 1 sub-determinants. This is extremely slow.

A much better method is called LU decomposition. In this, we write a matrix A as product of two

matrices L and U which are lower- and upper- triangular respectively. For example, for a 4×4 matrix,

we would find matrices so that

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

| {z }

A

=

l11 0 0 0

l21 l22 0 0

l31 l32 l33 0

l41 l42 l43 l44

| {z }

L

u11 u12 u13 u14

0 u22 u23 u24

0 0 u33 u34

0 0 0 u44

| {z }

U

Such a factorisation is not guaranteed to exist (and indeed is not unique), but typically it does. In this

project, you don’t really need to worry about this – your code will be tested with matrices for which

the LU decomposition exists. It is up to you to figure out how to calculate the determinant from the

LU decomposition!

Throughout the formulation, matrices will be represented by indices running between 1 ≤ i, j ≤ m, n.

However, in your code, you should stay consistent with Java notation and indices should start at 0

(i.e. 0 ≤ i, j ≤ m − 1, n − 1).

2 Programming Instructions

On the course web page for the project you will find files for the following classes. As with the previous

projects, the files have some predefined methods that are either complete or come with predefined

names and parameters. You must keep all names of public objects and methods as they are in the

templates. Other methods have to be filled in and it is up to you to design them properly. There are

five classes in this project:

• Matrix: a general class defining the basic properties and operations on matrices.

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MA117 Programming for Scientists: Project 3 Deadline: Noon, Monday 4th May 2020

• MatrixException: a subclass of the RuntimeException class which you should use to throw

matrix-related exceptions. This class is complete – you do not need to alter it.

• GeneralMatrix: a subclass of Matrix which describes a general m × n real matrix.

• TriMatrix: another subclass of Matrix which describes a n × n real tri-diagonal matrix.

• Project3: a completely separate class which will use Matrix and its subclasses to collect some

basic statistics involving random matrices.

Please note that unlike other projects, you may not assume that the data you receive will be valid.

Therefore you will need to check, amongst other things, that matrix multiplications are done using

matrices of valid sizes, the user is not trying to access matrix elements which are out of bounds, etc.

If something goes wrong, you are expected to throw a MatrixException.

The classes you need to work on are briefly described below.

2.1 The Matrix class

This is the base class from which you will build your specialised subclasses. Matrix is abstract – as

described in the lectures, this means that some of the methods are not defined, and they need to be

implemented in the subclasses. The general idea is that each subclass of Matrix can implement its

own storage schemes, whilst still maintaining various common methods inherent in all matrices.

In particular, the following functions are not abstract, and need to be filled in inside Matrix:

• the protected constructor function;

• toString, which should return a String representation of the matrix.

Additionally, the following abstract methods will be implemented by the subclasses of Matrix:

• getIJ and setIJ: accessor and mutator methods to get/set the ijth entry of the matrix.

• add: returns a new Matrix containing the sum of the current matrix with another.

• multiply(double a): multiply the matrix by a constant a ∈ R.

• multiply(Matrix B): multiply the matrix by another matrix. Note that this is intended to be

a left multiplication; i.e. A.multiply(B) corresponds to the multiplication AB.

• random(): fills current the matrix with random numbers, uniformly distributed between 0 and

1. For a tri-diagonal matrix, this should fill the three diagonals with random numbers.

In subclasses, you should pay attention to what type of matrix needs to be returned from each of the

functions. For example, when adding two GeneralMatrix objects the result should be a GeneralMatrix

(which is then typecast to a Matrix).

2.2 The GeneralMatrix class

GeneralMatrix represents a full m × n matrix and extends Matrix.

1. The matrix will be stored in a private two-dimensional array.

2. You should implement all of the functions mentioned above using the standard formulae from

linear algebra to do so, as well as the usual constructors, accessor and mutator methods.

3. You may choose whatever method you want to calculate the determinant of the matrix. However,

it is strongly recommended you use the provided decomp function, which will perform LU

decomposition for you since the algorithm is quite tricky for n × n matrices.

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MA117 Programming for Scientists: Project 3 Deadline: Noon, Monday 4th May 2020

To call decomp, you should pass it a double array d of length 1. It will return a new GeneralMatrix

storing both L = (lij ) and U = (uij ). For instance, when n = 5, the matrix returned is

u11 u12 u13 u14 u15

l21 u22 u23 u24 u25

l31 l32 u33 u34 u35

l41 l42 l43 u44 u45

l51 l52 l53 l54 u55

The reason we can store it in this compact form is that the algorithm insists that lii = 1 for

every i, and so this information can be omitted from the array.

On exit, the double inside the array you passed in will have a value of 1 or −1. You should

multiply the calculated determinant by this value so that it has the correct sign. This constant

arises because the decomposition algorithm will flip rows in the matrix to aid with singular

matrices, thus changing the sign of the determinant.

As a result, if you explicitly perform the multiplication LU, you probably won’t get the original

matrix back again, but rather a permutation of it. For example, consider a matrix J which is a

slightly altered identity matrix.

J =

1 0 0 0

0 1 0 0

0 1 1 0

0 0 0 1

decompose J

−−−−−−−−−−→ LU =

1 0 0 0

0 1 1 0

0 1 0 0

0 0 0 1

6= J

In the algorithm, one row was was swapped, so d[0] will be −1.

2.3 The TriMatrix class

TriMatrix represents a tri-diagonal matrix of size n×n and extends Matrix. The constructor therefore

only accepts a single parameter.

1. Tri-diagonal matrices are never stored in full two-dimensional arrays because they are sparse –

that is, most of the entres are zero. Instead, we use three arrays of doubles: diag, upper and

lower. These store the diagonal, upper-diagonal and lower-diagonal elements respectively. In

this form, the matrix looks like

T =

d1 u1

l2 d2 u2

l3 d3 u3

l4 d4

.

.

.

.

.

.

.

.

. un−1

ln dn

diag should therefore be of length n, whereas upper and lower should be of length n − 1.

2. For this class, you will need to implement your own decomp method to perform LU decomposition,

which should not be copied from GeneralMatrix, since the algorithm for a tri-diagonal matrix is

very simple to derive. First, we assume that the diagonal elements of the lower-diagonal matrix

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MA117 Programming for Scientists: Project 3 Deadline: Noon, Monday 4th May 2020

L are 1. Then, you should show that the matrix product

1

l

∗

2

1

l

∗

3

1

.

.

.

.

.

.

l

∗

n 1

| {z }

L

d

∗

1 u

∗

1

d

∗

2 u

∗

2

.

.

.

.

.

.

d

∗

n−1 u

∗

n−1

d

∗

n

| {z }

U

is tri-diagonal. Finally, set the product equal to the matrix T above, and equate co-efficients to

find a difference relation for each of the d

∗

i

, u

∗

i

and l

∗

i

. Just like the decomp method above, you

can then store the matrix in a compact form inside a TriMatrix.

2.4 The Project3 class

The final part of this project is to generate some simple statistics on random matrices. Here, the

definition of random is that each co-efficient of the matrix M will have Mij ∼ U(0, 1) (i.e. a uniformly

distributed random number between 0 and 1). X = det(M) is a random variable: the question is, how

is X distributed? In particular, you will estimate the variance σ

2 = var(X) by generating a number

of random matrices of various sizes, and then calculate the determinant of each of the samples.

Project3 contains two functions to aid you in this endeavour. It is not meant to be challenging –

indeed, it is probably the easiest part of the assignment!

• matVariance(): This function will be passed a Matrix object and an integer Nsamp. It should

generate random matrices Mi for 1 ≤ i ≤ Nsamp by calling the random function on the passed

matrix. The variance is estimated by

σ

2 = E(X2

) − [E(X)]2 =

1

Nsamp

N

Xsamp

i=1

det(Mi)

2 −

1

Nsamp

N

Xsamp

i=1

det(Mi)

2

You should not store each of the random samples, as this will consume a huge amount of

memory for large values of Nsamp.

• main: Your main function should not be a tester in this class. Instead, for 2 ≤ n ≤ 50, it should

create a n × n GeneralMatrix and a TriMatrix, and pass these to matVariance() to calculate

the variance of the distribution for this value of n. For each n, you should generate 15, 000

general matrix samples and 150, 000 tri-diagonal samples.

Ensure that you test matVariance() intensively before running with large numbers of samples.

Start with a small number of samples at first to ensure you are not encountering infinite loops,

etc. My solution code completes this in around a minute (on my laptop), so this should be your

aim. You should print this information out to the terminal. On each line, print out

n var1 var2

where n is the value of n, var1 is the variance found for the GeneralMatrix and var2 is the

variance found for the TriMatrix.

Finally, you should plot two graphs with the data you find; one for the general matrix and one for the

tri-diagonal. Along the x-axis plot the matrix size, and along the y-axis, the logarithm of the variance.

To save your output to a file, on daisy you can run the command

java Project3 > variance.data

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MA117 Programming for Scientists: Project 3 Deadline: Noon, Monday 4th May 2020

and then transfer this file to your computer – for more information, see the week 12 and 15 lab notes,

where you did something similar. Once you have this on your computer, you can then issue the

following commands in Matlab to produce the plots:

load ’variance.data’

subplot(211)

semilogy(variance(:,1), variance(:,2), ’r’)

subplot(212)

semilogy(variance(:,1), variance(:,3), ’b’)

orient landscape

saveas(figure(1), ’VarGraph.pdf’)

This will create two subplots; the top one containing the general matrix variance, the bottom the

tri-diagonal matrix. Finally it saves the plots as a PDF file, which can be opened in Adobe Reader or

similar viewers. You should add labels to the plot.

3 A note on efficiency

Your code will not, generally, be tested for efficiency, and will not be tested for very large matrices; at

most, you will be given a 100×100 matrix. However, it perfectly possible to calculate the determinant

of such a matrix in much less than a second using the methods outlined here. Bear in mind that you

will need a certain amount of efficiency in your code in order to complete the Project3 class.

Whilst it is certainly possible to write all of this code on daisy, I heartily encourage you to do

your initial testing on your laptop and desktop machines. Not only do they provide a more friendly

development environment, but if you inadvertantly run code with an infinite loop, it will not have an

impact on other users.

Finally, the Project3 class can be quite time-consuming, but shouldn’t take more than a couple of

minutes to run. You should test this on your own machine if possible; if not, then reduce the number

of samples generated at first to get an initial indication of the time it will take to run. Remember that

if your code is taking minutes to calculate determinants of small matrices, something is very wrong!

4 Submission

You should submit, using the Tabula system, the following four files: Matrix.java, GeneralMatrix.java,

TriMatrix.java and Project3.java, as well as a PDF containing both of your plots called VarGraph.pdf.

I will not accept any other format for this plot (Word, Excel, etc). Before you do that you should test

that all your methods work properly (use the method main you implement in each class).

There will be a large number of tests performed on your classes. This should allow for some partial

credit even if you don’t manage to finish all tasks by the deadline. Each class will be tested individually

so you may want to submit even a partially finished project. In each case, however, be certain that you

submit Java files that compile without syntax error. Submissions that do not compile will be marked

down. As before you can re-submit solutions as many times as you wish before the deadline; however,

ensure that you re-submit all files.

Finally, please ensure that you keep back-up copies of your work. Lost data do not present a valid

excuse for missing the deadline.

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