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Com S 228 Decoding an Archived Message
创建日期：2024-06-26 14:49:18
标签： Com S 228
Decoding an Archived Message

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Com S 228  Decoding an Archived Message


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!×#, 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 ?$ 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 ? operations. For large values of ?, this significantly improves the performance of our code.

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

? is a dense ? × ? matrix which, in general, has no special structure and no zero entries. ? is a tri-diagonal matrix, where all entries are zero apart from along the diagonal and upper and lower diagonals. Note that although ? is only a 5 × 5 matrix, your classes should represent a general ? × ? 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 trickiest 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 ? × ? matrix requires ?(?!) operations, since for each ? × ? matrix, we must calculate the values of the ? − 1 sub-determinants. This is extremely slow.

A much better method is called LU decomposition. In this, we write a matrix ? as product of two matrices ? and ? which are lower- and upper- triangular respectively. For example, for a 4×4 matrix, we would find matrices so that

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 ≤ ?, ? ≤ ?, ?.
However, in your code, you should stay consistent with Java notation and indices should start at 0
(i.e. 0 ≤ ?, ? ≤ ? − 1, ? − 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, parameter types and return types of public objects and methods as they are in the templates. Other methods must 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.

• 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 ? × ? real matrix.

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

• Project3: a 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.


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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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