Lab03

Laboratory 3 - Example - Classes/Structures, Inheritance, Properties, Access Modifiers #

Introduction #

The task for today’s laboratory is to create a simple library responsible for matrix operations.

Note #

To proceed to the next stages, you must uncomment the individual lines like #define STAGE0X located at the beginning of the Program.cs file. No other interference with this file is allowed.

Starting code #

Matrices

Stage 1 (4 points) #

In the Matrix folder, create an abstract class Matrix that will have:

  • a constructor that takes the number of rows and columns of the matrix as int
  • a public property Rows, which returns the number of rows in the matrix
  • a public property Columns, which returns the number of columns in the matrix
  • an abstract indexer that takes a row and column number. It allows reading and writing a float value to a specific cell of the matrix.
  • an overridden ToString method, so that matrices are printed in the same way as in the sample program output

Create the following classes that inherit from Matrix: DenseMatrix and SparseMatrix.

DenseMatrix should store elements in the form of a rectangular array.

SparseMatrix should store only the non-zero elements of the matrix in a dictionary. This implementation reduces the amount of memory needed when storing sparse matrices, i.e., those in which a significant number of values are zero.

Both classes should have a constructor that accepts the number of rows and columns. Newly initialized matrices should be filled with zeros.

The program should check the validity of the arguments. In case of invalidity, it should throw an ArgumentException with an appropriate comment.

Hints #

Scoring #

2 points for each derived class.

Stage 2 (3 points) #

To the Matrix class, add abstract methods Identity, Transpose, and Norm. Implement them in the derived classes.

The Identity method takes nothing and returns nothing. It only fills the matrix to be an identity matrix. In the case of a non-square matrix, the method should throw an InvalidOperationException.

The Transpose method returns the transposed matrix.

The Norm method returns the norm of the matrix defined as \[ ||A|| = \sqrt{\sum_i \sum_j A_{i,j}^2}. \]

Implementations in the SparseMatrix class should have a linear time complexity with respect to the number of non-zero elements of the matrix.

Scoring #

1 point for each method.

Stage 3 (3 points) #

To the Matrix class, add an abstract method GetInstance that takes the number of rows and columns and returns an object of type Matrix. This method should not be callable from outside this class and its derived classes. Implement this method in the derived classes in such a way that each class returns an instance of its own class with the requested size.

In the Matrix class, implement the following operator overloads:

  • matrix addition operator
  • matrix subtraction operator
  • matrix multiplication operator
  • matrix-scalar multiplication operator

In the implementation, use the GetInstance method. For functions that take one matrix, it should return a matrix of the same type. For functions that take two matrices, the returned implementation should be the same as the left (first) argument.

Hints #

Scoring #

1 for the GetInstance method, 0.5 points for each overload.

Stage 4 (2 points) #

In the final stage, we will use the interface we built during the previous stages. You need to create a MatrixAlgorithms class, and in it, a static method PseudoInverse that takes a Matrix and a maximum number of iterations, which has a default value of 1000. Use \( 10^{-7} \) for the value of \( \epsilon \) .

For the purpose of this lab, we will only deal with square matrices.

Algorithm:

\( A \) - input square matrix

\( I \) - identity matrix

\( \alpha = \frac{1}{||A||^2} \)
\( A1 = \alpha \cdot A^T \)
\( \text{for } i=1...maxIter: \)
\( \qquad A2 = A1 + A1 \cdot (I - A\cdot A1) \)
\( \qquad \text{if } ||A2-A1|| < \epsilon: \)
\( \qquad \qquad \text{break} \)
\( \qquad A1 = A2 \)
\( \text{return A2} \)

Scoring #

Correct implementation of the algorithm - 2 points. Partial implementation - 1 point

Example solution #

Matrices

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