Question

(a) Write a MATLAB script that stores a matrix in compressed column form. (b) Write a symmetric version of your program, that is, a script for symmetric matrices, where if \(a_{i, j}\) is stored \((i \neq j)\), then \(a_{j, i}\) is not. Try your programs on a \(5 \times 5\) example of your choice.

Short answer

- The main purpose of these MATLAB scripts is to store matrices in a compressed format, specifically compressed column form and without duplicates for symmetric matrices, to save memory and computational resources. Which MATLAB function is used to find the non-zero elements and their corresponding positions in a matrix? - The `find()` function is used to find the non-zero elements and their corresponding positions in a matrix. How does the first script store a matrix in compressed column form? - The first script stores a matrix in compressed column form by creating a new matrix containing the non-zero elements, and their row and column numbers as its columns. How does the second script store a symmetric matrix without storing duplicate elements? - The second script stores a symmetric matrix without storing duplicate elements by first getting the upper triangle of the matrix (including the diagonal), finding the non-zero elements and their positions, and then creating a matrix containing this information.

Step-by-step solution

Create a MATLAB script storing a matrix in compressed column form

First, create a new script file in MATLAB. Then, follow these steps to store a matrix in compressed column form (using the variable 'A' for the original matrix and 'compressedA' for the compressed form): 1. Initialize a sample 5x5 matrix `A` with some non-zero values. 2. Find the non-zero elements and their corresponding positions in the matrix. 3. Create a new matrix `compressedA` with as many rows as non-zero elements and three columns. 4. Store the non-zero elements, their row number, and the column number in the `compressedA` matrix. Here's the MATLAB code for this: ```MATLAB % Compressed column storage script A = [1 0 0 0 5; 0 2 0 0 0; 0 0 3 0 0; 0 0 0 4 0; 5 0 0 0 5]; % Example matrix [nzRows, nzCols, nzVals] = find(A); % Find non-zero elements and positions compressedA = [nzRows, nzCols, nzVals]; % Store non-zero elements in compressed form ```

Create a MATLAB script storing a symmetric matrix, without duplicates

First, create a new script file in MATLAB. Then, follow these steps to create a script that stores a symmetric matrix, without duplicates: 1. Initialize a sample 5x5 symmetric matrix `B` with some non-zero values and ensure it's symmetric. 2. Find the non-zero elements in the upper triangle matrix (including the diagonal) and their corresponding positions. 3. Create a new matrix `compressedB` with as many rows as non-zero elements in the upper triangle and three columns. 4. Store the non-zero elements, their row number, and the column number in the `compressedB` matrix. Here's the MATLAB code for this: ```MATLAB % Symmetric matrix storage script B = [2 3 0 4 0; 3 1 6 0 7; 0 6 5 0 0; 4 0 0 6 9; 0 7 0 9 8]; % Example symmetric matrix upperB = triu(B); % Get upper triangle of B (including diagonal) [nzRows, nzCols, nzVals] = find(upperB); % Find non-zero elements & positions in the upper triangle compressedB = [nzRows, nzCols, nzVals]; % Store non-zero elements in compressed form ``` Now you have two MATLAB scripts that store a matrix in compressed column form and a symmetric matrix, without storing duplicate elements, respectively. You can test these scripts using any \(5 \times 5\) matrix example of your choice.

Key concepts

Compressed Column Storage

Compressed Column Storage (CCS) is a technique used to store sparse matrices in a more memory-efficient way. A sparse matrix is one that has a majority of its elements as zero, and so, storing only the non-zero elements can lead to significant memory savings.

In CCS, the non-zero values, along with their row indices and starting positions of each column in the value array, are stored. Typically, this is done using three arrays: one for the non-zero values, one for the row indices of these values, and one for the column pointers that indicate where each column starts in the other two arrays.

MATLAB makes it easy to compress a matrix into this format using the 'find' function. Relevant to the exercise, a MATLAB script was written that uses this function to extract non-zero elements and then formats them into a compressed form, saving space and enhancing computational efficiency during matrix operations.

Symmetric Matrix Storage

In the context of symmetric matrices, where the element (aij) is equal to the element (aji), redundant storage can be eliminated by only storing elements from either the upper or lower triangle of the matrix. This is what we refer to as Symmetric Matrix Storage.

The script provided in the exercise takes advantage of MATLAB's 'triu' function to extract the upper triangular part of a symmetric matrix, including the diagonal. This step is key as it ensures no duplicate storage for the symmetric elements. 'find' is then used again to locate non-zero elements within this upper triangle, and these are stored, thus negating the need to store the symmetrical lower triangle and effectively halving the storage requirement for a symmetric matrix.

MATLAB Programming

MATLAB, short for Matrix Laboratory, is a programming environment that is widely used for numerical computing. It is particularly beloved by engineers and scientists for its powerful matrix capabilities and its ease of use when it comes to mathematics and data visualization.

The scripts in the exercise exemplify practical MATLAB programming techniques. These include creating matrices, utilizing built-in functions for processing matrices such as 'find' and 'triu', and condensing data into a more manageable and efficient format. MATLAB's high-level language enables the creation of scripts for complex operations like matrix compression with relatively simple and readable code, which is also an opportunity to teach efficiency and conciseness in programming.

Sparse Matrix Representation

Sparse matrix representation is an essential concept in computational mathematics where the aim is to efficiently store and compute matrices that have a significant number of zero entries. In many real-world problems, matrices can be massive, and most of their elements can be zero. Storing all these zeros is unnecessarily memory-intensive.

Therefore, sparse matrix formats like Compressed Sparse Column (CSC) or Compressed Sparse Row (CSR) were designed to store only the non-zero elements and their locations. These formats are proficient in memory usage as well as performance when carrying out matrix operations, such as multiplication or inversion. In the MATLAB scripts given, the 'find' function is used to facilitate sparse storage by returning the row indices, column pointers, and values of the non-zero elements, neatly demonstrating the application of sparse matrix representation in a programming context.