ComputingEigenvaluesAndEigenvectors

Project 2: Computing Eigenvalues and Eigenvectors using Normalized

Power Iteration Algorithm and Inverse Iteration Algorithm

This project was assigned for the Numerical Methods for Electrical Engineering (EE 242) course in the Spring 2021 semester.

Run on Terminal

g++ main.cpp -std=c++11 -o test
test A.txt 1e-6 X.txt

Project Goals

In this project, you will be implementing the normalized power iteration algorithm and the inverse iteration algorithm to find the largest and the smallest eigenvalues and corresponding eigenvectors of a given matrix A, where A is a real square matrix. Using normalized power iteration, you can find the dominant eigenvalue of A with the corresponding eigenvector. The inverse iteration algorithm is equivalent to the power iteration method applied to 𝐴⁻¹. Therefore, using inverse iteration algorithm the smallest eigenvalue of A is the reciprocal of the dominant eigenvalue of 𝐴⁻¹ and the corresponding eigenvector can be found. You should solve a system of linear equations instead of finding 𝐴⁻¹ directly. (Hint: You can use your first projects to solve the linear system of equations, but this time you should write your program as an object-oriented one.)

Your program should read A from an input file and output the dominant eigenvalue, its corresponding eigenvector and the smallest eigenvalue with its corresponding eigenvector as a text file.

Programming Details

Your program should,

• have three command-line arguments for the parameters. (Command line arguments can be thought of as the inputs of the main function.) The first argument is the name of the file you read the matrix from, the second argument is the tolerance, which will be used in the normalized power iteration algorithm and inverse iteration algorithm, and the third argument is the name of your output file,
• use dynamically allocated memory to store the matrix,
• print out an error message and quit if it detects any problems,
• calculate the eigenvalues and corresponding eigenvector and write them in a text file.

About Object-Oriented Programming

In this project, you will use matrices and matrix operations. In order to implement this project as an object-oriented program; for example, you can declare a class (an object) named Matrix, and implement all matrix operations such as multiplication, addition, transpose, and solving a system of linear equations (LU factorization and backward substitution) as methods of this class. This way, you don’t need to have a lot of complex loops in your program.