Here I have written the code to calculate the Eigenvalues of a Matrix(Used for Calculating the Principal Components and Loading Vectors), using the QR-Algorithm.
I have mainly used 3 variants:
- Theoretical QR-Algorithm: Which Fails for Large Matrices and is Slower.
- Practical QR-Algorithm: Which is Better than the Theoretical QR-Algorithm, which involves reducing Matrix to Upper-Hessenberg(Tridiagonal for Symmetric Matrix) Form and then Applying QR-Algorithm.
- Practical QR-Algorithm with Rayleigh Shifts: Which helps in breaking symmetry, sometimes.
I have added the code to calculate the Sample Variance-Covariance Matrix also.
The Hessenberg Reduction of a Matrix and the QR-Decomposition of a Matrix are also been performed and the respective codes are present.
For Both the above Reduction and Decompositions the Householder Matrices are used.
Usage(Recommended):
- Run the Eigenvalues_of_Covariance_Matrix.R(which computes the covariance matrix from the input matrix and finds it's eigenvalues and eigenvectors) or Eigenvalue_Matrix.R R Scrpit(which finds the eigenvalues and eigenvectors of the input matrix) Files from RStudio.
- Execute All the Lines of Code from the Script Files in RStudio. In this way we can change the input parameters of the function for better result.
Alternate Usage:
- Run the R script file using "Rscript Eigenvalues_of_Covariance_Matrix.R" from command line(For PCA)(For Eigenvalue Calculation of Covariance Matrix).
- Run the R script file using "Rscript Eigenvalue_Matrix.R" from command line(For Eigenvalue Calculation of a Matrix).
- A GUI Prompt to select the .csv file containing data matrix is presented. Select the .csv file containing data matrix(Example - Matrix2.csv)(without column names i.e. header and without row names).
- Prints out the number of steps after which convergence is obtained, along with the Eigenvalues and Eigenvectors.(If Convergence is not attained, then change the inputs of tol, n, method in the end of the file where function is called in the Script).
Tip:
- Sometimes the path to the RScript.exe in the bin folder of the installed directory of R is not added to path. Add it if it is already not added in order to use the Rscript command.
- Follow the Comments Mentioned near the end of each of the above Script Files. They contain examples on how to run the functions, and what are the inputs to the function.