Eigenvalues and Singular Values Recent Development with Applications

Abstract

The theory and computation of eigenvalues and eigenvectors are key tools in applied mathematics and scientific computing. They are widely utilized due to their effectiveness in solving various problems. This thesis focuses on studying existing approaches for finding eigenvalues and exploring their applications in image processing. The present thesis proposes a numerical technique to solve the fractional eigenvalue problem that is based on the Legendre wavelet. It reduces the computational complexity of the given problem by turning it into a set of algebraic equations. This technique also utilizes the exact fractional integration of the Legendre wavelet, enabling the computation of both real and complex eigenvalues and their corresponding eigenfunctions. The accuracy of the approximated solution is improved by increasing the Legendre wavelet parameters. Moreover, the numerical convergence has also been investigated by analyzing their performance through several examples. Following that, an eigenfunction approach has also been introduced in this thesis to solve multi-order fractional differential equations where the solution is expressed as a linear combination of eigenfunctions. Further, the application of eigenvalues and eigenvectors is studied in image processing. A singular value is a term that is related to the absolute value of the eigenvalue. Three methods have been introduced using singular value decomposition for enlarging images. These methods effectively transfer the detailed information of the low-resolution images to the high-resolution images by using interpolation on feature vectors which are obtained from the singular value decomposition of given images. For grayscale images, singular value decomposition and cubic spline interpolation-based method is proposed, which also includes image compression using a few dominant feature vectors (eigenvectors). The interpolated points for interpolation are obtained by using increments on the given feature vectors.