Eigenproblem Basics and Algorithms
Some might say that the <i>eigenproblem</i> is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the <i>ansatz</i> of solving many linear and nonlinear problems....
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-11-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/15/11/2046 |
| _version_ | 1797457718645096448 |
|---|---|
| author | Lorentz Jäntschi |
| author_facet | Lorentz Jäntschi |
| author_sort | Lorentz Jäntschi |
| collection | DOAJ |
| description | Some might say that the <i>eigenproblem</i> is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the <i>ansatz</i> of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided. |
| first_indexed | 2024-03-09T16:25:03Z |
| format | Article |
| id | doaj.art-dd21236138d84f468fe856e55bbdde2d |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T16:25:03Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-dd21236138d84f468fe856e55bbdde2d2023-11-24T15:08:53ZengMDPI AGSymmetry2073-89942023-11-011511204610.3390/sym15112046Eigenproblem Basics and AlgorithmsLorentz Jäntschi0Department of Physics and Chemistry, Technical University of Cluj-Napoca, Muncii 103-105, 400641 Cluj-Napoca, RomaniaSome might say that the <i>eigenproblem</i> is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the <i>ansatz</i> of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.https://www.mdpi.com/2073-8994/15/11/2046algorithmscharacteristic polynomialeigendecompositioneigenfunctioneigenpaireigenproblem |
| spellingShingle | Lorentz Jäntschi Eigenproblem Basics and Algorithms Symmetry algorithms characteristic polynomial eigendecomposition eigenfunction eigenpair eigenproblem |
| title | Eigenproblem Basics and Algorithms |
| title_full | Eigenproblem Basics and Algorithms |
| title_fullStr | Eigenproblem Basics and Algorithms |
| title_full_unstemmed | Eigenproblem Basics and Algorithms |
| title_short | Eigenproblem Basics and Algorithms |
| title_sort | eigenproblem basics and algorithms |
| topic | algorithms characteristic polynomial eigendecomposition eigenfunction eigenpair eigenproblem |
| url | https://www.mdpi.com/2073-8994/15/11/2046 |
| work_keys_str_mv | AT lorentzjantschi eigenproblembasicsandalgorithms |