Definition Why do we study it ? Is the Behavior system based or nodal based?

1. Introduction • Definition • Why do we study it ? • Is the Behavior system based or nodal based? • What are the real time applications • How do we calculate these values • What are the applications and applicable areas • A lot many definitions and new way of looking at the things • Matrix theories and spaces : an overview Numerical approach for large-scale Eigenvalue problems

2. T sin T m m m x2 T x1 T l l l 2l 2l 2l T sin A xi+1- xi xi+1 C B xi a c)The approximation for sin T sin Vibration of beads perpendicular to string Understanding with Mechanical Engineering a)Three beads on a string b)The forces on bead 1 Governing equations are Numerical approach for large-scale Eigenvalue problems

3. 1 2 1 1 0 -1 1 1 -1 (Modes of vibration) Physical Interpretation Physical interpretation: Numerical approach for large-scale Eigenvalue problems

4. In many physical applications we often encounter to study system behavior : The problem is called the Eigenvalue problem for the matrix A. is called the Eigenvalue (proper value or characteristic value) value u is called the Eigenvector of A. System can be written as To have a non-trivial solution for ‘u’ order of A Degree of polynomial in General Formulation - discussions Numerical approach for large-scale Eigenvalue problems

5. A A has distinct eigenvalues A has multiple eigenvalues Eigenvectors are linearly independent Eigenvectors are linearly independent Eigenvectors are linearly dependent semi simple matrix (diagonalizable) Non - semi simple (non - diagonalizable) Distinguishion Numerical approach for large-scale Eigenvalue problems

6. Spectral Approximation Set of all Eigenvalues : Spectrum of ‘A’ Single Vector Iteration • Power method • Shifted power method • Inverse power method • Rayliegh Quotient iteration Jacobi method Subspace iteration technique Deflation Techniques • Wielandt deflation with one vector • Deflation with several vectors • Schur - Weilandt deflation • Practical deflation procedures Projection methods • Orthogonal projection methods • Rayleigh - Ritz procedure • Oblique projection methods Numerical approach for large-scale Eigenvalue problems

7. Power method • It’s a single vector iteration technique • This method only generates only dominant eigenpairs • It generates a sequence of vectors This sequence of vectors when normalized properly, under reasonably mild conditions converge to a dominant eigenvector associated with eigenvalue of largest modulus. Methodology: Start : Choose a nonzero initial vector Iterate : for k = 1,2,…… until convergence, compute Numerical approach for large-scale Eigenvalue problems

8. Why and What's happening in power method To apply power method, our assumptions should be Numerical approach for large-scale Eigenvalue problems

9. Eigenvalues of A …. Eigenvalues of (A- I) Shifted Power method Numerical approach for large-scale Eigenvalue problems

10. Inverse power method-Shifted Inverse power method Basic idea is that Advantages 1.Least dominant eigenpair of A 2.Faster convergence rate Shifted Inverse power method: The same mechanism follows like in the shifted power method and only thing is that we will achieve faster convergence rates in comparison. Numerical approach for large-scale Eigenvalue problems

11. Rayliegh Quotient Numerical approach for large-scale Eigenvalue problems

12. Deflation Techniques Definition : Manipulate the system After finding out the largest eigenvalue in the system,displace it in such away that next larger value is the largest value in the system and apply power method. Weilandt deflation technique It’s a single vector deflation technique. Numerical approach for large-scale Eigenvalue problems

13. Deflation with several vectors: It uses the Schur decomposition Numerical approach for large-scale Eigenvalue problems

14. Schur - Weilandt Deflation Numerical approach for large-scale Eigenvalue problems

15. Projection methods Suppose if matrix ‘A’ is real and the eigenvalues are complex.consider power method where dominant eigenvalues are complex and but the matrix is real. Although the usual sequence Numerical approach for large-scale Eigenvalue problems

16. Orthogonal projection methods Numerical approach for large-scale Eigenvalue problems

17. What exactly is happening in orthogonal projection Suppose v is the guess vector, take successive two iterations in the power method. Form an orthonormal basis X = [v|Av]. Do the Gram - Schmidt process to QR factorize X. Say v is nX1 and Av is ofcourse of nX1. So ‘X’ is now nX2 matrix. Q = [q1|q2] where q1 = q1/norm(q1); q2 = projection onto q1; now Q is perfectly orthonormalized. q1,q2 form an orthonormal basis which spans x,x, which are corresponding eigenvectors as it converges. Numerical approach for large-scale Eigenvalue problems

18. Rayleigh - Ritz procedure Numerical approach for large-scale Eigenvalue problems

19. Oblique projection method Numerical approach for large-scale Eigenvalue problems

20. Jacobi method Jacobi method finds all the eigenvalues and vectors at a time. The inverse of orthogonal matrix is its own transpose. Numerical approach for large-scale Eigenvalue problems

21. Jacobi method…. Numerical approach for large-scale Eigenvalue problems

22. Subspace Iteration Techniques • It is one of the most important methods in the structural engineering Bauer’s method Numerical approach for large-scale Eigenvalue problems

23. Applications and Applicable areas • Problems related to the analysis of vibrations • usually symmetric generalized eigenvalue problems • Problems related to stability analysis • usually generates non - symmetric matrices • Structural Dynamics • Electrical Networks • Combustion processes • Macroeconomics • Normal mode techniques • Quantum chemistry • chemical reactions • Magneto Hydrodynamics Numerical approach for large-scale Eigenvalue problems