Mar Numerical approach for large-scale Eigenvalue problems 1 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 Introduction

Mar Numerical approach for large-scale Eigenvalue problems 2 Understanding with Mechanical Engineering l 2l l mmm l x2x2 x1x1 T T T T sin a)Three beads on a string b)The forces on bead 1 Vibration of beads perpendicular to string x i+1 a T sin xixi x i+1 - x i A CB c)The approximation for sin Governing equations are

Mar Numerical approach for large-scale Eigenvalue problems (Modes of vibration) Physical interpretation: Physical Interpretation

Mar 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’ Degree of polynomial in order of A General Formulation - discussions

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

Mar 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 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 Jacobi method Subspace iteration technique

Mar 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

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

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

Mar 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.

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

Mar 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.

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

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

Mar 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

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

Mar 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.

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

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

Mar 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.

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

Mar 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

Mar 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