MODULE 6B | Iterative Methods in Linear Algebra MODULE 6B Iterative Methods in Linear Algebra Eigenvalue Algorithms Power Method | Shifted-Inverse Power Method CE 25 Mathematical Methods in Civil Engineering II 22 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra Eigenvalue Algorithms Matrix Eigenvalue Problem โช Matrix eigenvalue problems are concerned with vector equations of the form: ๐จ {๐} = ๐{๐} EIGENVECTOR (characteristic vector) EIGENVALUE (characteristic value) ๐จ: the given SQUARE matrix (in this case, all matrices to be handled are square matrices) CE 25: Mathematical Methods in CE II 23 Examples of eigenvalue problem encountered in civil engineering are: (a) buckling in Mechanics of Materials, and (b) determining mode shapes in Structural Dynamics (pertinent to vibration in structures) 23 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra EIGENVALUE ALGORITHMS Matrix Eigenvalue Problem RECALL Theorem on Eigenvalues and Eigenvectors: Theorem 1. The eigenvalues of a square matrix A are the roots of the characteristic equation of A. Hence, an n x n matrix has at least one and at most n numerically different eigenvalues. CE 24: Mathematical Methods in CE II 24 Previously on CE 24 24 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra EIGENVALUE ALGORITHMS Matrix Eigenvalue Problem Analytical solution determines eigenvalues ๐ first as the roots of p ๐ = det ๐จ − ๐๐ฐ = 0. The eigenvectors ๐ are then computed for each eigenvalue be getting the solution of ๐จ − ๐๐ฐ ๐ = 0. ๐จ๐ = ๐๐ ๐จ๐ = ๐๐ฐ๐ ๐จ๐ − ๐๐ฐ๐ = ๐ ๐จ − ๐๐ฐ ๐ = ๐ CE 24: Mathematical Methods in CE II 25 Previously on CE 24 25 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra Eigenvalue Algorithms Dominant Eigenpair Assume that the ๐ × ๐ matrix ๐จ has n distinct eigenvalues ๐1 , ๐2 , … , ๐๐ and they are ordered in decreasing magnitude; that is, ๐1 > ๐2 > ๐3 > โฏ > |๐๐ | Eigenvalue ๐1 is called the dominant eigenvalue. Eigenvector ๐๐ corresponding to ๐1 is called a dominant eigenvector. CE 25: Mathematical Methods in CE II 26 The eigenvalue and eigenvector together is called an eigenpair. 26 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra Eigenvalue Algorithms Power Method If ๐ฅ0 is chosen appropriately, then the sequences ๐๐ = ๐ฅ1๐ ๐ฅ2๐ … ๐ฅ๐๐ ๐ and {๐๐ } can be generated recursively by ๐๐ = ๐จ ๐๐ and 1 {๐๐+๐ } = {๐๐ } ๐๐+1 where ๐ ๐๐+1 = max ๐๐ will converge to the dominant eigenvector ๐๐ and dominant eigenvalue ๐1 , respectively. CE 25: Mathematical Methods in CE II 27 Here, a variable in bold is a matrix. 27 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra Eigenvalue Algorithms Limitation of Power Method โช The power method only determines the dominant eigenpair. โช Power method can be used to determine eigenpairs of matrices with distinct eigenvalues. If two or more eigenvalues of a matrix ๐จ have equal magnitude, the method may not converge. CE 25: Mathematical Methods in CE II 28 These limitations will be addressed by the other method that we will be discuss in this module. 28 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra Eigenvalue Algorithm EXAMPLE 6B-3 Use the power method to find the dominant eigenpair for the matrix 0 11 −5 ๐จ = −2 17 −7 −4 26 −10 CE 25: Mathematical Methods in CE II 29 29 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra Eigenvalue Algorithm EXAMPLE 6B-3 Use the power method to find the dominant eigenpair for the matrix 0 11 −5 ๐จ = −2 17 −7 −4 26 −10 ๐ ๐ = ๐จ ๐๐ 1 {๐๐+๐ } = {๐๐ } ๐๐+1 ๐๐+1 = max ๐๐ ๐ FIRST ITERATION Step 1. Select initial guess, say ๐๐ = 1 Step 2. Compute ๐ฆ0 . 0 ๐๐ = −2 −4 11 17 26 1 1 ๐ 1เต −5 1 6 2 −7 1 = 8 = 12 2เต 3 12 −10 1 1 ๐๐ Step 3. Identify ๐1 . For this case, ๐1 = 12. Step 4. Compute ๐๐ . ๐๐ = 1 12 1เต 6 2 8 = 2เต 3 12 1 ๐1 CE 25: Mathematical Methods in CE II 30 As a numerical method, we also give an initial guess to {๐ฅ}. One iteration consists of: (a) Computing for {๐ฆ๐ } (b) Determining absolute max element in {๐ฆ๐ } → this will be ๐๐+1 (c) Dividing {๐ฆ๐ } by ๐๐+1 to get {๐ฅ๐+1 } 30 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra Eigenvalue Algorithm EXAMPLE 6B-3 Use the power method to find the dominant eigenpair for the matrix 0 11 −5 ๐จ = −2 17 −7 −4 26 −10 FIRST ITERATION SECOND ITERATION Step 1. Compute ๐ฆ0 . 1เต 2 ๐1 {๐๐ } = 12 2เต 3 1 ๐๐ ๐ 0 = −2 −4 11 17 26 −5 −7 −10 7เต 1เต 3 2 16 10 เต3 = ๐2 ๐๐ = 2เต = 3 3 16เต 1 3 {๐ฅ๐ }๐ ๐๐ 0 Steps 2 and 3. Identify ๐2 and compute ๐ฅ2 . ๐ 1.00000 1.00000 1.00000 1 12.0000 0.50000 0.66667 1.00000 2 5.33333 0.43750 0.62500 1.00000 6.66667 7เต 16 5เต 8 1 Step 4. Compute error. Terminate accordingly. ๐๐ = |๐๐ − ๐๐−1 | CE 25: Mathematical Methods in CE II 31 Error can only be computed after the second iteration because we are comparing the ๐๐ values. 31 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra Eigenvalue Algorithm EXAMPLE 6B-3 Use the power method to find the dominant eigenpair for the matrix 0 11 −5 ๐จ = −2 17 −7 −4 26 −10 1 2 ๐1 {๐๐ } = 12 2 3 1 9 ๐3 {๐๐ } = 2 5 12 11 18 1 ๐2 {๐๐ } = 16 3 38 ๐4 {๐๐ } = 9 7 16 5 8 1 ๐ 0 31 76 23 38 1 ๐ 1.00000 1.00000 1.00000 1 12.0000 0.50000 0.66667 1.00000 2 5.33333 0.43750 0.62500 1.00000 6.66667 3 4.50000 0.41667 0.61111 1.00000 0.83333 4 4.22222 0.40789 0.60526 1.00000 0.27778 5 4.10526 0.40385 0.60256 1.00000 0.11696 6 4.05128 0.40190 0.60127 1.00000 0.05398 4.00000 0.40000 0.60000 1.00000 0.00000 … 20 CE 25: Mathematical Methods in CE II 32 (๐ฅ๐ )๐ ๐๐ 32 CE 25 – Mathematical Methods in Civil Engineering II MODULE 6B | Iterative Methods in Linear Algebra Eigenvalue Algorithm EXAMPLE 6B-3 Use the power method to find the dominant eigenpair for the matrix 0 11 −5 ๐จ = −2 17 −7 −4 26 −10 The dominant eigenpair of matrix A is ๐ 0 0.4 ๐1 = 4, {๐๐ } = 0.6 1 (๐ฅ๐ )๐ ๐๐ ๐ 1.00000 1.00000 1.00000 1 12.0000 0.50000 0.66667 1.00000 2 5.33333 0.43750 0.62500 1.00000 6.66667 3 4.50000 0.41667 0.61111 1.00000 0.83333 4 4.22222 0.40789 0.60526 1.00000 0.27778 5 4.10526 0.40385 0.60256 1.00000 0.11696 6 4.05128 0.40190 0.60127 1.00000 0.05398 4.00000 0.40000 0.60000 1.00000 0.00000 … 20 CE 25: Mathematical Methods in CE II 33 After 20 iterations, we arrive at this dominant eigenpair. 33 CE 25 – Mathematical Methods in Civil Engineering II