Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion A Homotopy Method for Computing All Isolated Solvents of the Quadratic Matrix Equation AX 2 + BX + C = 0 Yongwen Hou Bo Yu School of Mathematical Sciences, Dalian University of Technology ASCM Beijing Oct. 26-28, 2012 HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Outline Introduction 1 Homotopy method for quadratic matrix equation 2 More results for special problems 3 Numerical results and conclusion 4 HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Quadratic matrix equation The unilateral quadratic matrix equation P ( X ) = AX 2 + BX + C = 0 , (QME) is considered, whose coefficients A , B , and C ∈ C n × n , and the matrix solution X is called a solvent . The corresponding quadratic eigenvalue problem is: P ( λ ) v = ( λ 2 A + λ B + C ) v = 0 , (QEP) λ ∈ C and v � = 0 ∈ C n , where λ and v are eigenvalue and eigenvector respectively. HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Relations between solvents and eigenvalues. Given a solvent X , satisfying P ( X ) = AX 2 + BX + C = 0 , then, P ( λ ) can be divided by the linear term X − λ I on the right: λ 2 A + λ B + C = − ( B + AX + λ A )( X − λ I ) , and thus, eigenvalues of P ( λ ) are those of X and those of a generalized eigenvalue problem ( B + AX ) v = − λ Av . HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Relations between solvents and eigenvalues. For n eigenpairs { λ i , v i } n i =1 of P ( λ ), where the eigenvectors v 1 , ..., v n are linearly independent, denoting V = [ v 1 , ..., v n ] and Λ = diag ( λ 1 , ..., λ n ) , then AV Λ 2 + BV Λ + CV = 0 is satisfied. Multiplied by V − 1 on the right, we have AV Λ 2 V − 1 + BV Λ V − 1 + C = 0 , which indicates V Λ V − 1 is a solvent of P ( X ). HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Solvents of quadratic matrix equation P ( X ) can have no solvents, a finite positive number, or infinitely many. For example, the equation X 2 − J n ( λ ) , where J n ( λ ) ( n > 1) is a Jordan block with the eigenvalue λ , has no solvents when λ = 0, and precisely two solvents when λ � = 0. And the equation X 2 − I n has infinite many solvents, including two isolated solvents and 2 n − 2 families of solvents. HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Existing algorithms Newton’s method is attractive for its local quadratic convergence, if a good approximation Z 0 of the desired solvent Z ∗ is given. It generates a sequence of matrices converging to Z ∗ : � Solve P ′ Z k ( E k ) = − P ( Z k ) for E k k = 0 , 1 , 2 , ... Update Z k +1 = Z k + E k X ( E ) : C n × n → C n × n is the Fr´ where P ′ e chet derivative of P at X in the direction E along X . HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Existing algorithms Bernoulli’s method, used to find the dominant or minimal solvent (if there is one) of P ( X ), is generalized from the case of the quadratic scalar equations. Dominant solvent: ( AZ k + B ) Z k − 1 + C = 0 , for k = 1 , 2 , ... k →∞ Z k = Z dom . lim Minimal solvent: ( AZ k − 1 + B ) Z k + C = 0 , for k = 1 , 2 , ... k →∞ Z k = Z min . lim HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Existing algorithms Definition Suppose P ( λ ) has exactly 2 n eigenvalues | λ 1 | ≥ | λ 2 | ≥ ... ≥ | λ 2 n | , and denote the set of eigenvalues of a matrix Z by λ ( Z ), a solvent Z 1 of P ( X ) is a dominant solvent if λ ( Z 1 ) = { λ 1 , ..., λ n } , and a solvent Z 2 is a minimal solvent if λ ( Z 1 ) = { λ n +1 , ..., λ 2 n } . HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Eigenvalue technique, used to construct all diagonalizable solvents . Theorem ( Higham et al., 2001) Suppose Q ( λ ) = M λ 2 + L λ + K has p distinct eigenvalues { λ i } p i =1 , with n ≤ p ≤ 2 n, and that the corresponding set of p eigenvectors { λ i } p i =1 satisfies the Haar condition (that is, every subset of n of them is linearly independent). Then there are at � p � least different solvents of Q ( X ) , and exactly this many if n p = 2 n, which are given by X = W diag ( µ i ) W − 1 , W = [ w 1 , ..., w n ] , where the eigenpaires { µ i , w i } n i =1 are chosen among the eigenvpairs { λ i , v i } p i =1 of Q. HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Existing algorithms N. J. H igham and H.-M. K im , Numerical analysis of quadratic matrix equation, IMA Journal of Numerical Analysis, 20 (2000), pp. 499-519. N. J. H igham and H.-M. K im , Solving a quadratic matrix equation by Newton’s method with exact line searches, SIAM Journal on Matrix Analysis and Applications, 23 (2001), pp. 303-316. HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion The aim–finding all isolated solvents We focus on locating all isolated solvents of P ( X ), of which there exists a neighborhood not containing other solvents. It is necessary to consider the element-wise form of P ( X ), which is a polynomial system of n 2 equations and n 2 variables, and denoted by: p ( x ) = 0 . Here, we set x = vec( X ) = ( x 11 , ..., x n 1 , ..., x 1 n , ..., x nn ) T , and p = vec( P ) = ( p 11 , ..., p n 1 , ..., p 1 n , ..., p nn ) T . HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Example 1: The element-wise form of the equation � − 1 − 6 � � � 0 12 X 2 + X + = 0 , 2 − 9 − 2 14 can be written as x 2 11 + x 12 x 21 − x 11 − 6 x 21 x 11 x 21 + x 12 x 22 − x 12 − 6 x 22 + 12 p ( x ) = = 0 . x 21 x 11 + x 22 x 21 + 2 x 11 − 9 x 21 − 2 x 21 x 12 + x 2 22 + 2 x 12 − 9 x 22 + 14 HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Homotopy method for solving polynomial systems Given a system f ( z ) : C r → C r , we design a start system g ( z ) = 0, and construct a homotopy: h ( z , t ) = (1 − t ) g ( z ) + tf ( z ) , satisfying Triviality: The solutions of start system g ( x ) = 0 are known. Smoothness: The solution set of h ( x , t ) = 0 with t ∈ [0 , 1) consists of a finite number of smooth paths emanating from isolated solutions of g ( x ) = 0, each parameterized by t . Accessibility: Every isolated solution of h ( x , 1) = p ( x ) = 0 is reached by a path. HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Homotopy method for solving polynomial systems HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
Introduction Homotopy method for quadratic matrix equation More results for special problems Numerical results and conclusion Step1. Given a start solution z 0 at t 0 = 0, a stopping criteria ǫ > 0, and an initial step h 0 . Step2. For i = 1 , 2 , ... Predict: Predict solution (˜ z i , t i ) such that: t i = t i − 1 + h i − 1 z i = z i − 1 − h z ( z i − 1 , t i − ) − 1 h t ( z i − 1 , t i − 1 ) h i − 1 ˜ Correct: Hold t constant, and correct z i , t i ) − 1 h (˜ z i = ˜ ˜ z i − h (˜ z i , t i ) until � h (˜ z i , t i ) � < ǫ , then let z i = ˜ z i . Adjust: Cut the step length in half on failure of the corrector, and double it if several corrections at the current step size have been successful. Terminate At t = 1, refine z to high accuracy by Newton iteration. HM for Solving AX 2 + BX + C = 0 Yongwen Hou, Bo Yu
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