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On Computing Multiple Eigenvalues , the Jordan Structure and their Sensitivity

On Computing Multiple Eigenvalues , the Jordan Structure and their Sensitivity. Zhonggang Zeng Northeastern Illinois University. Sixth International Workshop on Accurate Solution of Eigenvalue Problems (IWASEP VI). May 23, 2006, Penn State Univ. l l l l. + + +

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On Computing Multiple Eigenvalues , the Jordan Structure and their Sensitivity

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  1. On Computing Multiple Eigenvalues, theJordan Structure andtheirSensitivity Zhonggang Zeng Northeastern Illinois University Sixth International Workshop on Accurate Solution of Eigenvalue Problems (IWASEP VI) May 23, 2006, Penn State Univ

  2. l l l l + + + + + + + + + + + + + + + + AU = U l l l + + + l Objectives: - Multiple eigenvalues - Staircase decomposition Jordan decomposition A = XJX-1follows, if necessary - Sensitivity 1

  3. JCF computing … … 1966: Kublanovskaya 1970: Varah 1970: Ruhe 1976: Golub & Wilkinson 1979: Van Dooren 1980: Kagstrom & Ruhe 1983: Demmel 1987: Demmel & Kagstrom 1993: Demmel & Kagstrom 1997: Edelman, Elmroth & Kagstrom … … Main difficulty: accuracy of multiple eigenvalues 2

  4. 310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 030 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 030 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0310 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 031 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 03 JCF Ill-posed Problem: Jordan Canonical Form (JCF) 0 10 3 0 -1 -1 -4 0 0 -5 -5 0 1 0 0 -1 0 -5 -1 0 -3 -1 0 5 9 -1 3 -2 -1 1 1 -2 -2 1 -1 1 1 2 -1 -1 1 0 -1 1 3 1 7 2 -2 -11 1 0 6 -4 -3 6 0 5 -1 0 -3 -2 -1 0 0 0 -1 1 5 2 3 1 -1 0 0 0 0 0 -1 0 1 2 0 0 1 0 -1 1 -4 -2 -9 -2 6 19 -2 0 -8 8 6 -8 1 -7 1 -2 4 4 2 0 0 -1 0 -1 1 -1 1 2 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 9 -2 4 -3 3 3 1 -2 -2 1 0 1 2 1 -1 -1 1 0 -1 0 1 0 1 0 0 -2 0 3 4 0 0 3 0 2 0 0 -1 0 0 0 0 0 1 -4 -2 0 1 4 1 0 3 5 4 0 -2 0 0 1 0 3 1 0 1 1 -1 1 -2 1 -1 3 -1 -1 -3 3 0 -3 0 -2 -1 0 1 0 0 0 0 0 5 2 6 2 -3 -16 1 0 12 -5 -1 12 0 9 -1 0 -5 -3 -2 0 0 0 -1 4 0 1 -2 -4 -1 0 0 -5 -4 3 4 0 -1 -2 0 -3 -1 0 -1 -2 1 0 1 0 0 -2 0 0 2 0 0 2 3 2 0 0 -1 0 0 0 0 0 0 -1 4 -3 3 -1 1 1 0 0 0 0 -2 3 3 1 0 0 0 0 0 1 0 2 12 -1 2 -7 0 0 2 -4 -3 2 -3 2 4 6 -1 -2 0 0 -1 3 -4 -1 -5 -2 2 12 -1 0 -7 4 3 -7 0 -6 1 3 4 2 1 0 0 0 0 11 8 1 -2 -12 -3 0 6 -9 -8 6 1 5 0 -1 0 -7 -2 0 -3 -1 -2 0 7 -2 5 1 -1 1 -2 0 0 -2 0 -1 1 1 0 4 3 -1 -1 0 3 2 6 2 -2 -7 1 0 2 -5 -4 2 -2 2 0 3 -1 -3 1 2 0 2 5 -12 -10 2 -3 1 5 -1 0 6 6 0 0 0 -2 -1 0 6 0 3 5 0 4 -9 0 1 0 1 4 -1 0 4 4 0 -4 0 0 4 0 4 0 1 6 4 2 0 2 0 0 -3 0 0 3 0 0 3 0 3 0 0 -2 0 0 0 0 3 3

  5. The computed roots: + + + + Ill-posed problem: root-finding with coefficients in hardware precision: 4

  6. William Kahan (1972) They are sensitive to arbitrary perturbation, but may not be sensitive to structure preserving perturbation. Are multiple roots/eigenvalues really sensitive to perturbations? 5

  7. perturbed polynomial p original polynomial q projected polynomial with computed roots pejorative manifold A two staged algorithm: Determine the multiplicity structure (or manifold) Reformulate / solve least squares problem well-posed, and even well conditioned 6

  8. Stage II results: The backward error: 6.16 x 10-16 Refined roots 1.000000000000000 1.999999999999997 3.000000000000011 3.999999999999985 For polynomial with (inexact ) coefficients in machine precision Stage I results: The backward error: 6.05 x 10-10 Computed roots multiplicities 1.000000000000353 20 2.000000000030904 15 3.000000000176196 10 4.000000000109542 5 ACM TOMS, 2004: Z. Zeng, Algorithm 835 -- MultRoot Math. Comp. 2005: Z. Zeng, Computing multiple roots of inexact polynomials 7

  9. l 1 l 1 l l 1 l J = l 1 l l l l l l + + + + + + + + + + + + + + + + Kublanovskaya (1966) Ruhe, Kagstrom, Wilkinson,Golub, Demmel, … lI+S= l l l + + + l Staircase form with Weyr characteristics: [4,3,1] JCF computing: For a given matrix A Jordan form with Segre characteristics: [3,2,2,1] find X, J: AX = XJ find U, S, l AU = U(lI+S), UHU=I 8

  10. AY- Y(lI+S) = 0 C*Y – I = 0 is nonsingular if the Jordan structure is correct (Zeng and Li) The problem is well-posed, may even well-conditioned 9

  11. Gauss-Newton iteration on system Example: A 50x50 matrix with eigenvalues 1, 2, 3 with multiplicity 20, 15, 5 respectively 10

  12. A two-stage strategy for computing JCF: Stage I: determine the Jordan structure and an initial approximation Stage II: Solve the reformulated least squares problem at each eigenvalue, subject to the structural constraint, using the Gauss-Newtion iteration. 11

  13. Minimal polynomials: A rank/nulspace problem eigenvalue Segre characteristics l1 5 3 2 2 l2 3 3 1 l3 2 ---------------------------------------------------------------------------------------------------------------------------------------------------- 10 6 3 2 Let v be a coefficient vector of p1. Then for a random x 12

  14. leads to p2(l), p3(l), … recursively Rank/nulspace leads to p1(l) (the 1st minimal polynomial) By a Hessenberg reduction, with A1 = A, and a randomx being the first column of Q, 13

  15. Minimal polynomials Ill-posed root-finding with approximate data Root-finding JCF structure Computing multiple roots of inexact polynomials, Z. Zeng, Math Comp 2005 Rank-revealing 14

  16. Example: 100x100 matrix A with multiple eigenvalues 1, -1, 2, -2 50 simple eigenvalues: random 15

  17. -- space of matrices perturbed matrix original matrix “nearest” matrix on P manifold When matrix A is possibly approximate: Stage I: Determine the manifold (matrix bundle) Stage II: Solve the well-posed least squares problem 16

  18. has some known weaknesses • may substantially underestimate the error • may fail to discriminate a multiple eigenvalue Sensitivity of an eigenvalue(simple or multiple) Condition number At a multiple eigenvalue, k(l)= infinity by definition, not by computation 17

  19. Error estimate = 0.0000042 Actual error = 0.0022 The condition number fails to reveal ill-posedness Example: >> eig(A) ans = 2.00078250207872 + 0.00056834208954i 2.00078250207872 - 0.00056834208954i 1.99970128002318 + 0.00092011990651i 1.99970128002318 - 0.00092011990651i 1.99903243579620 2.00000000000000 >> eig(A+E) ans = 1.99785628939037 1.99891738496176 + 0.00185657043184i 1.99891738496176 - 0.00185657043184i 2.00107196168888 + 0.00187502026186i 2.00107196168888 - 0.00187502026186i 2.00216501730835 >> norm((x*y')/(x'*y)) ans = 14.04290360613186 18

  20. Condition number log10(error bound) log10(error) Example: 19

  21. For l = 0: Demmel’s bound (1987) Wilkinson’s bound (1984) For l = 0.1: Kahan’s bound (1972) What is the minimum perturbation ||E|| such that an eigenvalue of A increases its multiplicity? 20

  22. Wanted: a measurement of sensitivity • reliable in identifying a multiple eigenvalue • can be estimated/computed efficiently 21

  23. cost: O(n2) At an eigenpair ( l , x ) ofA, define a condition number t(l) = infinity at a multiple eigenvalue the distance ||E|| to the nearest bundle with l being a double eigenvalue speculation: 22

  24. Condition numbers Condition numbers Example: log10 (new error bound) log10(error bound) log10(error) Example: >> eig(A) ans = 2.00078250207872 + 0.00056834208954i 2.00078250207872 - 0.00056834208954i 1.99970128002318 + 0.00092011990651i 1.99970128002318 - 0.00092011990651i 1.99903243579620 2.00000000000000 23

  25. define If A+E is the nearest matrix with (m+1)-fold l if multiplicity m is wrong Sensitivity of an m-fold eigenvalue After obtaining an m-fold eigenvalue l in staircase decomposition 24

  26. Staircase decomposition for a double eigenvalue: -0.00000000025159 0.00000003957120 -0.00000000049184 -0.00000098980397 0.00000008765356 0.00001538421795 -0.00000205331981 -0.00017578611054 0.00003066233014 0.00154219061561 -0.00033766919246 -0.01044234247091 0.00284564181729 0.05361151873827 -0.01836110020880 -0.19998617218797 0.08857192169246 0.49723892540802 -0.30283241761169 -0.66937659896642 0.65751125003779 0.13279795424950 -0.68394528652648 0.49403745520757 -0.00000000025159 0.00000003957120 -0.00000000049184 -0.00000098980397 0.00000008765356 0.00001538421795 -0.00000205331981 -0.00017578611054 0.00003066233014 0.00154219061561 -0.00033766919246 -0.01044234247091 0.00284564181729 0.05361151873827 -0.01836110020880 -0.19998617218797 0.08857192169246 0.49723892540802 -0.30283241761169 -0.66937659896642 0.65751125003779 0.13279795424950 -0.68394528652648 0.49403745520757 0.03864934375529 -0.88858158392945 0 0.03864934375529 = F Example: 12x12 Frank matrix Eigenvalue Condition number 0.03102744447368 18281388.1 0.04950888583700 38773448.2 0.08122648015585 26647311.8 0.14364690493016 6701300.4 0.28474967198534 560311.0 0.64350532103739 14466.8 1.55398870911557 216.1 3.51185594858010 6.9 6.96153308556711 1.7 12.31107740086854 3.1 20.19898864587709 5.0 32.22889150157213 3.3 >> F = frank(12) F = 12 11 10 9 8 7 6 5 4 3 2 1 11 11 10 9 8 7 6 5 4 3 2 1 0 10 10 9 8 7 6 5 4 3 2 1 0 0 9 9 8 7 6 5 4 3 2 1 0 0 0 8 8 7 6 5 4 3 2 1 0 0 0 0 7 7 6 5 4 3 2 1 0 0 0 0 0 6 6 5 4 3 2 1 0 0 0 0 0 0 5 5 4 3 2 1 0 0 0 0 0 0 0 4 4 3 2 1 0 0 0 0 0 0 0 0 3 3 2 1 0 0 0 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0 0 0 1 1 + R 25

  27. Frank matrix is near matrices of an eigenvalue with multiplicity m = 2, l = 0.03864934375529, distance = 3.9e-12, t2(l) = 1.34e+07 m = 3, l = 0.05043386836727, distance = 4.7e-10, t3(l) = 2.73e+05 m = 4, l = 0.07030194271661, distance = 3.9e-08, t4(l) = 9.04e+03 m = 5, l = 0.10767505748315, distance = 2.1e-06, t5(l) = 5.40e+02 m = 6, l = 0.18704749378810, distance = 7.1e-05, t5(l) = 6.48e+01 26

  28. Condition estimator - Approximate (or power iteration) - QR decomposition Q R = - Inverse iterations: - Condition number: 27

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