Jordan Canonical Form Zhonggang Zeng, Northeastern Illinois Univ. ( - PowerPoint PPT Presentation

A Numerical Method for Computing the Jordan Canonical Form Zhonggang Zeng, Northeastern Illinois Univ. ( Joint work with T. Y. Li, Michigan State Univ ) 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 3 0 -1 -1 -4 0 0 -5 -5 0 1 0 0 -1 0 -5 -1 0 -3 -1 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 9 -1 3 -2 -1 1 1 -2 -2 1 -1 1 1 2 -1 -1 1 0 -1 1 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 7 2 -2 -11 1 0 6 -4 -3 6 0 5 -1 0 -3 -2 -1 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -4 -2 -9 -2 6 19 -2 0 -8 8 6 -8 1 -7 1 -2 4 4 2 0 0 -1 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 2 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 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 JCF 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 -2 0 3 4 0 0 3 0 2 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 -1 1 -2 1 -1 3 -1 -1 -3 3 0 -3 0 -2 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 1 0 1 0 0 -2 0 0 2 0 0 2 3 2 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 2 12 -1 2 -7 0 0 2 -4 -3 2 -3 2 4 6 -1 -2 0 0 -1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 -4 -1 -5 -2 2 12 -1 0 -7 4 3 -7 0 -6 1 3 4 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 11 8 1 -2 -12 -3 0 6 -9 -8 6 1 5 0 -1 0 -7 -2 0 -3 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 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 RANMEP 2008, Jan 5, 2008, NCTS, Taiwan 2 0 2 0 0 -3 0 0 3 0 0 3 0 3 0 0 -2 0 0 0 0 3

Theorem: Every matrix corresponds to a Jordan Canonical Form − = 1 A X J X ⎡ ⎤ J λ ⎡ 1 ⎤ 1 ⎢ ⎥ k ⎢ ⎥ λ J O ⎢ ⎥ ⎢ ⎥ = 2 = k J J ⎢ ⎥ ⎢ ⎥ k 1 O O ⎢ ⎥ ⎢ ⎥ λ ⎣ ⎦ ⎣ ⎦ J k l However, it is extremely difficult to compute JCF with numerical computation 1

Problem : Jordan Canonical Form (JCF) 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 3 0 -1 -1 -4 0 0 -5 -5 0 1 0 0 -1 0 -5 -1 0 -3 -1 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 9 -1 3 -2 -1 1 1 -2 -2 1 -1 1 1 2 -1 -1 1 0 -1 1 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 7 2 -2 -11 1 0 6 -4 -3 6 0 5 -1 0 -3 -2 -1 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 0 0 -4 -2 -9 -2 6 19 -2 0 -8 8 6 -8 1 -7 1 -2 4 4 2 0 0 -1 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 JCF 0 -1 1 -1 1 2 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 -2 0 3 4 0 0 3 0 2 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 0 0 -1 1 -2 1 -1 3 -1 -1 -3 3 0 -3 0 -2 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 1 0 1 0 0 -2 0 0 2 0 0 2 3 2 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 0 2 12 -1 2 -7 0 0 2 -4 -3 2 -3 2 4 6 -1 -2 0 0 -1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 -4 -1 -5 -2 2 12 -1 0 -7 4 3 -7 0 -6 1 3 4 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 11 8 1 -2 -12 -3 0 6 -9 -8 6 1 5 0 -1 0 -7 -2 0 -3 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 -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 { } λ = ( ) 3 A − λ + 15 E ( 10 ) A 2

Computing JCF is an ill-posed problem λ λ 1 1 λ λ 1 1 λ λ λ λ A = X X -1 1 1 A λ λ λ λ 1 1 λ λ λ 3 λ λ λ 4 λ 2 → ~ = + ε Under arbitrary perturbation A A A E λ 5 λ λ 1 λ 1 λ 8 λ 6 λ 2 ~ = ~ ~ − λ 3 λ 7 1 λ 4 A X X λ 5 λ 6 λ 7 λ 8 Can we recover the lost eigenvalue and JCF of A using perturbed matrix Ã ? 3

Background and motivation • The new JCF algorithm is motivated by our recent works on solving ill-posed problems in numerical polynomial algebra (approximate GCD, factorization, multiplicity structure, polynomial elimination, etc.) • Numerical polynomial algebra and numerical algebraic geometry are fertile application fields of numerical linear algebra • JCF computing has direct application in numerical irreducible factorization of multivariate polynomials. The preliminary implementation of the JCF algorithm is part of ApaLab (http://www.neiu.edu/~zzeng) 4

The pioneer work (Kublanovskaya-Ruhe): 1. Compute eigenvalues of A with Francis QR algorithm 2. Identify an eigenvalue cluster, and use the mean λ as an approximation to a multiple eigenvalues 3. Use a sequence of SVD and unitary similarity transformations on A - λ I to obtain a local staircase form 4. Repeat on other multiple eigenvalues. Reference: Kublanovskaya (1968) Ruhe (1970) Golub & Wilkinson (1976) Kågström & Ruhe (1980) Beelen & Van Dooren (1988) The main difficulty: accuracy of the multiple eigenvalue 5

>> J = find(abs(e-1)<0.2); >> sum(e(J))/20 ans = 1.00000000000000 >> K = find(abs(e-2)<0.2); >> sum(e(K))/15 ans = 2.00000000000000 If the cluster is properly identified, the center is usually be very good approximation to the multiple eigenvalue 6

7 The cluster mean can still be sensitive:

Can you solve ( x - 1.0 ) 100 = 0 x 100 - 100 x 99 + 4950 x 98 - 161700 x 97 + 3921225 x 96 - ... - 100 x + 1 = 0 8

Jordan Canonical Form Zhonggang Zeng, Northeastern Illinois Univ. ( - PowerPoint PPT Presentation

A Numerical Method for Computing the Jordan Canonical Form Zhonggang Zeng, Northeastern Illinois Univ. ( Joint work with T. Y. Li, Michigan State Univ ) 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1

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