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Eigen-decomposition of a class of Infinite dimensional tridiagonal matrices

Eigen-decomposition of a class of Infinite dimensional tridiagonal matrices. G.V. Moustakides: Dept. of Computer Engineering, Univ. of Patras, Greece B. Philippe: IRISA-INRIA, France. Definition of the problem. From finite to infinite dimensions.

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Eigen-decomposition of a class of Infinite dimensional tridiagonal matrices

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  1. Eigen-decomposition of a classofInfinite dimensional tridiagonal matrices G.V. Moustakides: Dept. of Computer Engineering, Univ. of Patras, GreeceB. Philippe: IRISA-INRIA, France

  2. Definition of the problem. From finite to infinite dimensions. Reduction of the problem to a finite dimensional d.e. and eigen-decomposition problem, using Fourier transform. Numerical aspects regarding the solution of the d.e. and the computation of the final (infinite dimensional) eigenvectors. Conclusion. Outline

  3. } I:D:A: r: identity matrixdiagonal matrixgeneral matrix real scalar real matrices of dimensions NN Definition of the problem

  4. Eigen-decomposition Eigenvalues:There is an infinite number. Eigenvectors:There is an infinite number and each eigenvector is of infinite size. Goal:To reduce the infinite dimensional eigen-decomposition problem into a finite one.

  5. From finite to infinite dimensions QKhas dimensions: (2K+1)N (2K+1)N, therefore we have (2K+1)N eigenvalue-eigenvector pairs. Typical values: N = 100-1000, K = 5-10.

  6. i(k,l)hasdimensions: N 1. k,l= -K,…,K, i=1,…,N. QKhas dimensions: (2K+1)N (2K+1)Ni(k)hasdimensions: (2K+1)N 1.k = -K,…,K, i = 1,…,N.

  7. Consider now the infinite dimensional problem by lettingK   Ai(k,l+1) + (D+lrI)i(k,l) + Ati(k,l-1) = i(k)i(k,l) Ai(k,l+1) + Di(k,l) + Ati(k,l-1) = (i(k) -lr)i(k,l)

  8. Key Idea i(k) = i + kr without loss of generality assume 0  i r i(k,l) = i(l-k) Ai(l-k+1)+Di(l-k)+Ati(l-k-1) = (i-(l-k)r)i(l-k) Reduction to finite dimensions Ai(k,l+1)+Di(k,l)+Ati(k,l-1) = (i(k)-lr)i(k,l) A,D:NNi(k,l): N1i=1,…,N, k,l= -,…, Ai(n+1)+(D-iI)i(n)+Ati(n-1) = -nri(n)

  9. Ai(n+1)+(D-iI)i(n)+Ati(n-1) = -nri(n) i , {i(n)}, i=1,…,N, 0  i r

  10. Important Fourier Transform Let …, x(-2), x(-1), x(0), x(1), x(2),… be a real sequence. Then we define its Fourier Transform as

  11. Ai(n+1)+(D-iI)i(n)+Ati(n-1) = -rni(n)

  12. We needi and i(0) to solve it. i() as being the Fourier transform of a (vector) sequence isnecessarily periodic with period 2.

  13. Let Z() be the transition matrix of the d.e., that is then we know that X()= Z()X0. The solution X() is periodic if and only if X(2)=X(0) Theorem Consider the following linear system of d.e.

  14. Steps to obtain (i ,{i (n)}), i=1,…,N • Compute the transition matrix () from the d.e. • Find the eigenvalue-eigenvector pairs i,i(0) of • Form the desired eigenvalue-FT(eigenvector) pairs as • Use Inverse Fourier Transform to recover the final infinite eigenvector {i(n)} from i().

  15. Numerical aspects • Numerical solution of the d.e. • Eigen-decomposition of (2). • Computation of the Inverse Fourier Transform of i() where

  16. Numerical solution of the d.e. One can show that () is unitary, therefore any numerical solution should respect this structure. A possible scheme is

  17. Pade 1 Pade 2 1 Step Integration 3 Step Integration. Yoshida scheme

  18. Pade 1, 3 step intgr. Pade 1, 1 step intgr. Pade 2, 1 step intgr. Pade 2, 3 step intgr.

  19. Eigen-decomposition of (2) Since (2) is unitary there are special eigen-decomposition algorithms that require lower computational complexity than the corresponding algorithm for the general case. From this problem we obtain the pairs i,i(0), i=1,…,N. Using the solution () of the differential equation we can compute the Discrete Fourier Transform of the eigenvectors Notice that we obtain a sampled version of the required Fourier transform.

  20. Ifx(n)=0 for n < 0andn M, then the Fourier Transform is equal then the finite sequence x(n), n=0,…, M-1, can be completely recovered from a sampled version of the Fourier transform. Specifically we need only the samples Inverting the Fourier Transform Let …, x(-2), x(-1), x(0), x(1), x(2),… with Fourier Transform

  21. Inverce discrete Fourier Transform Complexity O(M 2). For M=2m popular Fast Fourier Transform (FFT).Complexity O(M log(M)). Apply Inverse Discrete Fourier Transform to i(n), this will yield the desired vectors i(n). If only a small number of i(n) is significant, then we apply Inverse Discrete Fourier Transform only to a subset of the vectors i(n) produced by the solution of the d.e.

  22. Conclusion • We have presented as special infinite dimensional eigen- decomposition problem. • With the help of the Fourier Transform this problem was transformed into a d.e. followed by an eigen-decomposition both of finite size. • We presented numerical techniques that efficiently solve all subproblems of the proposed solution.

  23. E n D Questions please ?

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