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

2 Outline  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.

3 Definition of the problem I:D:A:r:I:D:A:r: identity matrix diagonal matrix general matrix real scalar real matrices of dimensions N  N }

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

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

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

7 Consider now the infinite dimensional problem by letting K   A  i (k,l+1) + (D+lrI)  i (k,l) + A t  i (k,l-1) = i (k)  i (k,l) A  i (k,l+1) + D  i (k,l) + A t  i (k,l-1) = ( i (k) -lr)  i (k,l)

8 Reduction to finite dimensions A  i (k,l+1)+D  i (k,l)+A t  i (k,l-1) = ( i (k)-lr)  i (k,l) A,D: N  N  i (k,l): N  1 i=1,…,N, k,l= - ,…,  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)+A t  i (l-k-1) = ( i -(l-k)r)  i (l-k) A  i (n+1)+(D- i I)  i (n)+A t  i (n-1) = -nr  i (n)

9 i, {  i (n)}, i=1,…,N, 0  i  r

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

11 A  i (n+1)+(D- i I)  i (n)+A t  i (n-1) = -rn  i (n)

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

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

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15 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 (  ).

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

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

18 3 Step Integration. Yoshida scheme 1 Step Integration Pade 1 Pade 2

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

20 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.

21 Inverting the Fourier Transform Let …, x(-2), x(-1), x(0), x(1), x(2),… with Fourier Transform If x(n)=0 for n < 0 and n  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

22 Complexity O(M 2 ). For M=2 m 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. Inverce discrete Fourier Transform

23 ConclusionConclusion  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.

24 E n D Questions please ?