Validated computation tool for the Perron-Frobenius eigenvalue using graph theory

Transcription

1 Validated computation tool for the Perron-Frobenius eigenvalue using graph theory Kaori Nagatou Kyushu University, JAPAN [ Joint work with Yutaka Ishii, Kyushu University ] AGA Seminar in INI, March 13, 2007

2 Contents Introduction Perron-Frobenius eigenvalue Matrix & Graph (strong connectivity) Tarjan s algorithm Enclosing of the Perron-Frobenius eigenvalue Numerical examples Conclusion 2

3 Introduction Matrix Eigenvalue Problem Find λ and x 0 such that Ax = λx A : real, large, nonnegative, sparse, non-symmetric Aim: Enclosing of a special eigenvalue (Perron-Frobenius eigenvalue) 3

4 We want to provide a numerical computational tool: Input nonnegative matrix Output rigorous upper and lower bounds of Perron-Frobenius eigenvalue 4

5 Self-validating methods Numerical verification Computer-assisted proofs Numerical computations with guaranteed accuracy It guarantees the existence (and uniqueness) of the solutions It guarantees the residual between exact solution and approximate solution The errors occurred in a computing process in a computer are assured. 5

6 INTLAB - Interval Laboratory, the Matlab toolbox for verified computations ( S. M. Rump, ) VERIFYEIG Verification of eigencluster near (lamda,xs) [L,X] = VerifyEig(A,lambda,xs,B) An eigenvalue cluster near lambda is enclosed, where xs(:,i), i=1:k is an approximation to the corresponding invariant subspace. If B is specified, the generalized eigenproblem A*x = lambda*b*x is treated. On output, the complex interval L contains (at least) k eigenvalues of A, and the complex n k matrix X includes a base for the corresponding invariant subspace. 6

7 Ex. Maximum eigenvalue of A (real symmetric matrix) λ : approximate maximum eigenvalue μ (1 + δ) λ, δ > 0 T X μi A, CC X T Y CC X ρ max N 1 k N j = 1 Y ( ) kj λ ( A) max λ+ ρ 7

8 Matrix eigenvalue problem Numerical verification for differential equations Infinite dimensional eigenvalue problem Lu = λu L : Elliptic differential operator, e.g. non-commutative harmonic oscillator 1-D Schroedinger operator etc. 8

9 Perron-Frobenius eigenvalue A n n 1 2 Let be a real matrix and the eigenvalues of A. λ, λ,, λn be spectral radius ρ( A) = max λ i 1 i n Simplicity of eigenvalue λ is geometrically simple corresponding eigenspace is one dimensional λ is algebraically simple λ is a simple root of the characteristic polynomial 9

10 A is reducible P : permutation matrix s.t. t PAP A A = O A22 A, A : square matrices In case of n = 1 A = O is reducible A is irreducible A is not reducible 10

11 P A : permutation matrix P t PAP A1 * * A * O O A 2 = m A1 A2 A m,,, : irreducible components of Irreducible, nonnegative or 1 1 zero matrix A 11

12 Perron-Frobenius Theorem A Let be a nonnegative and irreducible matrix. Then ρ ( A) > 0 A is an eigenvalue of, which is both geometrically and algebraically simple. Moreover the corresponding eigenvector is positive. λ ( A) ρ( A): PF Perron-Frobenius eigenvalue (P-F eigenvalue) Perron root, Perron eigenvalue, Frobenius eigenvalue 12

13 Outline of proof n n S = x x 0, xi = 1 i= 1 n n 1 Tx = Ax, θ ( x) = aij x j > 0 θ ( x) i= 1 j= 1 bounded, convex, closed, non-empty T : S S continuous by Brouwer s fixed point theorem x0 S such that Tx0 = x0 λ θ( x ) > 0 Ax = λx

14 A For a nonnegative matrix with irreducible components A,, 1 Am P-F eigenvalue of A λ PF A λ PF 1 i m ( ) max ( A) i If μ is another eigenvalue of A, then of course μ λ ( A) PF 14

15 Matrix and graph Graph G = ( V, E) V : a set of vertices, E : a set of edges e= (, v w) e v 1 1 v2 e 5 e 6 e 4 v 3 e 2 e 8 e 7 v4 v5 e 3 directed graph v 1 e 1 v 2 e 3 e 2 v3 e v4 4 undirected graph 15

16 vw, V path p : v w sequence of vertices and edges leading from v to w graph G is connected there is a path between every pair of vertices graph is strongly connected G for each pair of vertices there exist paths : 1 vw, V p v w and p : w v 2 16

17 Strong connectivity G = ( V, E) Let be a directed graph and. v w (equivalent) p v there is a closed path : v vw, V which contains w Let the distinct equivalence classes under this equivalence relation be V ( i 1,, n) and let i = { } G = ( V, E ), E = ( v, w) E v, w V Then: i i i i i G i (i) Each is strongly connected (ii) No G i is a proper subgraph of a strongly connected subgraph of G. G i : strongly connected components (SCC) of G 17

18 Matrix A a = a a an1 a = ( ) ij i, j 1,, n 11 1n nn 1 i 2 j n Associated graph G = ( V, E) given by V = n E = i j a ij { 1,, }, {(, ) 0} 18

19 Irreducibility & strong connectivity A is irreducible G is strongly connected A,, : 1 A m irreducible components of A G,, : 1 G m strongly connected components of G 19

20 The connections G,, G A,, A and 1 m 1 P-F eigenvalue of a matrix A m P-F eigenvalues of irreducible components Ai have been observed in Ishii, Y., Sands D.: Piecewise affine and projective maps in several dimensions, Ⅱ: Rigorous entropy computation, Preprint (2007). and applied to compute the topological entropy of some piecewise affine homeomorphism of the plane 20

21 Tarjan s algorithm Tarjan, R.: Depth-first search and linear graph algorithms, SIAM J. Comput. 1, no. 2, pp (1972). Efficient algorithm to find all the strongly connected components of any given finite directed graph G by using depth-first search Works correctly on any finite directed graph (, E ) O V Requires space and time 21

22 Depth First Search (DFS) Consider the following choice rule on searching a graph: When selecting an edge to traverse, always choose an edge emanating from the vertex most recently reached which still has unexplored edges. tree root 22

23 BEGIN INTEGER i ; PROCEDURE STRONGCONNECT(v) BEGIN LOWLINK(v) := NUMBER(v):= i := i+1; put v on stack of points; FOR w in the adjacency list of v DO BEGIN IF w is not yet numbered THEN BEGIN STRONGCONNECT(w); LOWLINK(v) := min(lowlink(v), LOWLINK(w);) END ELSE IF NUMBER(w) < NUMBER(v) DO BEGIN if w is on stack of points THEN LOWLINK(v) := min(lowlink(v), NUMBER(w)); END END If(LOWLINK(v) = NUMBER(v)) THEN BEGIN start new strongly connected component; WHILE w on top of point stack satisfies NUMBER(w) <= NUMBER(v) DO delete w from point stack and put w in current component; END END i:= 0; empty stack of points; FOR w a vertex IF w is not yet numbered THEN STRONGCONNECT(w); END 23

24 LOWLINK( v): the smallest vertex which is in the same strongly component as v. v is the root of some strongly connected component of LOWLINK( v) = v G graph G SCCs G1, G2, G3 6 24

25 MATLAB Code for Tarjan s Algorithm % Function to find all strongly connected components function [G,N,L,c,k,sccI,stack]=scc(v,G,N,L,c,k,sccI,stack) UNDEFINED=0; c=c+1; k=k+1; N(v)=c; % Visit number for the vertex v L(v)=c; % LOWLINK(v) stack(k)=v; for w=list(v) % w is in the adjacency list of v if N(w)==UNDEFINED [G,N,L,c,k,sccI,stack]=scc(w,G,N,L,c,k,sccI,stack); L(v)=min(L(v),L(w)); elseif N(w)<N(v) for z=stack if w==z L(v)=min(L(v),N(w)); end end end end if L(v)==N(v) scci=scci+1; j=1; m=1; for t=stack if (t>0 & N(t)>=N(v)) G(sccI,j)=t; %SCC component j=j+1; stack(m)=-1; end m=m+1; end end 25

26 Enclosing of Perron-Frobenius eigenvalue Let A be a nonnegative matrix. For any vector x= ( x ), x > 0 ( i = 1,, n) n i i= 1 i we have ( Ax) ( Ax) min ( A) max x i λpf 1 i n x 1 i n i i i (1) Finding a good vector x Getting sharp bounds for λ ( A) PF 26

27 Algorithm of validated computation for P-F eigenvalue nonnegative matrix A graph G Strongly connected components of G Irreducible components of A P-F eigenvalue enclosing of each irreducible component P-F eigenvalue enclosing of A 27

28 Numerical examples Example 1 n n A = tridiagonal matrix There are two eigenvalues (±) which have the same maximum absolute value Power method is not valid to obtain good approximate eigenvector 28

29 n =1024 A Apply (1) for itself λ ( A) [1.0, ] PF ( s) Apply (1) for each irreducible components of A ( s) λpf( A) [ , ] MATLAB Version (R14) on DELL Inspiron 5150 (Intel Pentium4 2.8GHz) 29

30 Example 2 xk ( ) = 0 or 1 ( k= 1,, n) x(1) x(2) x(3) x(4) xn ( /2 1) xn ( /2) A= xn (/21) + xn (/2+ 2) xn ( /2+ 3) xn ( /2+ 4) x( n 1) xn ( ) 30

31 Results n= 1024, b= 0.1 Lozi map L ab, x 1 a x + by : y x a, b \{0}: parameters 1.8 a SCC λ ( A) PF 369 [ , ] Time (s) [ , ] [ , ] [ , ] [ , ] a 2.06 λ ( A) = 2 PF 31

32 Conclusion We have developed a tool for rigorous computation of Perron-Frobenius eigenvalue for nonnegative matrix. (This program can be used to judge the irreducibility of a matrix) Improvements Making use of the sparsity Parallel computing We are planning to release our tool in near future. 32

33 References Ishii, Y., Sands D.: Piecewise affine and projective maps in several dimensions, Ⅱ: Rigorous entropy computation, Preprint (2007). Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge (1995). Tarjan, R.: Depth-first search and linear graph algorithms, SIAM J. Comput. 1, no. 2, pp (1972). 33