Roberto Castelli Basque Center for Applied Mathematics

A rigorous computational method to enclose the eigendecomposition of interval matrices Roberto Castelli Basque Center for Applied Mathematics Joint work with: Jean-Philippe Lessard Basque / Hungarian Workshop on Numerical Methods for Large Systems

Why Interval entries? Uncertainty of measurement - inaccuracy of data collection To consider variability of natural phenomena Qualitative indicators with statistical ranges Errors in computations...

Computation of bundles of periodic orbits ẏ = g(y) γ(t + T )=γ(t) γ 3T/4 (0) γ T/2 (0) Γ Γ = {γ(t) :t [0,T)} γ 0 (0) γ θ (t) =γ(t + θ) γ T/4 (0)

Computation of bundles of periodic orbits Consider γ(t + T )=γ(t) Γ = {γ(t) :t [0,T)} γ θ (t) =γ(t + θ) Φ θ (t) ẏ = g(y) γ 0 (0) Linearize around the periodic orbit: A θ (t) = g(γ θ (t)) Y = A θ (t)y Y (0) = I γ 3T/4 (0) γ T/4 (0) y = A θ (t)y : T- periodic matrix function γ T/2 (0) a fundamental matrix solution of the non-autonomous linear system Definition: The monodromy matrix is defined to be Φ θ (T ). Definition: The eigenvalues of Φ θ (T ) are called the Floquet multipliers. dim W s (Γ) = number of Floquet multipliers with modulus less than one. dim W u (Γ) = number of Floquet multipliers with modulus greater than one. Γ

Computation of bundles of periodic orbits

Rigorous Computations The goal of rigorous computations is to construct algorithms that provide an approximate solution to the problem f(x)=0 together with precise and possibly efficient bounds within which the exact solution is guaranteed to exist in the mathematically rigorous sense. Ingredients

Rigorous Computations The goal of rigorous computations is to construct algorithms that provide an approximate solution to the problem f(x)=0 together with precise and possibly efficient bounds within which the exact solution is guaranteed to exist in the mathematically rigorous sense. 1. Smoothness of the solutions Ingredients

Rigorous Computations The goal of rigorous computations is to construct algorithms that provide an approximate solution to the problem f(x)=0 together with precise and possibly efficient bounds within which the exact solution is guaranteed to exist in the mathematically rigorous sense. 1. Smoothness of the solutions Ingredients 2. Spectral methods, choice of suitable Banach spaces

Rigorous Computations The goal of rigorous computations is to construct algorithms that provide an approximate solution to the problem f(x)=0 together with precise and possibly efficient bounds within which the exact solution is guaranteed to exist in the mathematically rigorous sense. 1. Smoothness of the solutions Ingredients 2. Spectral methods, choice of suitable Banach spaces 3. Finite dim. projection, bounds on the truncation error terms (Analytical estimates)

Rigorous Computations The goal of rigorous computations is to construct algorithms that provide an approximate solution to the problem f(x)=0 together with precise and possibly efficient bounds within which the exact solution is guaranteed to exist in the mathematically rigorous sense. 1. Smoothness of the solutions Ingredients 2. Spectral methods, choice of suitable Banach spaces 3. Finite dim. projection, bounds on the truncation error terms (Analytical estimates) 4. Theorems: Banach Fixed Point Theorem, Conley Index...

Rigorous Computations The goal of rigorous computations is to construct algorithms that provide an approximate solution to the problem f(x)=0 together with precise and possibly efficient bounds within which the exact solution is guaranteed to exist in the mathematically rigorous sense. 1. Smoothness of the solutions Ingredients 2. Spectral methods, choice of suitable Banach spaces 3. Finite dim. projection, bounds on the truncation error terms (Analytical estimates) 4. Theorems: Banach Fixed Point Theorem, Conley Index... 5. Numerical analysis (Newton s method, Fast Fourier transform)

Rigorous Computations The goal of rigorous computations is to construct algorithms that provide an approximate solution to the problem f(x)=0 together with precise and possibly efficient bounds within which the exact solution is guaranteed to exist in the mathematically rigorous sense. 1. Smoothness of the solutions 2. Spectral methods, choice of suitable Banach spaces 3. Finite dim. projection, bounds on the truncation error terms (Analytical estimates) 4. Theorems: Banach Fixed Point Theorem, Conley Index... 5. Numerical analysis (Newton s method, Fast Fourier transform) 6. Interval Arithmetic Ingredients

Banach Fixed Point Argument Topological Arguments Rigorous integration Some Techniques - Sarah Day, Jean-Philippe Lessard, and Konstantin Mischaikow. Validated continuation for equilibria of PDEs. SIAM J. Numer. Anal., 45(4):1398--1424, 2007 - Marcio Gameiro, Jean-Philippe Lessard, and Konstantin Mischaikow. Validated continuation over large parameter ranges for equilibria of PDEs. Math. Comput. Simulation, 79(4):1368--1382, 2008. - Roberto Castelli and Jean-Philippe Lessard. Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits. Submitted., 2012. - Piotr Zgliczynski and Konstantin Mischaikow. Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation. Found. Comput. Math., 1(3):255--288, 2001 - Gianni Arioli, Piotr Zgliczynski, Symbolic dynamics for the Henon-Heiles Hamiltonian on the critical level. J. Diff. Equations 171, 173--202, 2001 - Piotr Zgliczynski. C1 -Lohner algorithm. Found. Comput. Math., 2(4):429--465, 2002. - Tomasz Kapela and Carles Simo. Computer assisted proofs for nonsymmetric planar choreographies and for stability of the Eight. Nonlinearity, 20(5):1241{1255, 2007.

Rigorous Computations: Banach Fixed Point Theorem Let (B, B ) be a Banach space, that is a complete normed vector space. Assume the existence of a contraction mapping T on B, that is a mapping such that 1. T (B) B; 2. there exists 0 < κ < 1 such that, for every x, y B, T (x) T (y) B κx y B. Then there exists a unique x B such that T (x )=x.

Rigorous Computations: Banach Fixed Point Theorem Let (B, B ) be a Banach space, that is a complete normed vector space. Assume the existence of a contraction mapping T on B, that is a mapping such that 1. T (B) B; 2. there exists 0 < κ < 1 such that, for every x, y B, T (x) T (y) B κx y B. Then there exists a unique x B such that T (x )=x. can be verified by strict inequalities

Rigorous Computations: Banach Fixed Point Theorem Let (B, B ) be a Banach space, that is a complete normed vector space. Assume the existence of a contraction mapping T on B, that is a mapping such that 1. T (B) B; 2. there exists 0 < κ < 1 such that, for every x, y B, T (x) T (y) B κx y B. Then there exists a unique x B such that T (x )=x. can be verified by strict inequalities can be verified using interval arithmetic!

Rigorous computations: Formulation in Banach Space

Rigorous computations: Formulation in Banach Space F (u, ν) =0 (E.g : Differential Equations) Knowledge about regularity x Ωs = f(x, Spectral methods ν) =0 x ν : modes : parameter { (x k ) k sup ωk x s k < k }

Rigorous computations: Formulation in Banach Space F (u, ν) =0 (E.g : Differential Equations) Knowledge about regularity x Ωs = f(x, Spectral methods ν) =0 x ν : modes : parameter { (x k ) k sup ωk x s k < k } x Consider such that f (m) ( x, ν 0 ) 0. Finite dim. reduction f(x, ν) =0 T ν (x) =x Newton-like operator at x

Rigorous computations: Formulation in Banach Space F (u, ν) =0 (E.g : Differential Equations) Knowledge about regularity x Ωs = f(x, Spectral methods ν) =0 x ν : modes : parameter { (x k ) k sup ωk x s k < k } x Consider such that f (m) ( x, ν 0 ) 0. Finite dim. reduction f(x, ν) =0 T ν (x) =x 8) Newton-like operator at x T ν :Ω s Ω s T ν (x) =x Jf(x, ν) J D x f( x, ν 0 ) 1 The chances of contracting a small set B around x depends on the magnitude of the eigenvalues of. J

Q: How to find a ball B x (r) such that T : B x (r) B x (r) is a contraction? ν ˆx x B x (r) Ω s

Q: How to find a ball B x (r) such that T : B x (r) B x (r) is a contraction? ν B x (r) = x + B(r) ˆx x B x (r) Ω s

Q: How to find a ball B x (r) such that T : B x (r) B x (r) is a contraction? ν B x (r) = x + B(r) Ball of radius r centered at 0 in the space Ω s ˆx x B x (r) Ω s

Q: How to find a ball B x (r) such that T : B x (r) B x (r) is a contraction? ν B x (r) = x + B(r) Ball of radius r centered at 0 in the space Ω s ˆx x B x (r) Ω s A: Radii polynomials Suppose the bounds Y, Z(r) satisfy D x T ν ( x + b)c T ν ( x) x Y Z(r) k k sup b,c B(r) p k (r) =Y + Z(r) r w s k

Q: How to find a ball B x (r) such that T : B x (r) B x (r) is a contraction? ν B x (r) = x + B(r) Ball of radius r centered at 0 in the space Ω s ˆx x B x (r) Ω s A: Radii polynomials Suppose the bounds Y, Z(r) satisfy D x T ν ( x + b)c T ν ( x) x Y Z(r) k k Lemma: If there exists r>0such that p k (r) < 0 for all k, then there is a unique ˆx B x (r) s.t. f(ˆx, ν) =0. proof. Banach fixed point theorem. sup b,c B(r) p k (r) =Y + Z(r) r w s k

Rigorous Computations An Example : Compute solution of x 2 2=0

Rigorous Computations An Example : Compute solution of x 2 2=0

Rigorous Computations An Example : Compute solution of x 2 2=0 Ω = R x =1.41 Choose the Banach Space Compute num. solution Define the fixed point operator x Define Define radii polynomial Find r sup b,c B(r) T ( x) x Y k D x T ( x + b)c Z(r) k

Rigorous Computations An Example : Compute solution of x 2 2=0 Ω = R x =1.41 Choose the Banach Space Compute num. solution Define the fixed point operator x Define Define radii polynomial Find r sup b,c B(r) T ( x) x Y k D x T ( x + b)c Z(r) k

Rigorous Computations An Example : Compute solution of x 2 2=0 Ω = R x =1.41 Df( x) 1 0.35 T (x) =x 0.35(x 2 2) Choose the Banach Space Compute num. solution Define the fixed point operator Define Define radii polynomial Find r sup b,c B(r) x T ( x) x Y k D x T ( x + b)c Z(r) k

Rigorous Computations An Example : Compute solution of x 2 2=0 Ω = R x =1.41 Df( x) 1 0.35 T (x) =x 0.35(x 2 2) Choose the Banach Space Compute num. solution Define the fixed point operator Define Define radii polynomial Find r sup b,c B(r) x T ( x) x Y k D x T ( x + b)c Z(r) k

Rigorous Computations An Example : Compute solution of x 2 2=0 Ω = R x =1.41 Df( x) 1 0.35 T (x) =x 0.35(x 2 2) T ( x) x 0.0042 DT( x + b)c 0.013 r +0.7 r 2 Choose the Banach Space Compute num. solution Define the fixed point operator Define Define radii polynomial Find r sup b,c B(r) x T ( x) x Y k D x T ( x + b)c Z(r) k

Rigorous Computations An Example : Compute solution of x 2 2=0 Ω = R x =1.41 Df( x) 1 0.35 T (x) =x 0.35(x 2 2) T ( x) x 0.0042 DT( x + b)c 0.013 r +0.7 r 2 Choose the Banach Space Compute num. solution Define the fixed point operator Define Define radii polynomial Find r sup b,c B(r) x T ( x) x Y k D x T ( x + b)c Z(r) k

Rigorous Computations An Example : Compute solution of x 2 2=0 Ω = R x =1.41 Df( x) 1 0.35 T (x) =x 0.35(x 2 2) T ( x) x 0.0042 DT( x + b)c 0.013 r +0.7 r 2 p(r) =0.7 r 2 0.987 r +0.0042 Choose the Banach Space Compute num. solution Define the fixed point operator Define Define radii polynomial Find r sup b,c B(r) x T ( x) x Y k D x T ( x + b)c Z(r) k

Rigorous Computations An Example : Compute solution of x 2 2=0 Ω = R x =1.41 Df( x) 1 0.35 T (x) =x 0.35(x 2 2) T ( x) x 0.0042 DT( x + b)c 0.013 r +0.7 r 2 p(r) =0.7 r 2 0.987 r +0.0042 Choose the Banach Space Compute num. solution Define the fixed point operator Define Define radii polynomial Find r sup b,c B(r) x T ( x) x Y k D x T ( x + b)c Z(r) k

Rigorous Computations An Example : Compute solution of x 2 2=0 Ω = R x =1.41 Df( x) 1 0.35 T (x) =x 0.35(x 2 2) T ( x) x 0.0042 DT( x + b)c 0.013 r +0.7 r 2 p(r) =0.7 r 2 0.987 r +0.0042 I =[0.00427, 1.4057], r I,p(r) < 0 Choose the Banach Space Compute num. solution Define the fixed point operator Define Define radii polynomial Find r sup b,c B(r) x T ( x) x Y k D x T ( x + b)c Z(r) k 2 1.41 ± 0.00427

Eigendecomposition of interval matrices Notation: bold case interval normal case scalar A A A i,j =[a i,j a i,j ], Â is the center of A

Eigendecomposition of interval matrices Notation: bold case interval normal case scalar A A A i,j =[a i,j a i,j ], Â is the center of A GOAL: enclosing eigendecomposition Given A IC n n construct a set of balls B i C C n, i =1...n such that for any A A and any i =1...n there exists x i =(λ i,v i ) B i s.t Av i = λ i v i, where x i is unique up to a scaling factor of v i.

Some literature Tetsuro Yamamoto. Error bounds for computed eigenvalues and eigenvectors. Numer. Math., 34 (2):189--199, 1980. Siegfried M. Rump and Jens-Peter M. Zemke. On eigenvector bounds. BIT, 43(4):823--837,2003. Milan Hladik, David Daney, and Elias Tsigaridas. Bounds on real eigenvalues and singular values of interval matrices. SIAM J. Matrix Anal. Appl., 31(4):2116--2129, 2009/10. G. Mayer. Result verification for eigenvectors and eigenvalues. In Topics in validated computations (Oldenburg, 1993), volume 5 of Stud. Comput. Math., pages 209--276. North- Holland, Amsterdam, 1994. G. Alefeld and H. Spreuer. Iterative improvement of componentwise error bounds for invariant subspaces belonging to a double or nearly double eigenvalue. Computing, 36(4):321--334, 1986. Siegfried M. Rump. Computational error bounds for multiple or nearly multiple eigenvalues. Linear Algebra Appl., 324(1-3):209--226, 2001. Special issue on linear algebra in self-validating methods.

Rigorous computation of eigendecomposition: scalar case Define the problem f(x)=0

Rigorous computation of eigendecomposition: scalar case Define the problem f(x)=0 Suppose ( λ, v) is an approximate eigenpair of A, i.e.a v λ v f(x) =A x =(λ,v 1,v 2,...,v k 1,v k+1,...,v n ) v 1. v ḳ. v n f(x) :C n C n λ v 1. v ḳ. v n, v k = max{ v i } f(x) =0 (λ,v) is an eigenpair of A.

Rigorous computation of eigendecomposition: scalar case Define the problem f(x)=0 Suppose ( λ, v) is an approximate eigenpair of A, i.e.a v λ v f(x) =A x =(λ,v 1,v 2,...,v k 1,v k+1,...,v n ) v 1. v ḳ. v n f(x) :C n C n λ v 1. v ḳ. v n, v k = max{ v i } f(x) =0 (λ,v) is an eigenpair of A. Note that the roots of f(x) are isolated

Rigorous computation of eigendecomposition: scalar case Numerical solution : x =( λ, v 1, v 2,..., v k 1, v k+1,..., v n ) Banach space: Ω = {x C n : x Ω < }, x Ω = max{ x i }, i Operator T(x): T (x) =x Rf(x) v 1. R Df( x) 1, Df( x) = v ḳ (A I n )ˆk.. v n

Rigorous computation of eigendecomposition: scalar case Definition of the bounds Y, Z(r) ( T(x)=x-Rf(x))

Rigorous computation of eigendecomposition: scalar case Definition of the bounds Y, Z(r) Y T ( x) x Y := Rf( x) ( T(x)=x-Rf(x))

Rigorous computation of eigendecomposition: scalar case Definition of the bounds Y, Z(r) Y T ( x) x Y := Rf( x) Z(r) sup b,c B(r) DT( x + b)c ( T(x)=x-Rf(x)) DT( x + b)c (I n R Df( x))c + R[(Df( x) Df( x + b))c] r I n R Df( x) 1 n + r 2 2 R ˆ1 n ( b, c r) Z(r) :=rz 0 + r 2 Z 1

Rigorous computation of eigendecomposition: scalar case Definition of the bounds Y, Z(r) Y T ( x) x Y := Rf( x) Z(r) sup b,c B(r) DT( x + b)c ( T(x)=x-Rf(x)) DT( x + b)c (I n R Df( x))c + R[(Df( x) Df( x + b))c] r I n R Df( x) 1 n + r 2 2 R ˆ1 n ( b, c r) Z(r) :=rz 0 + r 2 Z 1 p k (r) =(Y + Z(r)) k r.

Rigorous computation of eigendecomposition: scalar case Definition of the bounds Y, Z(r) Y T ( x) x Y := Rf( x) Z(r) sup b,c B(r) DT( x + b)c ( T(x)=x-Rf(x)) DT( x + b)c (I n R Df( x))c + R[(Df( x) Df( x + b))c] r I n R Df( x) 1 n + r 2 2 R ˆ1 n ( b, c r) Z(r) :=rz 0 + r 2 Z 1 p k (r) =(Y + Z(r)) k r. Suppose that for any r I, p k (r) < 0 for any k. Then, for any r I, thereexistsagenuine eigenpair x =(λ,v) of A in the ball B x (r) around ( λ, v)

Rigorous computation of eigendecomposition: extension to interval matrix Consider interval matrix A and interval arithmetics

Rigorous computation of eigendecomposition: extension to interval matrix Consider interval matrix A and interval arithmetics A IC n n, f : C n IC n, Df IC n n

Rigorous computation of eigendecomposition: extension to interval matrix Consider interval matrix A and interval arithmetics A IC n n, f : C n IC n, Df IC n n Define x =( λ, v) a numerical eigenpair of  Let R be a numerical inverse of ˆDf and define T = x Rf(x)

Rigorous computation of eigendecomposition: extension to interval matrix Consider interval matrix A and interval arithmetics A IC n n, f : C n IC n, Df IC n n Define x =( λ, v) a numerical eigenpair of  Let R be a numerical inverse of ˆDf and define T = x Rf(x) p k (r) < 0 for any A A there exists a unique (λ,v) so that λ λ <r, v j v j <r, v k = v k and Av = λv.

The Matlab Code 1. function r=enc_eig(a,v,l) 2. n= size(v,1); 3. iv=intval(v); 4. [m,iv]=max(abs(v)); 5. O=ones(n,1); 6. df=intval(a)-intval(l)*speye(n); 7. F=df*iV; 8. df(:,iv)=-iv; 9. R=inv(mid(df)); 10. Y=sup(abs(R*F)); 11. Z0=sup(abs(eye(n)-R*df))*O; 12. O(IV)=0; 13. Z1=2*abs(R)*O; 14. rd=[z1,z0-ones(n,1),y]; % coefficients of radii polynomials [r^2,r,const] 15. r=evaluate_intersection_int(rd); % evaluate where all the polynomials are negative 16. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 17. function neg_inter=evaluate_intersection_int(rd) 18. i2=intval(2); 19. rd=intval(rd); 20. neg_inter=nan; 21. Rm=(-rd(:,2)-sqrt(rd(:,2).^i2-intval(4)*rd(:,1).*rd(:,3)))./(i2*rd(:,1)); 22. Rp=-Rm-rd(:,2)./rd(:,1); 23. ISCOMP=find(isreal(Rp)+isreal(Rm)<2, 1); 24. if isempty(iscomp)==0 25. 'error: a polynomial is positive everywhere' 26. return 27. end 28. intersection=[max(rm.sup),min(rp.inf)]; 29. if intersection(1)<intersection(2) 30. neg_inter=intersection(1); 31. end 32. end 33. end 34. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 35. Definition of radii polynomilas Compute intersections of intervals

Some computational results Enclosure of one eigenpair for large matrices D = diag( N/2,...,N/2), X = rand(size(d)), A = X D X 1 [q, w] =eig(a) x =(q(:,k),w(k, k)) N r time (s) 10 4.3107e-14 0.0073 100 2.4589e-10 0.0114 500 3.5151e-09 0.2586 1000 2.3199e-08 1.6478

Some computational results Enclosure of the Eigendecomposition for large intervals D = diag(0,e ik 2π 5 ), X = rand(size(d)) + i rand(size(d)) Â = X D X 1, A = Â + 1rad

Some computational results Comparison of the computational time verifyeig - verification method, Brouwer s fix p. verifynlss - Krawczyk operator for the computation of zeros of nonlinear functions  is a matrix with eigenvalue 0 and N equispaced on the unit circle. On the left rad = 10 15

Some computational results Comparison of the computational time Our approach is satisfactory from the point of view of accuracy of the result and the algorithm is faster then verifyeig, especially for small N. This achievement has been possible thanks to the analytical estimates, in the form of radii polynomials, that allow minimizing the number of computations done with intervals.

Thank you Köszönöm - Eskerrik asko for your attention