Problem Applications Lemma Results Examples On the convergence of rational Ritz values Applications to rational interpolation of rational functions Bernhard Beckermann Laboratoire Painlev´ e Universit´ e de Lille 1 France AECDSS Luminy, Sept. 2009 Joint work with Stefan G¨ uttel (TU Freiberg) & Raf Vandebril (KU Leuven)
Problem Applications Lemma Results Examples 1 Do zeros of OP/ORF do approach discrete spectrum of orthogonality? 2 Two applications 3 A basic Lemma 4 Assumptions and results 5 Some numerical examples
Problem Applications Lemma Results Examples 1 Do zeros of OP/ORF do approach discrete spectrum of orthogonality? 2 Two applications 3 A basic Lemma 4 Assumptions and results 5 Some numerical examples
Problem Applications Lemma Results Examples Our problem: orthogonal polynomials (OP) With Λ = { λ 1 < λ 2 < ... < λ N } ⊂ R , consider � w ( λ j ) 2 f ( λ j ) g ( λ j ) , � f , g � N = weights w ( λ j ) > 0, � 1 , 1 � N = 1, λ j ∈ Λ together with it’s n th OP p n with roots Θ = { θ 1 < ... < θ n } . ∀ j = 0 , ..., n − 1 : � p n , x j � N = 0 . deg p n = n , Question Does the set Θ approach Λ? For which λ ∈ Λ do we get small dist ( λ, Θ)?
Problem Applications Lemma Results Examples Our problem: orthogonal polynomials (OP) With Λ = { λ 1 < λ 2 < ... < λ N } ⊂ R , consider � w ( λ j ) 2 f ( λ j ) g ( λ j ) , � f , g � N = weights w ( λ j ) > 0, � 1 , 1 � N = 1, λ j ∈ Λ together with it’s n th OP p n with roots Θ = { θ 1 < ... < θ n } . ∀ j = 0 , ..., n − 1 : � p n , x j � N = 0 . deg p n = n , Question Does the set Θ approach Λ? For which λ ∈ Λ do we get small dist ( λ, Θ)? Asymptotic answer: Kuijlaars ’99, BB ’00 For n = n N , N → ∞ , n / N → t , rate lim sup dist ( λ k N , N , Θ N ) 1 / N ≤ exp( U µ ( λ ) − F ) if Λ N ∋ λ k N , N → λ and Λ = Λ N , Θ = Θ N , w = w N ”nice”. Work on discrete OP: Rakhmanov ’96, Dragnev & Saff ’97
Problem Applications Lemma Results Examples Our new problem: orthogonal rational functions (ORF) Consider n th ORF with poles Ξ = { ξ 1 , ..., ξ n } and roots Θ = { θ 1 < ... < θ n } . � , x j ∀ j = 0 , ..., n − 1 : � p n (1 − x deg p n = n , � N = 0 , q n ( x ) = ) . q n q n ξ j ξ j ∈ Ξ Question For which λ ∈ Λ do we get small dist ( λ, Θ)? Interaction between Λ and Θ?
Problem Applications Lemma Results Examples Our new problem: orthogonal rational functions (ORF) Consider n th ORF with poles Ξ = { ξ 1 , ..., ξ n } and roots Θ = { θ 1 < ... < θ n } . � , x j ∀ j = 0 , ..., n − 1 : � p n (1 − x deg p n = n , � N = 0 , q n ( x ) = ) . q n q n ξ j ξ j ∈ Ξ Question For which λ ∈ Λ do we get small dist ( λ, Θ)? Interaction between Λ and Θ? Similar asymptotic answer if poles far from support Λ For n = n N , N → ∞ , n / N → t , rate lim sup dist ( λ k N , N , Θ N ) 1 / N ≤ exp( U µ − ν ( λ ) − F ) if Λ N ∋ λ k N , N → λ and Λ = Λ N , θ = Θ N , w = w N , Ξ = Ξ N ”nice”.
Problem Applications Lemma Results Examples Answer with logarithmic potential theory OP: Constrained energy problem with external field Q = 0 ORF: Constrained energy problem with external field Q = − U ν (constrain σ asymptotics of supports, ν asymptotics of poles, more details later)
Problem Applications Lemma Results Examples Answer with logarithmic potential theory OP: Constrained energy problem with external field Q = 0 ORF: Constrained energy problem with external field Q = − U ν (constrain σ asymptotics of supports, ν asymptotics of poles, more details later) ... and what happens if poles are ∈ conv(Λ N ) \ Λ N ?
Problem Applications Lemma Results Examples 1 Do zeros of OP/ORF do approach discrete spectrum of orthogonality? 2 Two applications 3 A basic Lemma 4 Assumptions and results 5 Some numerical examples
Problem Applications Lemma Results Examples Approach (poles of) RF by lower order RF Consider the Markov function � w ( λ ) 2 1 φ ( z ) = z − λ = � z − · , 1 � N . λ ∈ Λ OP p n = denominator of n th Pad´ e approximant at ∞ . ORF p n / q n : p n denominator of [ n − 1 | n ]th rational interpolant r of φ at ξ 1 , ξ 1 , ξ 2 , ξ 2 , ..., ξ n , ξ n ∈ R (interpolation of value and first derivative).
Problem Applications Lemma Results Examples Approach (poles of) RF by lower order RF Consider the Markov function � w ( λ ) 2 1 φ ( z ) = z − λ = � z − · , 1 � N . λ ∈ Λ OP p n = denominator of n th Pad´ e approximant at ∞ . ORF p n / q n : p n denominator of [ n − 1 | n ]th rational interpolant r of φ at ξ 1 , ξ 1 , ξ 2 , ξ 2 , ..., ξ n , ξ n ∈ R (interpolation of value and first derivative). Error � φ ( z ) − r ( z ) = q n ( z ) 2 w N ( λ ) 2 p n ( λ ) 2 p n ( z ) 2 q n ( λ ) 2 z − λ λ ∈ Λ N Orthogonality: for j = 0 , 1 , ...., n − 1 � w N ( λ ) 2 p n ( λ ) λ j , x j q n ( λ ) 2 = � p n � N = 0 . q n q n λ ∈ Λ N
Problem Applications Lemma Results Examples Approach eigenvalues by rational Ritz values Given A ∈ R N × N symmetric, b ∈ R N , the rational Arnoldi method yields V ∈ R N × n with columns ONB of rational Krylov space span { q n ( A ) − 1 b , Aq n ( A ) − 1 b , ..., A n − 1 q n ( A ) − 1 b } . Question Does the set Θ of Ritz values := eigenvalues of V T n AV n approach spectrum of A ? Which eigenvalues are found by n th rational Ritz values?
Problem Applications Lemma Results Examples Approach eigenvalues by rational Ritz values Given A ∈ R N × N symmetric, b ∈ R N , the rational Arnoldi method yields V ∈ R N × n with columns ONB of rational Krylov space span { q n ( A ) − 1 b , Aq n ( A ) − 1 b , ..., A n − 1 q n ( A ) − 1 b } . Question Does the set Θ of Ritz values := eigenvalues of V T n AV n approach spectrum of A ? Which eigenvalues are found by n th rational Ritz values? Link: Ruhe 84-94, Meerbergen 01, Decker & Bultheel ’08, ... Θ is just the set of roots of ORF p n / q n for the scalar product � � f , g � N = ( g ( A ) b ) T ( f ( A ) b ) = w ( λ ) 2 f ( λ ) g ( λ ) , λ ∈ Λ Λ = spectrum of A , w ( λ ) eigencomponents of b .
Problem Applications Lemma Results Examples 1 Do zeros of OP/ORF do approach discrete spectrum of orthogonality? 2 Two applications 3 A basic Lemma 4 Assumptions and results 5 Some numerical examples
Problem Applications Lemma Results Examples dist ( λ, Θ) and a polynomial extremal problem Set as before support of orthogonality, weights w ( λ ) 2 > 0, Λ = { λ 1 < ... < λ N } { θ 1 < ... < θ n } Θ = zeros of n th ORF p n / q n , Ξ = { ξ 1 , ..., ξ n } ⊂ R poles of n th ORF p n / q n . LEMMA (BB’00) We have � N w ( λ j ) 2 ( λ j − θ 1 ) s ( λ j ) 2 � j =1 , j � =1 q n ( λ j ) 2 � θ 1 − λ 1 = min � deg( s ) < n , w ( λ k ) 2 q n ( λ 1 ) 2 s ( λ 1 ) 2 and, if λ k ∈ [ θ κ − 1 , θ κ ], then ( λ k − θ κ − 1 )( θ κ − λ k ) = � N w ( λ j ) 2 ( λ j − θ κ − 1 )( λ j − θ κ ) s ( λ j ) 2 � j =1 , j � = k q n ( λ j ) 2 � � deg( s ) < n − 1 min . w ( λ k ) 2 s ( λ k ) 2 q n ( λ k ) 2
Problem Applications Lemma Results Examples 1 Do zeros of OP/ORF do approach discrete spectrum of orthogonality? 2 Two applications 3 A basic Lemma 4 Assumptions and results 5 Some numerical examples
Problem Applications Lemma Results Examples Notion from potential theory Given a (signed) Borel measure µ , its logarithmic potential is defined as � 1 U µ ( z ) = log | x − z | d µ ( x ) . The mutual logarithmic energy of measures µ 1 and µ 2 is defined as � U µ 1 ( y ) d µ 2 ( y ) , I ( µ 1 , µ 2 ) = I ( µ ) := I ( µ, µ ) . The normalized counting measure for Ξ = { ξ 1 , ..., ξ n } is � χ N (Ξ) = 1 δ ξ j N ξ j ∈ Ξ
Problem Applications Lemma Results Examples Assumptions Set with n = n N such that n / N → t ∈ (0 , 1) support of orthogonality, weights w N ( λ j , N ) 2 > 0, Λ N = { λ 1 , N , ..., λ N , N } Θ N = { θ 1 , N , ..., θ n , N } zeros of n th ORF p n / q n , Ξ N = { ξ 1 , N , ..., ξ n , N } ⊂ R poles of n th ORF p n / q n . Assumptions: (A1) ∃ Λ compact including all Λ N , χ N (Λ N ) → σ , U σ ∈ C (Σ; R ). (A2) ∃ Ξ closed including all Ξ N , χ N (Ξ N ) → ν , U ν ∈ C (Σ; R ∪ {∞} ). (A3) � 1 , 1 � N = 1, lim N min j w N ( λ j , N ) 1 / N = 1. (A4) Weak separation: for all Λ N ∋ λ k N , N → λ � 1 lim sup lim sup log | λ j , N − λ k N , N | = 0 . δ → 0+ N →∞ 0 < | λ j , N − λ kN , N |≤ δ
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