Computation of Rational Szeg o-Lobatto Quadrature Formulas P. GONZ - PowerPoint PPT Presentation

Computation of Rational Szeg˝ o-Lobatto Quadrature Formulas P. GONZ´ ALEZ-VERA Departament of Mathematical Analysis. La Laguna University. 38271 La Laguna. Tenerife. Canary Islands. Spain. e-mail: pglez@ull.es Joint work with: A. Bultheel (Belgium) E. Hendriksen (The Netherlands) O. Njastad (Norway) P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

General aim Approximate calculation of integrals on the unit circle. � π n f ( e iθ ) ω ( θ ) dθ ≈ Af ( z α )+ Bf ( z β )+ � I ω ( f ) = λ j f ( z j ) = I n +2 ( f ) − π j =1 1 /α k z α z 2 z 1 A, B, λ j > 0 z β α k 1 z j � = z k , j � = k z 3 z n z j / ∈ { z α , z β } I ω ( f ) = I n +2 ( f ) with f in a subspace of rational functions with prescribed poles at α k and 1 /α k of as high dimension as possible. P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Contents I Introduction: Gauss, Gauss-Radau and Gauss-Lobatto formulas. II Periodic integrands: Szeg˝ o-type quadrature formulas. III Rational quadratures on the unit circle. IV Rational Szeg˝ o-Lobatto formulas. V Error and convergence. P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Gauss, Gauss-Radau and Gauss-Lobatto formulas � b I σ ( f ) = f ( x ) σ ( x ) , −∞ ≤ a < b ≤ + ∞ a There exist n distinct nodes { x j } n j =1 ⊂ ( a, b ) and positive weights { A j } n j =1 such that, n � I σ ( P ) = I n ( P ) = A j P ( x j ) , ∀ P ∈ P 2 n − 1 j =1 P 2 n − 1 = span { x k : 0 ≤ k ≤ 2 n − 1 } , P = span { x k , k = 0 , 1 , . . . } ⇓ Gauss, Gauss-Christtophel or Gaussian quadrature formulas. Maximal domain of validity Positivity of the weights ⇒ Stability and convergence. Efficient computation in terms of an eigenvalue problem involving certain tridiagonal Jacobi matrices. P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Gauss, Gauss-Radau and Gauss-Lobatto formulas [ a, b ] finite. [ a, b ] = [ − 1 , 1] . � +1 I σ ( f ) = f ( x ) σ ( x ) dx − 1 Theorem Given r, s ∈ { 0 , 1 } there exist positive weights A ( r,s ) , A ( r,s ) and + − { A ( r,s ) j =1 along with n distinct nodes { x ( r,s ) } n } ⊂ ( − 1 , 1) such j j that, n I n + r + s ( f ) = rA ( r,s ) f (1)+ sA ( r,s ) A ( r,s ) f ( x ( r,s ) � f ( − 1)+ ) = I σ ( f ) + − j j j =1 ∀ f ∈ P 2 n + r + s − 1 r + s = 0 ⇒ Gauss Formulas. r + s = 1 ⇒ Gauss-Radau Formulas. r + s = 2 ⇒ Gauss-Lobatto Formulas. ⇓ Gauss type Formulas P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Szeg˝ o-type quadratures � π I ω ( f ) = f ( θ ) ω ( θ ) dθ − π f : 2 π -periodic. ω ( θ ) : 2 π -periodic weighted function. n � I n ( f ) = λ j f ( θ j ); { θ j } ⊂ [ − π, π ) , θ j � = θ k if j � = k j =1 N � I ω ( T ) = I n ( T ) : T ( θ ) = ( a k cos kθ + bk sin kθ ) k =0 T : trigonometric polynomial with as high degree as possible. N ≤ n − 1 N = n − 1 ⇒ Quadrature formulas with the highest trigonometric precision degree. (Bi-orthogonal systems of trigonometric polynomials) P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Szeg˝ o-type quadratures Alternative approach Passing to the unit circle T = { z ∈ C : | z | = 1 } . Some notations D = { z ∈ C : | z | = 1 } , E = { z ∈ C : | z | > 1 } . p, q ∈ Z , p ≤ q , Λ p,q = span { z k : p ≤ k ≤ q } (Laurent polynomials) Λ ≡ space of all Laurent polynomials. m � ( a k cos kθ + b k sin kθ ) = L ( e iθ ) , L ∈ Λ − m,m T ( θ ) = k =0 � π n f ( e iθ ) ω ( θ ) dθ ≈ I n ( f ) = � I ω ( f ) = λ j f ( z j ) − π j =1 z j ∈ T , z j � = z k if j � = k . z j = e iθ j , θ j ∈ [ − π, π ) , θ j � = θ k . I ω ( f ) = I n ( f ) , ∀ f ∈ Λ − ( n − 1) , ( n − 1) . P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Szeg˝ o-type quadratures Some notations { ρ k } ∞ 0 : monic Szeg˝ o polynomials associated with ω ( θ ) , i.e. ρ 0 ( z ) = 1 , ρ n ( z ) = z n + · · · + δ n , n = 1 . 2 . . . . z = e iθ , n ≥ 1 , � π � ρ n ( z ) , z k � ω = − π ρ n ( z ) z k ω ( θ ) dθ = 0 , k = 0 , 1 , . . . , n − 1 . ρ ∗ n ( z ) = z n ρ n (1 /z ) (Reverse polynomial) Theorem � π − π f ( e iθ ) ω ( θ ) dθ , then I n ( f ) = � n Given I ω ( f ) = j =1 λ j f ( z j ) = = I ω ( f ) , ∀ f ∈ Λ − ( n − 1) , ( n − 1) , if and only if, 1 The nodes { z j } n j =1 are the zeros of: B n ( z, τ ) = zρ n − 1 ( z )+ τρ ∗ n − 1 ( z ) , | τ | = 1 . � − 1 � n − 1 | ρ k ( z j ) | 2 � 2 λ j = , j = 1 , . . . , n . � ρ k , ρ k � ω k =0 P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Szeg˝ o-type quadratures Szeg˝ o quadrature formulas ( W. B. Grags (1982,1991), Jones et als (1989)) Efficient computation in terms of an eigenvalue problem involving Hessemberg Matrices whose entries essentially depend on the numbers: δ k = ρ k (0) , ρ 0 = 1 , | δ k | < 1 , k = 1 , 2 , . . . (Verblunsky coefficients, Schur parameters, Reflection coefficients). P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Szeg˝ o-type quadratures Szeg˝ o-Lobatto Quadrature rules C. Jagels, L. Reichel (2007) A. Bultheel, L. Daruis, P. G-V (2009) � π f ( e iθ ) ω ( θ ) dθ I ω ( f ) = − π Given: z α � = z β on T , find n distinct nodes { z j } n j =1 ⊂ T , ∈ { z α , z β } and n + 2 positive weights A, B and { λ j } n z j / j =1 such that: n � I ω ( f ) = I n +2 ( f ) = Af ( z α ) + Bf ( z β ) + λ j f ( z j ) , ∀ f ∈ Λ − n,n . j =1 P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Szeg˝ o-type quadratures Basic approach ω ( θ ) � { ρ k } ∞ 0 � π − π e − ikθ ω ( θ ) dθ , k = 0 , ± 1 , ± 2 , . . . Available information: µ k = Recursion: ρ k ( z ) = zρ k − 1 ( z ) + δ k ρ ∗ k − 1 ( z ) , ρ 0 = 1 , δ k = ρ k (0) , k = 1 , 2 , . . . . (Levinson algorithm ). From ρ n ( z ) and given ˜ δ n +1 ∈ D . ( | ˜ δ n +1 | < 1 ). ρ n +1 ( z ) = zρ n ( z ) + ˜ δ n +1 ρ ∗ Define ˜ n ( z ) . Now take ˜ τ n +2 ∈ T , ( | τ n +2 | = 1 ) and consider ˜ ρ ∗ B n +2 ( z ) = z ˜ ρ n +1 ( z ) + ˜ τ n +2 ˜ n +1 ( z ) ˜ B n +2 ( z ) has n + 2 distinct zeros on T . AIM Determine ˜ δ n +1 ∈ D and ˜ τ n +2 ∈ T such that B n +2 ( z α ) = ˜ ˜ B n +2 ( z β ) = 0 P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Rational Quadrature Formulas on the Unit Circle Gauss-type Formulas: Exact integration of polynomials (all the poles at infinity) Szeg˝ o-type formulas: Exact integration of Laurent polynomials (all the poles at the origin and infinity) ⇓ � π − π f ( e iθ ) ω ( θ ) dθ ⇒ quadrature formulas exactly integrating Given more general rational functions with prescribed poles not on T . The poles: { α k } ⊂ D ; { 1 /α k } ⊂ E ⇓ Theory on Orthogonal Rational Functions ( A. Bultheel, E. Hendriksen, P. G-V, O. Njastad) P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Rational Quadrature Formulas on the Unit Circle The spaces of rational functions { α k } ∞ 1 ⊂ D Blaschke Factors ξ i ( z ) = η i α i − z α i 1 − α i z ; η i = | α i | if α i � = 0 ; η i = − 1 if α i = 0 , i=1,2,. . . Blaschke Products B 0 = 1 , B n = B n − 1 ξ n , n ≥ 1 . Set L n = span { B k : k = 0 , 1 , . . . , n } k k � � ω 0 = π 0 = 1; ω k = ( z − α j ); π k = (1 − α j z ) , k = 1 , 2 , . . . j =1 j =1 k ω k ( z ) � π k ( z ); γ k = ( − 1) k B k ( z ) = γ k η j ; j =1 � R = P � � P ( z ) � L = : P ∈ P n = R ( z ) = π n (1 − α 1 z ) · · · (1 − α n z ) P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Rational Quadrature Formulas on the Unit Circle The spaces of rational functions f ∗ ( z ) = f (1 /z ) ⇒ L n ∗ = { f : f ∗ ∈ L n } = { f = Q , Q ∈ P n } ω n � Q ( z ) � = f = ( z − α 1 ) · · · ( z − α n ) � P � R p,q = L p ∗ + L q = f = : P ∈ P p + q ω p π q = span { B k , k = − p, . . . , − 1 , 0 , 1 , . . . , q } 1 B − k = B k ∗ = B k α k = 0 , k = 1 , 2 , . . . , R p,q = Λ − p,q . P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Rational Quadrature Formulas on the Unit Circle � π f ( e iθ ) ω ( θ ) dθ I ω ( f ) = − π n � I n ( f ) = λ j f ( z j ) , z j ∈ T , z j � = z k if j � = k j =1 I ω ( R ) = I n ( R ) , ∀ R ∈ R p ( n ) ,p ( n ) , p ( n ) as large as possible. p ( n ) ≤ n − 1 p ( n ) = n − 1? P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009

Rational Quadrature Formulas on the Unit Circle Orthogonal Rational Functions n ≥ 1 , { ϕ j } n j =0 , ϕ j ∈ L j \L n − 1 an orthogonal basis for L n (Gram-Schmidt process on { B k } n 0 ) k � ϕ k ( z ) = a j B j ( z ) , a k � = 0 leading coefficient j =0 a k = 1 ⇒ ϕ k ( z ) ≡ monic. When the process is repeated ∀ n ≥ 1 ⇒ { ϕ k } ∞ 0 a sequence of ORF w. r. t. ω ( θ ) and the poles { α k } ∞ 1 . (Rational Szeg˝ o Functions) Drawback The zeros of ϕ n ( z ) lie in D . P. GONZ´ ALEZ-VERA, pglez@ull.es Luminy, September 2009