Computation of Rational Szegő-Lobatto Quadrature Formulas

Transcription

1 Computation of Rational Szegő-Lobatto Quadrature Formulas Joint work with: A. Bultheel (Belgium) E. Hendriksen (The Netherlands) O. Njastad (Norway) P. GONZÁLEZ-VERA Departament of Mathematical Analysis. La Laguna University La Laguna. Tenerife. Canary Islands. Spain.

2 General aim Approximate calculation of integrals on the unit circle. I ω (f) = π π f(e iθ )ω(θ)dθ Af(z α )+Bf(z β )+ n λ j f(z j ) = I n+2 (f) z 3 z 2 z α α k 1/α k z 1 z β 1 z n A, B, λ j > 0 zj zk, j k 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.

3 Contents I Introduction: Gauss, Gauss-Radau and Gauss-Lobatto formulas. II Periodic integrands: Szegő-type quadrature formulas. III Rational quadratures on the unit circle. IV Rational Szegő-Lobatto formulas. V Error and convergence.

4 Gauss, Gauss-Radau and Gauss-Lobatto formulas I σ (f) = b a f(x)σ(x), a < b + There exist n distinct nodes {x j } n (a, b) and positive weights {A j } n such that, n I σ (P ) = I n (P ) = A j P (x j ), P P 2n 1 P 2n 1 = span{x k : 0 k 2n 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.

5 Gauss, Gauss-Radau and Gauss-Lobatto formulas [a, b] finite. [a, b] = [ 1, 1]. Theorem I σ (f) = +1 1 f(x)σ(x)dx Given r, s {0, 1} there exist positive weights A (r,s) +, A(r,s) and {A (r,s) j } n along with n distinct nodes {x(r,s) j } ( 1, 1) such that, n I n+r+s (f) = ra (r,s) + f(1)+sa(r,s) f( 1)+ f P 2n+r+s 1 r + s = 0 Gauss Formulas. r + s = 1 Gauss-Radau Formulas. r + s = 2 Gauss-Lobatto Formulas. Gauss type Formulas A (r,s) j f(x (r,s) j ) = I σ (f)

6 Szegő-type quadratures I ω (f) = π π f(θ)ω(θ)dθ f: 2π-periodic. ω(θ): 2π-periodic weighted function. I n (f) = n λ j f(θ j ); {θ j } [ π, π), θ j θ k if j k I ω (T ) = I n (T ) : T (θ) = N (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)

7 Szegő-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 T (θ) = (a k cos kθ + b k sin kθ) = L(e iθ ), L Λ m,m I ω (f) = k=0 π π f(e iθ )ω(θ)dθ I n (f) = n λ j f(z j ) 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).

8 Szegő-type quadratures Some notations {ρ k } 0 : monic Szegő polynomials associated with ω(θ), i.e. ρ 0 (z) = 1,ρ n (z) = z n + + δ n, n = z = e iθ, n 1, ρ n (z), z k ω = π 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 Given I ω (f) = π π f(eiθ )ω(θ)dθ, then I n (f) = n λ jf(z j ) = = I ω (f), f Λ (n 1),(n 1), if and only if, 1 The nodes {z j } n are the zeros of: B n (z, τ) = zρ n 1 (z)+τρ n 1(z), τ = 1. 2 λ j = [ n 1 ] 1 ρ k (z j ) 2, j = 1,..., n. ρ k, ρ k ω k=0

9 Szegő-type quadratures Szegő 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).

10 Szegő-type quadratures Szegő-Lobatto Quadrature rules C. Jagels, L. Reichel (2007) A. Bultheel, L. Daruis, P. G-V (2009) I ω (f) = π π f(e iθ )ω(θ)dθ Given: z α z β on T, find n distinct nodes {z j } n T, z j / {z α, z β } and n + 2 positive weights A, B and {λ j } n 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.

11 Szegő-type quadratures Basic approach ω(θ) {ρ k } 0 Available information: µ k = π π e ikθ ω(θ)dθ, k = 0, ±1, ±2,... 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). Define ρ n+1 (z) = zρ n (z) + δ n+1 ρ n(z). Now take τ n+2 T, ( τ n+2 = 1) and consider B n+2 (z) = z ρ n+1 (z) + τ n+2 ρ 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

12 Rational Quadrature Formulas on the Unit Circle Gauss-type Formulas: Exact integration of polynomials (all the poles at infinity) Szegő-type formulas: Exact integration of Laurent polynomials (all the poles at the origin and infinity) Given π π f(eiθ )ω(θ)dθ quadrature formulas exactly integrating 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)

13 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 1 α i z ; η i = α 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 ω 0 = π 0 = 1; ω k = (z α j ); π k = ω k (z) k B k (z) = γ k π k (z) ; γ k = ( 1) k η j ; } { L = {R = Pπn : P P n = R(z) = k (1 α j z), k = 1, 2,... } P (z) (1 α 1 z) (1 α n z)

14 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 ω n, Q P n } = { f = R p,q = L p +L q = } Q(z) (z α 1 ) (z α n ) { f = P ω p π q : P P p+q = span{b k, k = p,..., 1, 0, 1,..., q} B k = B k = 1 B k α k = 0, k = 1, 2,..., R p,q = Λ p,q. }

15 Rational Quadrature Formulas on the Unit Circle I ω (f) = π π f(e iθ )ω(θ)dθ n I n (f) = λ j f(z j ), z j T, z j z k if j k 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?

16 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 Szegő Functions) Drawback The zeros of ϕ n (z) lie in D.

17 Rational Quadrature Formulas on the Unit Circle Theorem (Rational Szegő Quadratures) Let {ϕ k } 0 be the sequence of monic Rational Szegő Functions. For each n 1, take τ T and set X n (z) = X n (z, τ) = (z α n 1 )ϕ n 1 (z) + τ(1 α n 1 z)ϕ n 1 (z), (ϕ k (z) = B k(z)ϕ k (z) = B k (z)ϕ k (1/z)). Then, 1 X n (z) has exactly n distinct nodes z 1,..., z n which lie on T. 2 There exist positive numbers λ 1,..., λ n such that n π I n (f) = λ j f(z j ) = I ω (f) = f(e iθ )ω(θ)dθ, f R n 1,n 1 3 λ j = n 1 π ϕ k (z j ) 2 ϕ k 2, j = 1,..., n, ω k=0 ϕ k 2 ω = ϕ k, ϕ k ω = π π ϕ k(z)ϕ k (z)ω(θdθ), (z = e iθ ) 4 There can not exist an n-point quadrature rule with nodes on T to be exact either in R n,n 1 or R n 1,n.

18 Rational Quadrature Formulas on the Unit Circle An Illustrative Example ω(θ) = 1 2π, θ [ π, π], {α k} D. {ϕ} 0 : the Orthogonal Sequence of monic Rational Szegő Function. zb ϕ k, ϕ k ω = 1, k = 0, 1, 2,... ϕ n (z) = κ n(z) n z α n, κ n 0, n = 1, 2,... τ T I n (f) = n λ j f(z j ) = 1 2π π π z j : zω n 1 (z) + τπ n 1 (z) = 0 n 1 ω n 1 (z) = (z α k ); π n 1 (z) = λ j = k=1 f(e iθ )dθ, f R n 1,n 1 n 1 k=1 (1 α k z) 1 1 +, j = 1,..., n n 1 1 α k 2 k=1 z j α k 2 α k = 0, k = 1, 2,... z j : z n + τ = 0, λ j = 1 n, j = 1,..., n.

19 Rational Szegő-Lobatto Formulas Given x 1, x 2 T, x 1 x 2, n 1. Find n distinct nodes z 1,..., z n on T, z j / {x 1, x 2 } and positive weights A 1, A 2, λ 1,..., λ n such that: Ĩ n+2 (f) = A 1 f(x 1 ) + A 2 f(x 2 ) + f R p(n),p(n), p(n) as large as possible. n + 2 nodes on T p(n) n + 1. n λ j f(z j ) = I ω (f)

20 Rational Szegő-Lobatto Formulas p(n) = n + 1 Ĩn+2(f) a Rational Szegő Quadrature Formula γ n (x 1 ) = γ n (x 2 ) with γ n (x j ) = (α n 1 x j )ϕ n 1 (x j ) (1 α n 1 x j )ϕ n 1 (x T, j = 1, 2. j) ϕ n 1 (z): (n 1)-th monic Rational Szegő Function γ n (x 1 ) γ n (x 2 ) p(n) n If p(n) = n Ĩn+2(f) Rational Szegő-Lobatto Quadrature Formula.

21 Rational Szegő-Lobatto Formulas The Approach µ: positive Borel Measure on T; {α k } 1 D: The poles of the rational functions. {ϕ k } 0 : Sequence of Rational Szegő Functions. Recurrence Relation [( ) z αn 1 ϕ n(z) = ε n ϕ n 1(z) + 1 α nz ϕ n(z) = η nε n [ ( δn ε n ( δn ε n ) ( ) z αn 1 ϕ n 1(z) + 1 α nz ϕ n(z) = B n(z)ϕ n (z) = B n(z)ϕ n(1/z), n 1. B 0(z) = 1, B k (z) = ( 1) k k ε n 0; δ n < ε n, β n = δn ε n, β n < 1. ) ( ) ] 1 αn 1z ϕ n 1(z) 1 α nz ( ) 1 αn 1z ϕ n 1(z) 1 α nz η j (z α 1) (z α k ) (1 α 1z) (1 α k z), ηj = ] { αj α j, αj 0 1, α j = 0

22 Rational Szegő-Lobatto Formulas Theorem Let µ be a positive Borel measure on T, {ϕ n } 0 being its sequence of monic Rational Szegő Functions with respect to the fixed sequence of poles {α k } 1 D. Then, given β n+1 D, there exists a positive Borel measure µ on T such that if { ϕ k } 0 monic sequence, it holds that ϕ k (z) = ϕ k (z), k = 0, 1,..., n. [( ) ( z α ϕ n+1(z) = ε n n+1 ϕ n(z) + β n+1 ε n α n+1 z Taking dµ(θ) = ω(θ)dθ; d µ(θ) = ω(θ)dθ; 1 α nz 1 α n+1 z represents its ) ] ϕ n(z), I ω (R) = I ω (R), R R n,n

23 Rational Szegő-Lobatto Formulas Take τ n+2 T and set X n+2 (z) = (z α n+1 ) ϕ n+1 (z) + (1 α n+1 z) ϕ n+1(z) X n+2 (z) has n + 2 distinct zeros z 1,..., z n+2 on T and there exist positive numbers λ 1,..., λ n+2 such that Since n+2 Ĩ n+2 (f) = λ j f( z j ) = I ω (f), f R n+1,n+1. I ω (R) = I ω (R), f R n,n Ĩn+2(R) = I ω (R), R R n,n X n+2 (z) L n+2 depends on two arbitrary parameters β n+1 D and τ n+2 T.

24 Rational Szegő-Lobatto Formulas Theorem (Main Result) Given a natural number n and two distinct complex numbers x 1, x 2 T along with a weight function ω(θ) on T there exist β n+1 D and τ n+2 T such that: where and B n+2 (x 1 ) = B n+2 (x 2 ) = 0 B n+2 (z) = (z α n+1 ) ϕ n+1 (z) + (1 α n+1 z) τ n+2 ϕ n+1(z) ϕ n+1 (z) = ( ) ( ) z αn ϕ n (z) + 1 α n+1 z β 1 αn z n+1 ϕ 1 α n+1 z n(z) ϕ n (z) being the n-th monic Rational Szegő Function for ω(θ) and the given sequence of poles {α k } 1 D.

25 Rational Szegő-Lobatto Formulas Corollary Given x 1 and x 2 T (x 1 x 2 ) there exist n distinct nodes z 1,..., z n on T with z j / {x 1, x 2 }, j = 1,..., n and n + 2 positive weights A 1, A 2 and λ j, j = 1,..., n such that Ĩ n+2 (f) = A 1 f(x 1 ) + A 2 f(x 2 ) + n λ j f( z j ) = I ω (f) { } P (z) f R n,n = (z α 1 ) (z α n )(1 α 1 z) (1 α n z) ; P P 2n

26 Error and Convergence Error Assume x 1 x 2 both on T. Consider Ĩ n+2 (f) = A 1 f(x 1 ) + A 2 f(x 2 ) + n λ j f( z j ) and set E n+2 (f) = I ω (f) Ĩn+2(f). Suppose f analytic on a neighborhood of T and let U be the domain of analiticity of ω n (z)π n (z)f(z). Then, ( ( )) r n+1 E n+2 (f) 2 ω n π n f γr γr 1 r 2 g n(r) + R1 n 1 R 2 1 g n R where r < 1 < R such that γ r γ R U and g n (s) = 1 π n 1 2π 1 se iθ dθ, s < 1 α j 2 π a > 0; γ a = {z C : z = a}, f K = max z K { f(z) }

27 Error and Convergence Convergence For each n 1 take x 1,n, x 2,n T, x 1,n x 2,n n Ĩ n+2 (f) = A 1,n f(x 1,n ) + A 2,n f(x 2,n ) + λ j,n f( z j,n ) (Rational Szegő-Lobatto formula) Recall: Ĩ n+2 = I ω (f), f R n n, lim n Ĩ n+2 (f) = I ω (f)? Lemma Set R = n=0 R n,n, then R is dense in the class C(T) of continuous functions on T, iff k=1 (1 α k ) = + Theorem Set R ω (T) = {f : T C : fω integrable on T}; then lim Ĩ n+2 (f) = I ω (f) = n if k=1 (1 α k ) = + π π f(e iθ )ω(θ)dθ, f R ω (T)