Rational Gauss-Radau and rational Szegő-Lobatto quadrature on the interval and the unit circle respectively

Transcription

1 Rational Gauss-Radau and rational Szegő-Lobatto quadrature on the interval and the unit circle respectively Francisco Perdomo-Pio Departament of Mathematical Analysis. La Laguna University La Laguna. Tenerife. Canary Islands. Spain. Joint work with: Adhemar Bultheel and Karl Deckers

2 Aim of the talk Present relation between Rational Gauss-Radau and Rational Szegő-Lobatto quadrature formulas. Motivation interchange properties, analytic computations, reduces computational effort,...

3 Aim of the talk Present relation between Rational Gauss-Radau and Rational Szegő-Lobatto quadrature formulas. Motivation interchange properties, analytic computations, reduces computational effort,...

4 Preliminaries Interval I = [ 1, 1] C I = (C { }) \ I {α 1, α{ 2,...} C I p L n = n(x) n k=1 (1 x/α k) : p n P n }, n = 1, 2,... Unit circle T = {z C : z = 1} D = {z C : z < 1} {β 1, β 2 {,...} D p L n = n(z) : p n k=1 (1 β k z) n P n }, n = 1, 2,... Quadrature rules J µ (f) := K n f(t)dµ(t) λ j f(t j ) =: J n (f), j=1 {t j } n j=1 K where K = I or K = T, and t = x or t = z.

5 Preliminaries Interval I = [ 1, 1] C I = (C { }) \ I {α 1, α{ 2,...} C I p L n = n(x) n k=1 (1 x/α k) : p n P n }, n = 1, 2,... Unit circle T = {z C : z = 1} D = {z C : z < 1} {β 1, β 2 {,...} D p L n = n(z) : p n k=1 (1 β k z) n P n }, n = 1, 2,... Quadrature rules J µ (f) := K n f(t)dµ(t) λ j f(t j ) =: J n (f), j=1 {t j } n j=1 K where K = I or K = T, and t = x or t = z.

6 positive rational interpolatory quadrature rule on I µ a positive bounded Borel measure on I, {λ j } n j=1 R+ 0, J µ (f) = J n (f) at least for all f L n 1. Special cases Rational Gaussian quadrature formula, which has maximal domain of exactness: L n L c n 1 = {f(x) g(x) : f L n, g L n 1 }. Rational Gauss-Radau quadrature formula (i.e, one node x α fixed in advance), with domain of exactness: L n 1 L c n 1.

7 Construction quadratures Rational Gaussian quadrature nodes x k = zeros of ϕ n L n \ L n 1, where ϕ n µ L n 1 w.r.t. the inner product f, g µ = 1 1 f(x)g(x)dµ(x). λ k = { n 1 j=0 ϕ j(x k ) 2 } 1. Rational Gauss-Radau quadrature nodes x k = zeros of the quasi-orthogonal rational function (qorf) Q n,τn = ϕ n (x) + τ n 1 x/α n 1 1 x/α n ϕ n 1 (x), τ n C. λ k = { n 1 j=0 ϕ j(x k ) 2 } 1. In remainder we assume α n R I.

8 positive rational interpolatory quadrature rule on T µ a positive bounded Borel measure on T, { λ j } n j=1 R+ 0, J µ ( f) = J n ( f) for all f L p L p = {f(z) g(1/z) : f, g L p }, (n 1)/2 p n 1. Special cases Rational Szegő quadrature formula, which has maximal domain of exactness: L n 1 L (n 1). Rational Szegő-Lobatto quadrature formula (i.e., two nodes z α and z β fixed in advance) with domain of exactness: L n 2 L (n 2).

9 Construction quadratures Rational Szegő quadrature nodes z k = zeros of a para-orthogonal rational function (porf) Q n,ξn (z) = z β n 1 1 β n z φ n 1(z) + ξ n 1 β n 1 z 1 β n z φ n 1(z), where φ n µ Ln 1 w.r.t. the inner product f, g = π f(z) g(1/z)d µ(θ), µ π φ n(z) = φ n (1/z) n j=1 λk = { n 1 j=0 φ j(z k ) 2 } 1. z β k 1 β k z, and ξ n T.

10 Construction quadratures Rational Szegő-Lobatto quadrature nodes z k = zeros of a porf Q n, ξn (z) = z β n 1 φ 1 β n z n 1 (z) + ξ 1 β n 1 z n 1 β n z ( φ n 1 (z) = c z βn 2 k φ 1 β n 1 z n 2(z) + δ 1 β n 2 z n 1 δ n 1 D, and ξ n T. λk = { n 1 j=0 φ j (x k ) 2 } 1. In remainder we assume β n ( 1, 1). φ n 1 (z), where ) 1 β n 1 z φ n 2 (z),

11 Connection between I and T Joukowski transformation x = J(z) = 1 2 (z + z 1 ). {α 1, α 2,...} C I { ˆβ 1, ˆβ 2,... } D. ˆβ 2k = ˆβ 2k 1 = β k, α k = J(β k ). The measures µ on I µ on T a symmetric measure µ(e) = µ({cos θ, θ E [0, π)}) + µ({cos θ, θ E [ π, 0)}), The integrals I f(x)dµ(x) = 1 2 T f(z)d µ(z), f = f J.

12 Connection between I and T Theorem Sym {β 1,..., β n 1 } real or complex conjugate pairs, and µ positive symmetric Borel measure on T. Then, 1 zeros of Q n,ξn appear in complex conjugate pairs iff ξ n = ±1. 2 z = 1 is zero iff ξ n = 1 and z = 1 iff ξ n = ( 1) n+1. 3 Let J n ( f) be an n-point rational Szegő-Lobatto quadrature formula based on zeros of Q n,±1. Then for k = 1,..., n, the weight λ k = λ j if z j = z k. Theorem Rel µ related with µ by Joukowski transform, then ) ϕ k (x) = F (φ 2k (z), φ 2k (z) ( ) = G φ 2k 1 (z), φ 2k 1(z).

13 Relating rational Szegő-Lobatto and Gauss-Radau rules Theorem SL-GR µ and µ related by Joukowski transform. Let J 2n ( f) = λ α (z α ) f(z α ) + λ β f(zα ) + 2n 2 k=1 λ k f(zk ) be a 2n-point rational Szegő-Lobatto quadrature formula with fixed nodes {z α, z α } T \ { 1, +1} for J µ ( f). Set x α = J(z α ), λ α = λ α, x k = J(z k ) and λ k = λ k. Then the formula J n (f) = λ α f(x α ) + n 1 j=1 λ jf(x j ) coincides with the n-point rational Gauss-Radau formula based on the zeros of the qorf where τ n depends of x α Q n,τn (x) = H(z) Q2n,1 (z)

14 Relating rational Szegő-Lobatto and Gauss-Radau rules Proof From Theorem Sym it follows that Q2n,1 has 2n zeros (all different from 1 and 1), appearing in complex conjugate pairs, and λ α = λ β, λ n 1+k = λ k. If f L n 1 L c n 1 then f(z) = (f J)(z) ˆ L2n 2 ˆ L(2n 2) n 1 J n (f) = λ α f(x α ) + λ k f(x k ) = 1 2 { k=1 λα (z α ) f(z α ) + λ β f(zα ) + = 1 2 J 2n( f) = 1 2 J µ( f) = J µ (f). 2n 2 k=1 λk f(zk ) }

15 Relating rational Gauss-Radau and Szegő-Lobatto rules Theorem GR-SL Let µ and µ related by the Joukowski transformation. Let J n (f) = λ α f(x α ) + n j=1 λ kf(x k ) be an n-point rational Gauss-Radau quadrature formula with fixed node x α for J µ (f), based on the zeros of the qorf Q n,τn. Set x k = cos θ k, x α = cos θ α, and define z k = z n 1+k = e iθ k, λ k = λ n 1+k = λ k, k = 1,..., n 1. z α = e iθα, λ α = λ α. Then J 2n ( f) = λ α f(zα ) + λ 2n 2 α f(zα ) + λk f(zk ) k=1 coincides with the 2n-point rational Szegő-Lobatto quadrature formula with fixed nodes {z α, z α } for J µ ( f) based on the zeros of the porf Q2n, ξ2n with ξ 2n = 1, and with δ 2n 1 depending on τ n.

16 Relating rational Gauss-Radau and Szegő-Lobatto rules Outline of the proof It is known that the qorf is orthogonal to L n 1 w.r.t. a measure µ (not necessarily supported in I). We can express Q n,τn (x) in terms of ˆφ 2n 2 (z) and ˆφ 2n 2 (z) in two ways: by assuming that Q n,τn (x) = H(z) Q2n,1 (z), which is equivalent with assuming that the measure µ is supported on I; by making use of the definition of Q n,τn (x) together with Theorem Rel. By comparison we then obtain a relation between δ 2n 1 and τ n.

17 Relating rational Gauss-Radau and Szegő-Lobatto rules Next, we need the following lemma. Lemma {β 1,..., β n 1 } real or complex conjugate pairs, and µ positive symmetric Borel measure on T. φ c n 1(z) = b n z β n 1 z β n 1 φ n 1 (z) + a n 1 β n 1 z z β n 1 φ n 1 (1/z) a n and b n can be expressed in terms of δ n. Since φ c n 1 (z) is independent of δ n, so are a n and b n. Hence, the expressions for a n and b n remains the same when replacing δ n with δ n. As a result, we find that δ n C D, a n 0 δ n L D, a n = 0 with C a circle and L a line that depends of a n and b n.

18 Relating rational Gauss-Radau and Szegő-Lobatto rules Finally, we have the following lemma. Lemma Let τ n be such that Q n,τn (x) has all its zeros in ( 1, 1). Then the corresponding δ 2n 1 satisfies the condition given in the previous lemma.

19 Bibliography A. Bultheel, L. Daruis, P. González-Vera, A connection between quadrature formulas on the unit circle and the interval [ 1, 1] Journal of Computational and Applied Mathematics, 132 (1) (2001) A. Bultheel, P. González-Vera, E. Hendriksen, O. Njåstad, Orthogonal Rational Functions, volume 5 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, A. Bultheel, P. González-Vera, E. Hendriksen, O. Njåstad, Computation of rational Szegő-Lobatto quadrature formulas, Applied Numerical Mathematics, (2010). Submitted. K. Deckers, A. Bultheel, R. Cruz-Barroso, F. Perdomo-Pío, Positive rational interpolatory quadrature formulas on the unit circle and the interval, Applied Numerical Mathematics, Publish online 30 March K. Deckers, A. Bultheel, J. Van Deun, A generalized eigenvalue problem for quasi-orthogonal rational functions, Numerische Mathematik, (2010). Submitted. K. Deckers, J. van Deun, A. Bultheel, An extended relation between orthogonal rational functions on the unit circle and the interval [ 1, 1], Journal of Mathematical Analysis and Applications, 334 (2) (2007) J. van Deun, A. Bultheel, Orthogonal rational functions and quadrature on an interval, Journal of Computational and Applied Mathematics, 153 (1-2) (2003)