An Arnoldi Gram-Schmidt process and Hessenberg matrices for Orthonormal Polynomials

Transcription

1 [ 1 ] University of Cyprus An Arnoldi Gram-Schmidt process and Hessenberg matrices for Orthonormal Polynomials Nikos Stylianopoulos, University of Cyprus New Perspectives in Univariate and Multivariate OPs Banff International Research Station October 2010 Construction Asymptotics Matrices

2 [ 2 ] University of Cyprus Bergman polynomials {p n } on an archipelago G G1 GN Γ1 G2 ΓN Γ2 Γ j, j = 1,..., N, a system of disjoint and mutually exterior Jordan curves in C, G j := int(γ j ), Γ := N j=1γ j, G := N j=1g j. f, g := f (z)g(z)da(z), f L2 (G) := f, f 1/2 G The Bergman polynomials {p n } n=0 of G are the orthonormal polynomials w.r.t. the area measure on G: p m, p n = p m (z)p n (z)da(z) = δ m,n, G with p n (z) = λ n z n +, λ n > 0, n = 0, 1, 2,.... Construction Asymptotics Matrices

3 [ 3 ] University of Cyprus Construction of p n s Algorithm: Conventional Gram-Schmidt (GS) Apply the Gram-Schmidt process to the monomials 1, z, z 2, z 3,... Main ingredient: the moments µ m,k := z m, z k = z m z k da(z), m, k = 0, 1,.... G The above algorithm has been been suggested by pioneers of Numerical Conformal Mapping (like P. Davis and D. Gaier) as the standard procedure for constructing Bergman polynomials. It was subsequently used by researchers in this area (e.g. Burbea, Kokkinos, Papamichael, Sideridis and Warby). It has been even employed in the numerical conformal FORTRAN package BKMPACK of Warby. Construction Asymptotics Matrices Conventional GS Arnoldi GS

4 [ 4 ] University of Cyprus Instability Indicator The GS method is notorious for its instability. For measuring it, when orthonormalizing a system S n := {u 0, u 1,..., u n } of functions, the following instability indicator has been proposed by J.M. Taylor, (Proc. R.S. Edin., 1978): u n 2 L I n := 2 (G) min u span(sn 1 ) u n u 2, n N. L 2 (G) Note that, when S n is an orthonormal system, then I n = 1. When S n is linearly dependent then I n =. Also, if G n := [ u m, u k ] n m,k=1, denotes the Gram matrix associated with S n then, κ 2 (G n ) I n, where κ 2 (G n ) is the spectral condition number of G n. Construction Asymptotics Matrices Conventional GS Arnoldi GS

5 [ 5 ] University of Cyprus Instability of the Conventional GS In the single-component case N = 1, consider the monomials u j := z j, j = 0, 1,..., n. Then, for the conventional GS we have the following result: Theorem (N. Papamichael and M. Warby, Numer. Math., 1986.) Assume that the curve Γ is piecewise-analytic without cusps and let L := z L (Γ)/cap(Γ) ( 1), where cap(γ) denotes the capacity of Γ. Then, c 1 (Γ) L 2n I n c 2 (Γ) L 2n. Note that L = 1, iff G D and that I n is sensitive to the relative position of G w.r.t. the origin. When G is the 8 2 rectangle centered at the origin, then L = 3/ In this case, I and the method breaks down in MATLAB, for n = 25. Construction Asymptotics Matrices Conventional GS Arnoldi GS

6 [ 6 ] University of Cyprus The Arnoldi algorithm in Numerical Liner Algebra Let A C m,m, b C m and consider the Krylov subspace K k := span{b, Ab, A 2 b..., A k 1 b}. The Arnoldi algorithm produces an orthonormal basis {v 1, v 2,..., v k } of K k as follows: W. Arnoldi (Quart. Appl. Math., 1951) At the n-th step, apply GS to orthonormalize the vector Av n 1 (instead of A n 1 b) against the (already computed) orthonormal vectors {v 1, v 2,..., v n 1 }. Construction Asymptotics Matrices Conventional GS Arnoldi GS

7 [ 7 ] University of Cyprus The Arnoldi algorithm for OP s Let µ be a (non-trivial) finite Borel measure with compact support Σ := supp(µ) on C and consider the series of orthonormal polynomials p n (z, µ) := λ n (µ)z n +, λ n (µ) > 0, n = 0, 1, 2,..., generated by the inner product f, g µ = f (z)g(z)dµ(z). Arnoldi GS for Orthonormal Polynomials At the n-th step, apply GS to orthonormalize the polynomial zp n 1 (instead of z n ) against the (already computed) orthonormal polynomials {p 0, p 1,..., p n 1 }. Used in B. Gragg & L. Reichel, Linear Algebra Appl. (1987), for the construction of Szegö polynomials. Construction Asymptotics Matrices Conventional GS Arnoldi GS

8 [ 8 ] University of Cyprus Stability of the Arnoldi GS In the case of the Arnoldi GS, the instability indicator is given by: zp n 1 2 L I n := 2 (G) min p Pn 1 zp n 1 p 2, n N. L 2 (G) Theorem It holds, 1 I n z L (Σ) λ 2 n 1 (µ) λ 2 n(µ), n N. Typically: When dµ dz (Szegö polynomials), or dµ da (Bergman polynomials), then c 1 (Γ) λ n 1(µ) λ n (µ) c 2 (Γ), n N. When dµ w(x)dx on [a, b] R, this ratio tends to a constant. Construction Asymptotics Matrices Conventional GS Arnoldi GS

9 [ 9 ] University of Cyprus Zeros of Bergman polys: Three Disks Zeros of the Bergman polynomials p n, n = 140, 150 and 160. Theory in: Gustafsson, Putinar, Saff & St, Adv. Math., Construction Asymptotics Matrices Conventional GS Arnoldi GS

10 [ 10 ] University of Cyprus Single-component case N = 1 Ω Φ Δ G ζ ϕζ D 0 Γ Ω := C \ G Φ(z) = γz + γ 0 + γ 1 z + γ 2 +. z2 cap(γ) = 1/γ The Bergman polynomials of G: p n (z) = λ n z n +, λ n > 0, n = 0, 1, 2,.... Construction Asymptotics Matrices N = 1 Archipelago

11 [ 11 ] University of Cyprus Strong asymptotics when Γ is analytic Ω Φ Δ G D 0 ρ 1 Lρ Γ T. Carleman, Ark. Mat. Astr. Fys. (1922) If ρ < 1 is the smallest index for which Φ is conformal in ext(l ρ ), then n + 1 γ 2(n+1) π λ 2 = 1 α n, where 0 α n c 1 (Γ) ρ 2n, n where p n (z) = n + 1 π Φn (z)φ (z){1 + A n (z)}, n N, A n (z) c 2 (Γ) n ρ n, z Ω. Construction Asymptotics Matrices N = 1 Archipelago

12 [ 12 ] University of Cyprus Strong asymptotics when Γ is smooth We say that Γ C(p, α), for some p N and 0 < α < 1, if Γ is given by z = g(s), where s is the arclength, with g (p) Lipα. Then both Φ and Ψ := Φ 1 are p times continuously differentiable in Ω \ { } and \ { } respectively, with Φ (p) and Ψ (p) Lipα. P.K. Suetin, Proc. Steklov Inst. Math. AMS (1974) Assume that Γ C(p + 1, α), with p + α > 1/2. Then n + 1 γ 2(n+1) 1 π λ 2 = 1 α n, where 0 α n c 1 (Γ) n n, 2(p+α) where p n (z) = n + 1 π Φn (z)φ (z){1 + A n (z)}, n N, A n (z) c 2 (Γ) log n, z Ω. np+α Construction Asymptotics Matrices N = 1 Archipelago

13 [ 13 ] University of Cyprus Strong asymptotics for Γ non-smooth Theorem (St, C. R. Acad. Sci. Paris, 2010) Assume that Γ is piecewise analytic without cusps. Then, n + 1 γ 2(n+1) π λ 2 = 1 α n, where 0 α n c(γ) 1 n n, n N. and for any z Ω, where p n (z) = A n (z) n + 1 π Φn (z)φ (z) {1 + A n (z)}, c 1 (Γ) dist(z, Γ) Φ (z) 1 n + c 2 (Γ) 1 n, n N. Construction Asymptotics Matrices N = 1 Archipelago

14 [ 14 ] University of Cyprus Ratio asymptotics for λ n Corollary (St, C. R. Acad. Sci. Paris, 2010) n + 1 n λ n 1 λ n = cap(γ) + ξ n, where ξ n c(γ) 1 n, n N. The above relation provides the means for computing approximations to the capacity of Γ, by using only the leading coefficients of the Bergman polynomials. In addition it yiels: Corollary c 1 (Γ) I n c 2 (Γ), n N. Hence, under the assumptions of the previous theorem, the Arnoldi GS for Bergman polynomials, in the single component case, is stable. Construction Asymptotics Matrices N = 1 Archipelago

15 [ 15 ] University of Cyprus Leading coefficients in an archipelago G1 GN Γ1 G2 ΓN Γ2 Theorem (Gustafsson, Putinar, Saff & St, Adv. Math., 2009) Assume that every Γ j is analytic, j = 1, 2,..., N. Then, c 1 (Γ) Corollary n + 1 π 1 cap(γ) n+1 λ n c 2 (Γ) n + 1 c 3 (Γ) I n c 4 (Γ), n N. π 1, n N. cap(γ) n+1 Hence, the Arnoldi GS, for Bergman polynomials on an archipelago, is stable. Construction Asymptotics Matrices N = 1 Archipelago

16 [ 16 ] University of Cyprus The inverse conformal map Ψ Ω Φ Δ G Γ ζ ϕζ D 0 Recall that Φ(z) = γz + γ 0 + γ 1 z + γ 2 z 2 +, and let Ψ := Φ 1 : {w : w > 1} Ω, denote the inverse conformal map. Then, where Ψ(w) = bw + b 0 + b 1 w + b 2 +, w > 1, w 2 b = cap(γ) = 1/γ. Construction Asymptotics Matrices Faber Hessenberg Recurrence

17 [ 17 ] University of Cyprus Faber polynomials of G The Faber polynomial F n (z) (n N) of G, is the polynomial part of the Laurent series expansion of Φ n (z) at : ( ) 1 F n (z) = Φ n (z) + O, z. z The Faber polynomial of the 2nd kind G n (z), is the polynomial part of the expansion of the Laurent series expansion of Φ n (z)φ (z) at : ( ) 1 G n (z) = Φ n (z)φ (z) + O, z. z Note: G n (z) = F n+1 (z) n + 1. Construction Asymptotics Matrices Faber Hessenberg Recurrence

18 [ 18 ] University of Cyprus The Faber matrix G The Faber polynomials of the 2nd kind satisfy the recurrence relation, zg n (z) = bg n+1 (z) + n b k G n k (z), n = 0, 1,..., k=0 and induce the upper Hessenberg Toeplitz matrix: b 0 b 1 b 2 b 3 b 4 b 5 b 6 b b 0 b 1 b 2 b 3 b 4 b 5 0 b b 0 b 1 b 2 b 3 b b b 0 b 1 b 2 b 3 G = b b 0 b 1 b b b 0 b b b Construction Asymptotics Matrices Faber Hessenberg Recurrence

19 [ 19 ] University of Cyprus The upper Hessenberg matrix M The Bergman polynomials satisfy the recurrence relation, n+1 zp n (z) = a k,n p k (z), n = 0, 1,..., k=0 where a k,n are Fourier coefficients: a k,n = zp n, p k, and induce the (infinite) upper Hessenberg matrix: a 00 a 01 a 02 a 03 a 04 a 05 a 06 a 10 a 11 a 12 a 13 a 14 a 15 a 16 0 a 21 a 22 a 23 a 24 a 25 a a 32 a 33 a 34 a 35 a 36 M = a 43 a 44 a 45 a a 54 a 55 a a 65 a Construction Asymptotics Matrices Faber Hessenberg Recurrence

20 [ 20 ] University of Cyprus Eigenvalues = Zeros Note The eigenvalues of the finite n n Faber matrix are the zeros of the Faber polynomial G n (z). Note The eigenvalues of the finite n n Hessenberg matrix are the zeros of the Bergman polynomial p n (z). Construction Asymptotics Matrices Faber Hessenberg Recurrence

21 [ 21 ] University of Cyprus Main subdiagonal Consider the main subdiagonal of the Hessenberg matrix: a n+1,n = zp n, p n+1 = λ n z n+1 +, p n+1 = λ n z n+1, p n+1 = Since cap(γ) = b, it follows from the ratio asymptotics for λ n, that: λ n λ n+1. Lemma n + 2 n + 1 a n+1,n = b + O ( ) 1, n N. n That is, the main subdiagonal of the Hessenberg matrix tends to the main subdiagonal of the Faber matrix. Construction Asymptotics Matrices Faber Hessenberg Recurrence

22 [ 22 ] University of Cyprus M G More generally, using the theory on strong asymptotics for non-smooth curves we have: Theorem (Saff & St) Assume that Γ is piecewise analytic w/o cusps, then for any fixed k N {0}, ( ) n n + k + 1 a n,n+k = b k + O n, n. That is, the k-th diagonal of the Hessenberg matrix tends to the k-th diagonal of the Faber matrix. Construction Asymptotics Matrices Faber Hessenberg Recurrence

23 [ 23 ] University of Cyprus Ratio asymptotics for p n (z) Theorem (St, C. R. Acad. Sci. Paris, 2010) Assume that Γ is piecewise analytic without cusps. Then, for any z Ω, and sufficiently large n N, n + 1 p n+1 (z) = Φ(z){1 + B n (z)}, n + 2 p n (z) where B n (z) c 1 (Γ) dist(z, Γ) Φ (z) 1 n + c 2 (Γ) 1 n. Construction Asymptotics Matrices Faber Hessenberg Recurrence

24 [ 24 ] University of Cyprus Only ellipses carry finite-term recurrences for {p n } Definition We say that the polynomials {p n } n=0 satisfy a (M + 1)-term recurrence relation, if for any n M 1, zp n (z) = a n+1,n p n+1 (z) + a n,n p n (z) a n M+1,n p n M+1 (z). Theorem (St, C. R. Acad. Sci. Paris, 2010) Assume that: Γ = G is piecewise analytic without cusps; the Bergman polynomials {p n } n=0 satisfy a (M + 1)-term recurrence relation, with some M 2. Then M = 2 and Γ is an ellipse. The above theorem refines results of Putinar & St (CAOT, 2007) and Khavinson & St (Springer, 2010). Construction Asymptotics Matrices Faber Hessenberg Recurrence

25 [ 25 ] University of Cyprus Banded Hessenberg matrices are tridiagonal Corollary If the Hessenberg matrix is banded with constant bandwidth 3, then is tridiagonal. This result should put an end to the long search in Numerical Linear Algebra, for practical polynomial iteration methods, based on short-term recurrence relations of orthogonal polynomials,. Construction Asymptotics Matrices Faber Hessenberg Recurrence