formulas with complex poles

Transcription

1 Computing rational Gauss-Chebyshev quadrature formulas with complex poles Karl Deckers Joris Van Deun Adhemar Bultheel Department of Computer Science K.U.Leuven Augustus 7, 2006

2 Rational Gauss-Chebyshev quadrature Algorithm to compute the nodes and weights for rational Gauss-Chebyshev quadrature formulas. Gauss quadrature: 1 1 f(x)w(x)dx n λ nk f(x nk ) k=1 Chebyshev weight functions: w(x) = (1 x) a (1 + x) b, a, b { ± 1 } 2

3 Notations X I C Riemann sphere C { } I interval [ 1, 1] complement of I with respect to a set X A n sequence of poles {α 1,..., α n } C I L n space of rational functions with poles in A n

4 Back to the quadrature formula Theorem There exist a set of nodes x nk and weights λ nk, k = 1,..., n so that the quadrature formula 1 1 f(x)w(x)dx n λ nk f(x nk ) k=1 is exact for f L n 1 L n 1. In the special case in which α n is real, this quadrature formula is exact for f L n L n 1.

5 Nodes and weights nodes x nk = cos θ nk I satisfy F n (θ nk ) = πk, k = 1,..., n F n (θ) is strictly increasing with increasing θ [0, π] the nodes have to be computed numerically, e.g. using Newton s method

6 Nodes and weights weights the weights are given by λ nk = G n (x nk ), k = 1,..., n the weights can be computed straightforwardly

7 Introduction Asymptotic inflection point distribution Two methods for determining a set of initial values for Newton s method: Asymptotic Zero Distribution (AZD) Asymptotic Inflection Point Distribution (AIPD)

8 Asymptotic inflection point distribution Theorem Assume the sequence of poles is bounded away from I and the asymptotic distribution of the poles is given by a measure ν on (a subset of) C I, then the asymptotic distribution of the nodes is given by an absolutely continuous measure µ and the density of the nodes on [ 1, x] is given by t(x) = x 1 µ (u)du.

9 Asymptotic inflection point distribution distribution of the poles is known lim n α n = α R I θ (0) n,k = f AZD(t n,k ) f AZD (t) is the inverse of t(θ) {t n,k } n k=1 is a set of n equally distributed points in [0, 1] distribution of the poles is unknown t(θ) can be approximated by a finite sum t n (θ) we can use the cubic interpolating spline s AZD (t) to approximate the inverse of t n (θ) θ (0) n,k = s AZD(t n,k )

10 Asymptotic inflection point distribution

11 Asymptotic inflection point distribution Problem : does not work well for poles close to the boundary introducing large local maxima of df n(θ) dθ Example: a = [-.5+i*1e-3*ones(1,2),.75+i*1e-2*ones(1,4), 1.01,-2] w = 2 %w(x) = \sqrt((1-x)/(1+x))

12 Asymptotic inflection point distribution Asymptotic inflection point distribution define θ (0) b j l j = F n = f AIP D (α j ) θ bj θ (0) b j π approximate from the left θ (0) n,k = θ(0) b j, with j = arg max j (l j k) approximate from the right θ (0) n,k = θ(0) b j, with j = arg min j (l j k)

13 Asymptotic inflection point distribution Asymptotic inflection point distribution

14 Syntax for gqcorf [x,l,err,fail] = gqcorf(a,w) where x vector with the resulting nodes L vector with the resulting weights err (optional) to check whether the computations succeeded fail (optional) vector with indices of nodes/weights for which the computations failed (if any) a vector with poles w (optional) choice of weight function

15 Introduction Example: a = [-.5+i*1e-3*ones(1,2),.75+i*1e-2*ones(1,4), 1.01,-2] w = 2 %w(x) = \sqrt((1-x)/(1+x)) method x λ π n k=1 λ k i total found bisection AZD AIPD 17 2

16 Introduction

17 Introduction The complexity of the algorithm is of order ϑ(m n). If m << n the complexity is of order ϑ(n). Example: m = 5 n t n t n t

18 Introduction Example: n = 8192 m t m t m t