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Introduction Preliminaries Computing the nodes Numerical example

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

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

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

• k=1

λnkf(xnk)

◮ Chebyshev weight functions:

w(x) = (1 − x)a(1 + x)b, a, b ∈

• ±1

2

• Karl Deckers, Joris Van Deun, Adhemar Bultheel

Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

Notations

C Riemann sphere C ∪ {∞} I interval [−1, 1] XI complement of I with respect to a set X An sequence of poles {α1, . . . , αn} ⊂ C

I

Ln space of rational functions with poles in An

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

Back to the quadrature formula

Theorem

There exist a set of nodes xnk and weights λnk, k = 1, . . . , n so that the quadrature formula 1

−1

f(x)w(x)dx ≈

n

• k=1

λnkf(xnk) is exact for f ∈ Ln−1 · Ln−1. In the special case in which αn is real, this quadrature formula is exact for f ∈ Ln · Ln−1.

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

Nodes and weights

nodes

◮ xnk = cos θnk ∈ I satisfy Fn (θnk) = πk, k = 1, . . . , n ◮ Fn(θ) is strictly increasing with increasing θ ∈ [0, π] ◮ the nodes have to be computed numerically, e.g. using

Newton’s method

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Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

Nodes and weights

weights

◮ the weights are given by λnk = Gn(xnk), k = 1, . . . , n ◮ the weights can be computed straightforwardly

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example Asymptotic zero distribution Asymptotic inflection point distribution

Computing the nodes

Two methods for determining a set of initial values for Newton’s method:

◮ Asymptotic Zero Distribution (AZD) ◮ Asymptotic Inflection Point Distribution (AIPD)

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example Asymptotic zero distribution Asymptotic inflection point distribution

Asymptotic zero 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.

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example Asymptotic zero distribution Asymptotic inflection point distribution

Asymptotic zero distribution

distribution of the poles is known

◮ limn→∞ αn = α ∈ R I ◮ θ(0) n,k = fAZD(tn,k) ◮ fAZD(t) is the inverse of t(θ) ◮ {tn,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 tn(θ) ◮ we can use the cubic interpolating spline sAZD(t) to

approximate the inverse of tn(θ)

◮ θ(0) n,k = sAZD(tn,k)

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example Asymptotic zero distribution Asymptotic inflection point distribution

Asymptotic zero distribution

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Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example Asymptotic zero distribution Asymptotic inflection point distribution

Asymptotic zero distribution

Problem : does not work well for poles close to the boundary introducing large local maxima of dFn(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))

−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3 3.5 5 10 15 20 25 30

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example Asymptotic zero distribution Asymptotic inflection point distribution

Asymptotic inflection point distribution

define

◮ θ(0) bj = fAIPD(αj) ≈ θbj ◮ lj = Fn “ θ(0)

bj

” π

approximate from the left

θ(0)

n,k = θ(0) bj , with j = arg maxj(lj ≤ k)

approximate from the right

θ(0)

n,k = θ(0) bj , with j = arg minj(lj ≥ k)

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example Asymptotic zero distribution Asymptotic inflection point distribution

Asymptotic inflection point distribution

0.5 1 1.5 2 2.5 3 3.5 5 10 15 20 25 30

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

Numerical example

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

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

Numerical example

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 λk|

itotal found bisection 1.6 × 10−14 411 8 AZD 2.3 × 10−15 2.0 × 10−14 36 6 AIPD 17 2

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

Numerical example

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Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

Numerical example

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 8 0.02 256 0.14 8192 3.17 16 0.01 512 0.28 16384 6.35 32 0.02 1024 0.55 32768 12.72 64 0.05 2048 1.08 65536 25.65 128 0.07 4096 2.01 131072 51.26

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example

Numerical example

Example: n = 8192 m t m t m t 5 3.78 65 5.80 1025 43.41 9 4.25 129 8.17 2049 82.47 17 4.13 257 11.57 4097 169.41 33 4.96 513 24.80 8192 344.23

Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas