Computing rational Gauss-Chebyshev quadrature formulas with complex - PowerPoint PPT Presentation

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 n � f ( x ) w ( x ) dx ≈ λ nk f ( x nk ) − 1 k =1 ◮ Chebyshev weight functions: � ± 1 � w ( x ) = (1 − x ) a (1 + x ) b , a, b ∈ 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] X I complement of I with respect to a set X I A n sequence of poles { α 1 , . . . , α n } ⊂ C L n space of rational functions with poles in A n 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 x nk and weights λ nk , k = 1 , . . . , n so that the quadrature formula � 1 n � f ( x ) w ( x ) dx ≈ λ nk f ( x nk ) − 1 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 . Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Computing the nodes Numerical example 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 25 20 15 10 5 0 0 0.5 1 1.5 2 2.5 3 3.5 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 λ nk = G n ( x nk ) , k = 1 , . . . , n ◮ the weights can be computed straightforwardly Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Asymptotic zero distribution Computing the nodes Asymptotic inflection point distribution Numerical example 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 Asymptotic zero distribution Computing the nodes Asymptotic inflection point distribution Numerical example 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 I , then the asymptotic distribution of the nodes is subset of) C given by an absolutely continuous measure µ and the density of � x the nodes on [ − 1 , x ] is given by t ( x ) = − 1 µ ′ ( u ) du . Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Asymptotic zero distribution Computing the nodes Asymptotic inflection point distribution Numerical example Asymptotic zero distribution distribution of the poles is known I ◮ lim n →∞ α n = α ∈ R ◮ 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 ) Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Asymptotic zero distribution Computing the nodes Asymptotic inflection point distribution Numerical example Asymptotic zero distribution 25 25 20 20 15 15 10 10 5 5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Asymptotic zero distribution Computing the nodes Asymptotic inflection point distribution Numerical example Asymptotic zero 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)) 30 2.5 2 25 1.5 1 20 0.5 15 0 −0.5 10 −1 −1.5 5 −2 −2.5 0 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Asymptotic zero distribution Computing the nodes Asymptotic inflection point distribution Numerical example Asymptotic inflection point distribution define ◮ θ (0) b j = f AIPD ( α j ) ≈ θ b j “ ” θ (0) F n bj ◮ l 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 ) Karl Deckers, Joris Van Deun, Adhemar Bultheel Computing rational Gauss-Chebyshev quadrature formulas

Introduction Preliminaries Asymptotic zero distribution Computing the nodes Asymptotic inflection point distribution Numerical example Asymptotic inflection point distribution 30 25 20 15 10 5 0 0 0.5 1 1.5 2 2.5 3 3.5 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)) | π − � n method ∆ x ∆ λ k =1 λ k | found i total 1 . 6 × 10 − 14 bisection 0 0 411 8 2 . 3 × 10 − 15 2 . 0 × 10 − 14 AZD 0 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 30 25 20 15 10 5 0 0 0.5 1 1.5 2 2.5 3 3.5 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