Computing Gauss-Jacobi quadrature rules

Transcription

1 .... Computing Gauss-Jacobi quadrature rules Alex Townsend Joint work with Nick Hale NA Internal Seminar Trinity Term 2012

2 . Gauss Jacobi Quadrature An n-point Gauss Jacobi quadrature rule: 1 1 w(x)f (x)dx n w k f (x k ) k=1 with w(x) = (1 x) α (1 + x) β, α, β > 1. x k = simple roots of P (α,β) n. w k = C n,α,β [ ] 2, k = 1,..., n. (1 xk 2) P n (α,β) (x k ) Many formulae for the weights: Swarztrauber (2002), Yakimiv (1996).

3 . How many can you compute? How many Gauss Legendre nodes can be computed? Coalescing nodes GLR Newton n By Hand Iterative methods Golub Welsch How Many? Accurate Inaccurate Year

4 . Golub Welsch method A set of Jacobi polynomials satisfy a 3-term recurrence A n P n (x) = (x B n )P n 1 (x) C n P n 2 (x). The zeros of P n (x) are the eigenvalues of the tridiagonal matrix B n C n A n 1 B n 1 C n 1 A n 2 B n 2 C n A 2 B 2 C 2 A 1 B 1. It is a comrade matrix and hence, a strong linearisation (Mackay et al, 2007). It can be made symmetric tridiagonal. Requires O(n 2 ) operations for nodes and weights. Best not to compute weights from eigenvectors.

5 . Glaser Liu Rokhlin method A set of Jacobi polynomials satisfy a second-order differential equation (1 x 2 )P n (x) + a n (x)p n(x) + b n (x)p n (x) = 0. Predictor-corrector method: Once at a root, step to the next by predicting with Runge Kutta and then correcting with local Taylor approximations and Newton. Runge Kutta Local Taylor Newton iterates Requires O(n) operations for nodes and weights.

6 . Our method Newton s method in θ-space, x = cos(θ). Newton s method in θ space We will need:..1 Fast and accurate Jacobi polynomial evaluation asymptotic approximations...2 Fast and accurate evaluation of derivative recurrence relations...3 Sufficiently good initial guesses asymptotic approximations.

7 . What does a Legendre polynomial look like? Boundary Region: Bessel like Interior Region: Trig like cos(5π/6) cos(π/6) We use two asymptotic expansions: interior and boundary.

8 . Interior Asymptotic Formulae M 1 P n (cos θ) 2( 1) n m=0 ( 1 2 m )( m 1) 2 cos(αn,m ) n (2 sin θ) m+ 1 2 where α n,m = ( n + m + 1 2) θ ( m + 1 2) π 2 and C R n,m. (2 sin θ) M+ 1 2 n M R n,m Log error 10 0 Absolute error of interior asymptotic for n= Max Absolute error Convergence of Interior Asy formula 10 0 n=10 n=20 n=50 n=100 n=200 n= x Terms

9 . Boundary Asymptotic Formulae ( θ P n (cos θ) J 0 (ρθ) sin(θ) M m=0 A m (θ) ρ 2m ) M 1 + θj B m (θ) 1(ρθ) ρ 2m+1, m=0 where ρ = n Only the first few terms are known explictly. ( ) θo n 2M 3 c 2 n error = θ π ( ) 2 θ 3 O n 2M θ c n Absolute error of boundary asymptotic for n= Log error x

10 . Derivative Evaluation A recurrence relation for the derivative of P n is (1 x 2 )P n(x) = nxp n (x) + np n 1 (x), or in θ-space sin(θ) d dθ P n(cos(θ)) = n cos(θ)p n (cos(θ)) + np n 1 (cos(θ)). Many quantities can be reused for derivative evaluation.

11 . Initial Guesses Sufficiently good = quadratically clustering near endpoints. (Petras, 1998) Initial guesses come from the asymptotic formulae. Analogously, there are interior and boundary initial guesses. (Lether and Wenston, 1995) Max error in the initial guesses Caused by the 5th node changing from interior to boundary Max Error n = n

12 . Evaluating J 0 (ρθ) near Legendre nodes Inbuilt bessel evaluation only gets 15-digits of absolute accuracy, we need one bit more. Instead, we evaluate the asymptotic expansion: J 0 (ρθ) ( ) 1 ( 2 2 cos ω πρθ ( 1) k a 2k (ρθ) 2k sin ω ( 1) k a 2k+1 k=0 k=0 (ρθ) 2k+1 (1) where ω = ρθ 1 4π and do the argument reduction by ourselves. Relative error Absolute error Inbuilt Formula (1) )

13 . Gauss Legendre nodes and weights Gauss Legendre nodes and weights (α = 0, β = 0) Error in nodes for degree 1000 GLR Asy GW Absolute error in weights for degree GLR Asy GW

14 . Gauss Jacobi nodes and weights Gauss Jacobi nodes and weights (α = 0.1, β = 0.3). Error in nodes for degree 1000 Absolute Error in weights for degree GLR Asy GW GLR Asy GW

15 . Error as a quadrature rule Gauss quadrature integrates polynomials exactly. Approximate the error 1 n error = max w(x)x j dx w j=0,...,10 k x j k. 1 k= Error as a quadrature rule GLR Asy GW 10 3 CPU timings in array based language 10 2 GLR Asy GW Est. quad error Seconds n n

16 . Conclusions..1 Golub Welsch method can be accurate, but has O(n 2 ) complexity...2 GLR method is fast, but a little inaccurate for n > The method described here is fast and accurate for n > 200, but currently only implemented for Gauss Jacobi nodes and weights.