The circle theorem and related theorems for Gauss-type quadrature rules

Transcript

1 OP.circle p. / The circle theorem and related theorems for Gauss-type quadrature rules Walter Gautschi wxg@cs.purdue.edu Purdue University

2 OP.circle p. 2/ Web Site http : // archives/22/wxg/codes

3 OP.circle p. 3/ The circle theorem for Jacobi weight functions Jacobi weight function w(t) = w (α,β) (t) := ( t) α (+t) β, α >, β > Gauss-Jacobi quadrature formula f(t)w(t)dt = n ν= λg ν f(τ G ν ) + R G n (f) R G n (p) = for all p P 2n

4 OP.circle p. 3/ The circle theorem for Jacobi weight functions Jacobi weight function w(t) = w (α,β) (t) := ( t) α (+t) β, α >, β > Gauss-Jacobi quadrature formula f(t)w(t)dt = n ν= λg ν f(τ G ν ) + R G n (f) R G n (p) = for all p P 2n Davis and Rabinowitz (96) nλ G ν πw (α,β) (τ G ν ) (τ G ν ) 2, n

5 plots α, β =.75(.25).(.5)3., n = 2 : 5 : 4 n = 6 : 5 : β α.8.8 OP.circle p. 4/

6 plots α, β =.75(.25).(.5)3., n = 2 : 5 : 4 n = 6 : 5 : β α.8.8 open questions. true for more general weight functions? 2. what about Gauss-Radau? 3. limiting curves other than the circle? OP.circle p. 4/

7 OP.circle p. 5/ The circle theorem for weight functions in the Szegö class w S iff ln w(t) t 2 dt > Theorem If w S and /( t 2 w(t)) L [ ], where is any compact subinterval of (, ), then nλ ν /(πw(τ ν )) τ 2 ν on, n.

8 OP.circle p. 5/ The circle theorem for weight functions in the Szegö class w S iff ln w(t) t 2 dt > Theorem If w S and /( t 2 w(t)) L [ ], where is any compact subinterval of (, ), then nλ ν /(πw(τ ν )) τ 2 ν on, n. Remarks. equality holds if w(t) = ( t 2 ) /2 2. pointwise convergence almost everywhere holds if w S locally, supp w = [, ], ln w(t)dt >

9 OP.circle p. 6/ Proof Christoffel function Nevai (979) λ n (x; w) := min p P n p(x)= R p 2 (t)w(t)dt nλ n (x; w) πw(x) x 2, n uniformly for x λ n (τ G ν ; w) = λ G ν, ν =, 2,..., n

10 OP.circle p. 7/ Example Pollaczek weight function (not in S) w(t; a, b) = where 2 exp(ω arccos(t)) + exp(ωπ), t, a > b ω = ω(t) = (at + b)( t 2 ) /2

11 OP.circle p. 7/ Example Pollaczek weight function (not in S) w(t; a, b) = where 2 exp(ω arccos(t)) + exp(ωπ), t, a > b ω = ω(t) = (at + b)( t 2 ) / a = b = a = 4, b = n = 38(5)4

12 OP.circle p. 8/ Gauss-Radau formula f(t)w(t)dt = λ R f( ) + n ν= λ R ν f(τ R ν ) + R R n (f) Theorem Under the conditions of the previous theorem, the Gauss-Radau formula admits a circle theorem.

13 OP.circle p. 8/ Gauss-Radau formula f(t)w(t)dt = λ R f( ) + n ν= λ R ν f(τ R ν ) + R R n (f) Theorem Under the conditions of the previous theorem, the Gauss-Radau formula admits a circle theorem. Proof Let w R (t) = (t + )w(t). Then Define τ R ν are the zeros of π n ( ; w R ) l ν(t) = µ ν t τ R µ τ R ν τ R µ, ν =, 2,..., n

14 OP.circle p. 9/ proof (cont ) Gauss-Radau is interpolatory, λ R ν = = (t+)π n (t;w R ) (τ R ν +)(t τ R ν )π n(τ R ν ;w R ) w(t)dt (t+)l ν(t) τ R ν + w(t)dt.

15 OP.circle p. 9/ proof (cont ) Gauss-Radau is interpolatory, λ R ν = = (t+)π n (t;w R ) (τ R ν +)(t τ R ν )π n(τ R ν ;w R ) w(t)dt (t+)l ν(t) τ R ν + w(t)dt. Let λ ν, ν =, 2,..., n, be the Gauss weights for w R, λ ν = l ν(t)w R (t)dt = l ν(t)(t + )w(t)dt = (τ R ν + )λ R ν

16 OP.circle p. 9/ proof (cont ) Gauss-Radau is interpolatory, λ R ν = = (t+)π n (t;w R ) (τ R ν +)(t τ R ν )π n(τ R ν ;w R ) w(t)dt (t+)l ν(t) τ R ν + w(t)dt. Let λ ν, ν =, 2,..., n, be the Gauss weights for w R, Then λ ν = l ν(t)w R (t)dt = l ν(t)(t + )w(t)dt = (τ R ν + )λ R ν nλ R ν πw(τ R ν ) = nλ ν π(τ R ν +)w(τ R ν ) = nλ ν πw R (τν R ) (τν R ) 2

17 OP.circle p. / Example w(t) = t α ln(/t) on [, ], α > Gauss-Radau f(t)tα ln(/t)dt = λ f() + n ν= λ νf(τ ν ) + R n (f circle theorem (transformed to [, ]) nλ ν πτ α ν ln(/τ ν ) ( 2 )2 (τ ν 2 )2, n.

18 Example w(t) = t α ln(/t) on [, ], α > Gauss-Radau f(t)tα ln(/t)dt = λ f() + n ν= λ νf(τ ν ) + R n (f circle theorem (transformed to [, ]) nλ ν πτ α ν ln(/τ ν ) ( 2 )2 (τ ν 2 )2, n. plot (α =.75(.25).(.5)3) n = 2(5)4 n = 6(5)8 OP.circle p. /

19 OP.circle p. / Gauss-Lobatto formula f(t)w(t)dt = λl f( ) + n ν= λl ν f(τ L ν ) +λ L n+f() + R L n(f) Theorem Under the conditions of the previous theorem, the Gauss-Lobatto formula admits a circle theorem.

20 OP.circle p. / Gauss-Lobatto formula f(t)w(t)dt = λl f( ) + n ν= λl ν f(τ L ν ) +λ L n+f() + R L n(f) Theorem Under the conditions of the previous theorem, the Gauss-Lobatto formula admits a circle theorem. Proof Similar to Gauss-Radau: nλ L ν πw(τ L ν ) = nλ ν π( (τ L ν ) 2 )w(τ L ν ) = nλ ν πw L (τ L ν ) (τ L ν ) 2

21 OP.circle p. 2/ Gauss-Kronrod formula (GK) f(t)w(t)dt = n ν= λk ν f(τν G ) + n+ µ= λ K µ f(τ K µ ) + R K n (f)

22 OP.circle p. 2/ Gauss-Kronrod formula (GK) f(t)w(t)dt = n ν= λk ν f(τν G ) + n+ µ= λ K µ f(τ K µ ) + R K n (f) Theorem Assume (GK) exists with τµ K distinct nodes in (, ) and τµ K τν G, all µ, ν, and the Gauss formula for w admits a circle theorem. Then, under some additional (technical) conditions, 2nλ K ν πw(τν G ) (τν G ) 2, 2nλ K µ πw(τµ K ) (τµ K ) 2

23 OP.circle p. 2/ Gauss-Kronrod formula (GK) f(t)w(t)dt = n ν= λk ν f(τν G ) + n+ µ= λ K µ f(τ K µ ) + R K n (f) Theorem Assume (GK) exists with τµ K distinct nodes in (, ) and τµ K τν G, all µ, ν, and the Gauss formula for w admits a circle theorem. Then, under some additional (technical) conditions, 2nλ K ν πw(τν G ) (τν G ) 2, 2nλ K µ πw(τµ K ) (τµ K ) 2 Proof Similar to previous proofs.

24 Example Jacobi weight w (α,β), α, β [, 52 ) Peherstorfer and Petras (23) n = 2(5)4 6(5)8 α, β = (.4)2, β α OP.circle p. 3/

25 OP.circle p. 4/ Connection with potential theory density of the equilibrium measure ω [,] of the interval [, ] ω [,] (t) = π t, < t < 2 Christoffel function and equilibrium measure nλ n (t; w) w(t) ω [,] (t), n

26 OP.circle p. 4/ Connection with potential theory density of the equilibrium measure ω [,] of the interval [, ] ω [,] (t) = π t, < t < 2 Christoffel function and equilibrium measure nλ n (t; w) w(t) ω [,] (t), n In general, for E R a regular set and E a compact set on which w satisfies the Szegö condition (Totik, 2), nλ G ν w(τ G ν ) ω E (τ G ν ) on

27 OP.circle p. 5/ Example w(t) = weight function supported on two intervals t γ (t 2 ξ 2 ) p ( t 2 ) q, t [, ξ] [ξ, ], elsewhere,

28 OP.circle p. 5/ Example w(t) = weight function supported on two intervals t γ (t 2 ξ 2 ) p ( t 2 ) q, t [, ξ] [ξ, ], elsewhere, plot (for ξ = 2 and p = q = ± 2, γ = ±) n = 6(5)8

29 OP.circle p. 5/ Example w(t) = weight function supported on two intervals t γ (t 2 ξ 2 ) p ( t 2 ) q, t [, ξ] [ξ, ], elsewhere, plot (for ξ = 2 and p = q = ± 2, γ = ±) n = 6(5)8 density of equilibrium measure ω [, ξ] [ξ,] (t) = π t (t 2 ξ 2 ) /2 ( t 2 ) /2