Simultaneous Gaussian quadrature for Angelesco systems

Transcription

1 for Angelesco systems 1 KU Leuven, Belgium SANUM March 22, Joint work with Doron Lubinsky

2 Introduced by C.F. Borges in 1994

3 Introduced by C.F. Borges in 1994 (goes back to Angelesco 1918).

4 Introduced by C.F. Borges in 1994 (goes back to Angelesco 1918). r measures µ 1,..., µ r are given, one function f : R R.

5 Introduced by C.F. Borges in 1994 (goes back to Angelesco 1918). r measures µ 1,..., µ r are given, one function f : R R. We want to approximate r integrals f (x) dµ j (x), 1 j r by means of quadrature sums of the form N k=1 λ (j) k,n f (x k,n), 1 j r, and the quadrature needs to be correct for polynomials of degree as high as possible.

6 Multiple orthogonal polynomials Type II multiple orthogonal polynomial P n for the multi-index n = (n 1,..., n r ) N r is the monic polynomial of degree n = n 1 + n n r for which x k P n (x) dµ j (x) = 0, 0 k n j 1, 1 j r.

7 Multiple orthogonal polynomials Type II multiple orthogonal polynomial P n for the multi-index n = (n 1,..., n r ) N r is the monic polynomial of degree n = n 1 + n n r for which x k P n (x) dµ j (x) = 0, 0 k n j 1, 1 j r. Theorem If we choose as nodes the zeros x k, n of P n and if we take λ (j) k, n = l k, n (x) dµ j (x), (l k, n fundamental polynomials of Lagrange interpolation). Then n λ (j) k, n p(x k, n) = p(x) dµ j (x), 1 j r j=1 whenever p is a polynomial of degree at most n + n j 1.

8 Multiple orthogonal polynomials Type II multiple orthogonal polynomial P n for the multi-index n = (n 1,..., n r ) N r is the monic polynomial of degree n = n 1 + n n r for which x k P n (x) dµ j (x) = 0, 0 k n j 1, 1 j r. Theorem If we choose as nodes the zeros x k,rn of P (n,n,...,n) and if we take λ (j) k,rn = l k,rn (x) dµ j (x), (l k,rn fundamental polynomials of Lagrange interpolation). Then rn λ (j) k,rn p(x k,rn) = p(x) dµ j (x), 1 j r j=1 whenever p is a polynomial of degree at most (r + 1)n 1.

9 Angelesco systems An Angelesco system is a system of r measures such that supp(µ j ) [a j, b j ] and the intervals (a 1, b 1 ),..., (a r, b r ) are disjoint.

10 Angelesco systems An Angelesco system is a system of r measures such that supp(µ j ) [a j, b j ] and the intervals (a 1, b 1 ),..., (a r, b r ) are disjoint. < a 1 < b 1 a 2 < b 2 a r < b r <

11 Angelesco systems An Angelesco system is a system of r measures such that supp(µ j ) [a j, b j ] and the intervals (a 1, b 1 ),..., (a r, b r ) are disjoint. < a 1 < b 1 a 2 < b 2 a r < b r < Property The type II multiple orthogonal polynomial P n for an Angelesco system always exists and has n j zeros on (a j, b j ) for 1 j r: P n (x) = r p n,j (x). j=1

12 Angelesco systems An Angelesco system is a system of r measures such that supp(µ j ) [a j, b j ] and the intervals (a 1, b 1 ),..., (a r, b r ) are disjoint. < a 1 < b 1 a 2 < b 2 a r < b r < Property The type II multiple orthogonal polynomial P n for an Angelesco system always exists and has n j zeros on (a j, b j ) for 1 j r: P n (x) = r p n,j (x). In fact p n,j is an ordinary orthogonal polynomial of degree n j on [a j, b j ] for the measure i j p n,i(x) dµ j (x). j=1

13 Known results Property (Nikishin-Sorokin) The quadrature weights have the following properties λ (j) k,rn > 0, if x k,rn [a j, b j ],

14 Known results Property (Nikishin-Sorokin) The quadrature weights have the following properties λ (j) k,rn > 0, if x k,rn [a j, b j ], and λ (j) k,rn has alternating sign when x k,rn [a i, b i ] with i j.

15 Known results Property (Nikishin-Sorokin) The quadrature weights have the following properties λ (j) k,rn > 0, if x k,rn [a j, b j ], and λ (j) k,rn has alternating sign when x k,rn [a i, b i ] with i j. Furthermore λ (j) k,rn is positive for the zero in [a i, b i ] closest to [a j, b j ].

16 Simultaneous quadrature on two intervals We consider only two intervals [a 1, b 1 ] and [a 2, b 2 ], with b 1 a 2.

17 Simultaneous quadrature on two intervals We consider only two intervals [a 1, b 1 ] and [a 2, b 2 ], with b 1 a 2. We set P (n,n) (x) = ( 1) n p n (x)q n (x), with p n (x) = n 2n (x x j,2n ), q n (x) = ( 1) n (x x j,2n ), j=1 j=n+1 i.e., p n has zeros on [a 1, b 1 ] and q n has zeros on [a 2, b 2 ].

18 Simultaneous quadrature on two intervals Quadrature rules: 2n λ (1) b1 j,2n P(x j,2n) = P(x) dµ 1 (x), a 1 j=1 2n j=1 λ (2) b2 j,2n P(x j,2n) = P(x) dµ 2 (x), a 2 for every polynomial P of degree 3n 1.

19 Simultaneous quadrature on two intervals Quadrature rules: 2n λ (1) b1 j,2n P(x j,2n) = P(x) dµ 1 (x), a 1 j=1 2n j=1 λ (2) b2 j,2n P(x j,2n) = P(x) dµ 2 (x), a 2 for every polynomial P of degree 3n 1. Choose P(x) = π(x)q n (x), then n λ (1) j,2n π(x j,2n)q n (x j,2n ) = j=1 b1 for every polynomial π of degree 2n 1 a 1 π(x) q n (x) dµ 1 (x),

20 Simultaneous quadrature on two intervals Quadrature rules: 2n λ (1) b1 j,2n P(x j,2n) = P(x) dµ 1 (x), a 1 j=1 2n j=1 λ (2) b2 j,2n P(x j,2n) = P(x) dµ 2 (x), a 2 for every polynomial P of degree 3n 1. Choose P(x) = π(x)q n (x), then n λ (1) j,2n π(x j,2n)q n (x j,2n ) = j=1 b1 for every polynomial π of degree 2n 1 a 1 π(x) q n (x) dµ 1 (x), λ (1) j,2n q n(x j,2n ) = λ j,n (q n dµ 1 ), 1 j n, Gauss quadrature

21 Simultaneous quadrature on two intervals Quadrature rules: 2n j=1 2n j=1 λ (1) b1 j,2n P(x j,2n) = P(x) dµ 1 (x), a 1 λ (2) b2 j,2n P(x j,2n) = P(x) dµ 2 (x), a 2 for every polynomial P of degree 3n 1.

22 Simultaneous quadrature on two intervals Quadrature rules: 2n j=1 2n j=1 λ (1) b1 j,2n P(x j,2n) = P(x) dµ 1 (x), a 1 λ (2) b2 j,2n P(x j,2n) = P(x) dµ 2 (x), a 2 for every polynomial P of degree 3n 1. Choose P(x) = π(x)pn(x), 2 then 2n λ (1) b1 j,2n π(x j,2n)pn(x 2 j,2n ) = π(x) pn(x) 2 dµ 1 (x), a 1 j=n+1 for every polynomial π of degree n 1.

23 Simultaneous quadrature on two intervals Quadrature rules: 2n j=1 2n j=1 λ (1) b1 j,2n P(x j,2n) = P(x) dµ 1 (x), a 1 λ (2) b2 j,2n P(x j,2n) = P(x) dµ 2 (x), a 2 for every polynomial P of degree 3n 1. Choose P(x) = π(x)pn(x), 2 then 2n λ (1) b1 j,2n π(x j,2n)pn(x 2 j,2n ) = π(x) pn(x) 2 dµ 1 (x), a 1 j=n+1 for every polynomial π of degree n 1. Interpolatory quadrature formula for [a 1, b 1 ] with quadrature nodes on [a 2, b 2 ]

24 Possé-Chebyshev-Markov-Stieltjes inequalities Theorem (Lubinsky-WVA) Suppose 1 l n and g : (, x l,2n ] [0, ) has 2n continuous derivatives, with Then k=1 g (k) (x) 0, 0 k 2n. l 1 λ (1) xl,2n k,2n g(x k,2n) g(x) dµ 1 (x) a 1 l k=1 λ (1) k,2n g(x k,2n).

25 Possé-Chebyshev-Markov-Stieltjes inequalities Theorem (Lubinsky-WVA) Suppose 1 l n and g : (, x l,2n ] [0, ) has 2n continuous derivatives, with Then k=1 g (k) (x) 0, 0 k 2n. l 1 λ (1) xl,2n k,2n g(x k,2n) g(x) dµ 1 (x) a 1 l k=1 λ (1) k,2n g(x k,2n). Follows from the Possé-Chebyshev-Markov-Stieltjes inequalities for the measure q n dµ 1 and the fact that g/q n is completely monotonic on [a 1, b 1 ].

26 Chebyshev-Markov-Stieltjes inequalities Property For 1 l n 1 λ (1) xl+1,2n l,2n dµ 1 (x), λ (1) xl+1,2n l,2n + λ(1) l+1,2n dµ 1 (x), x l 1,2n x l,2n and n k=1 λ (1) k,2n b1 a 1 dµ 1 (x).

27 Chebyshev-Markov-Stieltjes inequalities Property For n + 1 l 2n 1 λ (2) xl+1,2n l,2n dµ 2 (x), λ (2) xl+1,2n l,2n + λ(2) l+1,2n dµ 2 (x), x l 1,2n x l,2n and 2n k=n+1 λ (2) k,2n b2 a 2 dµ 2 (x).

28 Potential theory Suppose that µ 1 > 0 a.e. on [a 1, b 2 ] and µ 2 > 0 a.e. on [a 2, b 2 ].

29 Potential theory Suppose that µ 1 > 0 a.e. on [a 1, b 2 ] and µ 2 > 0 a.e. on [a 2, b 2 ]. asymptotic distribution of the zeros of p n : 1 lim n n n f (x j,2n ) = j=1 b a 1 f (x) dν 1 (x), f C([a 1, b 1 ])

30 Potential theory Suppose that µ 1 > 0 a.e. on [a 1, b 2 ] and µ 2 > 0 a.e. on [a 2, b 2 ]. asymptotic distribution of the zeros of p n : 1 lim n n n f (x j,2n ) = j=1 b asymptotic distribution of the zeros of q n : 1 lim n n 2n j=n+1 g(x j,2n ) = a 1 f (x) dν 1 (x), f C([a 1, b 1 ]) b2 a g(x) dν 2 (x), g C([a 2, b 2 ])

31 Potential theory The limiting measures (ν 1, ν 2 ) satisfy a vector equilibrium problem ( ) I (ν 1 ) + I (ν 2 ) + I (ν 1, ν 2 ) = min I (τ 1 ) + I (τ 2 ) + I (τ 1, τ 2 ) over all probability measures τ 1 on [a 1, b 1 ] and τ 2 on [a 2, b 2 ], where I (τ i, τ j ) = and I (τ i ) = I (τ i, τ i ). bi bj a i a j log 1 x y dτ j(x) dτ i (y),

32 Potential theory Variational conditions for the potentials of ν 1 and ν 2 1 U(x; ν 1 ) = log x y dν 1 1(y), U(x; ν 2 ) = log x y dν 2(y).

33 Potential theory Variational conditions for the potentials of ν 1 and ν 2 1 U(x; ν 1 ) = log x y dν 1 1(y), U(x; ν 2 ) = log x y dν 2(y). 2U(x; ν 1 ) + U(x; ν 2 ) { = l 1, x [a 1, b ], > l 1, x (b, b 1 ],

34 Potential theory Variational conditions for the potentials of ν 1 and ν 2 1 U(x; ν 1 ) = log x y dν 1 1(y), U(x; ν 2 ) = log x y dν 2(y). { = l 1, x [a 1, b ], 2U(x; ν 1 ) + U(x; ν 2 ) > l 1, x (b, b 1 ], { = l 2, x [a, b 2 ], U(x; ν 1 ) + 2U(x; ν 2 ) > l 2, x [a 2, a ).

35 Potential theory Variational conditions for the potentials of ν 1 and ν 2 1 U(x; ν 1 ) = log x y dν 1 1(y), U(x; ν 2 ) = log x y dν 2(y). { = l 1, x [a 1, b ], 2U(x; ν 1 ) + U(x; ν 2 ) > l 1, x (b, b 1 ], { = l 2, x [a, b 2 ], U(x; ν 1 ) + 2U(x; ν 2 ) > l 2, x [a 2, a ). b = b 1 and a = a 2 b < b 1 and a = a 2 b = b 1 and a 2 < a three possibilities

36 Potential theory 1 b1 γn(q 2 n dµ 1 ) = pn(x) 2 q n (x) dµ 1 (x), a 1 1 γ 2 n(p n dµ 2 ) = b2 a 2 q 2 n(x) p n (x) dµ 2 (x)

37 Potential theory 1 b1 γn(q 2 n dµ 1 ) = pn(x) 2 q n (x) dµ 1 (x), a 1 1 γ 2 n(p n dµ 2 ) = b2 Asymptotic behavior of these norms a 2 q 2 n(x) p n (x) dµ 2 (x) lim γ n(q n dµ 1 ) 1/n = e l1/2, n lim γ n(p n dµ 2 ) 1/n = e l2/2. n Gonchar-Rakhmanov 1981, Nikishin-Sorokin 1988/1991

38 Estimates Property For 1 j n one has λ (1) j,2n λ m(x j,2n ; µ 1 ), m = 3n 2.

39 Estimates Property For 1 j n one has λ (1) j,2n λ m(x j,2n ; µ 1 ), m = 3n 2. If J 1 is a closed subinterval of (a 1, b 1 ) and µ 1 is absolutely continuous in an open neighborhood of J 1, while µ 1 is bounded from below by a positive constant there, then λ (1) j,2n C 1 n, x j,2n J 1.

40 Estimates Property Suppose b 1 < a 2. Then uniformly on compact subsets of (a 1, b ) λ (1) j,2n ( 1 + o(1) ) λ n/2 (x j,2n ; µ 1 ).

41 Estimates Property Suppose b 1 < a 2. Then uniformly on compact subsets of (a 1, b ) λ (1) j,2n ( 1 + o(1) ) λ n/2 (x j,2n ; µ 1 ). If J 1 is a closed subinterval of (a 1, b ) and µ 1 is absolutely continuous in an open neighborhood of J 1 and µ 1 is bounded above by a constant, then λ (1) j,2n C 2 n, x j,2n J 1.

42 Convergence results: positive weights Theorem (Lubinsky-WVA) Let f be a continuous function on [a 1, b 1 ] and f (b ) = 0, with [a 1, b ] = supp(ν 1 ). Then lim n n j=1 b λ (1) j,2n f (x j,2n) = f (x) dµ 1 (x). a 1

43 Convergence results: positive weights Theorem (Lubinsky-WVA) Let f be a continuous function on [a 1, b 1 ] and f (b ) = 0, with [a 1, b ] = supp(ν 1 ). Then lim n n j=1 b λ (1) j,2n f (x j,2n) = f (x) dµ 1 (x). a 1 The first quadrature rule converges for functions f that can be approximated by weighted polynomials (weight q n (x)), i.e., there exists a sequence of polynomials (R n ) n such that lim f R nq n [a1,b n 1 ] = 0.

44 Convergence results: positive weights Theorem (Lubinsky-WVA) Let f be a continuous function on [a 1, b 1 ] and f (b ) = 0, with [a 1, b ] = supp(ν 1 ). Then lim n n j=1 b λ (1) j,2n f (x j,2n) = f (x) dµ 1 (x). a 1 The first quadrature rule converges for functions f that can be approximated by weighted polynomials (weight q n (x)), i.e., there exists a sequence of polynomials (R n ) n such that lim f R nq n [a1,b n 1 ] = 0. These are continuous functions that vanish outside (a 1, b ) [Totik, 1994].

45 Convergence results: positive weights Theorem (Lubinsky-WVA) Let g be a continuous function on [a 2, b 2 ] and g(a ) = 0, with [a, b 2 ] = supp(ν 2 ). Then lim 2n n j=n+1 λ (2) b2 j,2n g(x j,2n) = g(x) dµ 2 (x). a

46 Convergence results: positive weights Theorem (Lubinsky-WVA) Let g be a continuous function on [a 2, b 2 ] and g(a ) = 0, with [a, b 2 ] = supp(ν 2 ). Then lim 2n n j=n+1 λ (2) b2 j,2n g(x j,2n) = g(x) dµ 2 (x). a The second quadrature rule converges for functions g that can be approximated by weighted polynomials (weight p n (x)), i.e., there exists a sequence of polynomials (S n ) n such that lim g S np n [a2,b n 2 ] = 0.

47 Convergence results: positive weights Theorem (Lubinsky-WVA) Let g be a continuous function on [a 2, b 2 ] and g(a ) = 0, with [a, b 2 ] = supp(ν 2 ). Then lim 2n n j=n+1 λ (2) b2 j,2n g(x j,2n) = g(x) dµ 2 (x). a The second quadrature rule converges for functions g that can be approximated by weighted polynomials (weight p n (x)), i.e., there exists a sequence of polynomials (S n ) n such that lim g S np n [a2,b n 2 ] = 0. These are continuous functions that vanish outside (a, b 2 ) [Totik, 1994].

48 Convergence results: alternating weights Theorem (Lubinsky-WVA) Suppose that µ 1 > 0 on [a 1, b 1 ] and µ 2 > 0 on [a 2, b 2 ], and b 1 < a 2. Then ) lim n λ(1) j,2n 1/n = exp (2U(x; ν 1 ) + U(x; ν 2 ) l 1, whenever x j,2n x [a, b 2 ].

49 Convergence results: alternating weights Theorem (Lubinsky-WVA) Suppose that µ 1 > 0 on [a 1, b 1 ] and µ 2 > 0 on [a 2, b 2 ], and b 1 < a 2. Then ) lim n λ(1) j,2n 1/n = exp (2U(x; ν 1 ) + U(x; ν 2 ) l 1, whenever x j,2n x [a, b 2 ]. The weights λ (1) j,2n (n + 1 j 2n) grow exponentially fast if 2U(x; ν 1 ) + U(x; ν 2 ) > l 1 The weights decrease to zero exponentially fast if 2U(x; ν 1 ) + U(x; ν 2 ) < l 1.

50 Example: two equal length intervals The function 2U(z; ν 1) + U(z; ν 2) for [ 3.26, 1] and [1, 3.26]

51 Example: two intervals of different length The function 2U(z; ν 1) + U(z; ν 2) for [ 1, 0] and [0, 1 4 ]

52 Example: two intervals of different length The function 2U(z; ν 1) + U(z; ν 2) for [ 1, 0] and [0, 1 4 ] (detail)

53 Convergence result: equal length intervals Theorem (Lubinsky-WVA) Suppose both intervals [a 1, b 1 ] and [a 2, b 2 ] are of the same size and let b 1 < a 2. If f is continuous on [a 1, b 1 ] and [a 2, b 2 ], then lim n lim n 2n j=1 2n j=1 λ (1) b1 j,2n f (x j,2n) = f (x) dµ 1 (x), a 1 λ (2) b2 j,2n f (x j,2n) = f (x) dµ 2 (x). a 2

54 Example: two intervals of equal length The potentials U(z; ν 1) and U(z; ν 2)

55 Example: two intervals of different length The potentials U(z; ν 1) and U(z; ν 2)

56 Hermite-Padé approximation Let g 1 (z) = b1 Hermite-Padé approximation to (g 1, g 2 ): a 1 dµ 1 (x) b2 z x, g dµ 2 (x) 2(z) = a 2 z x. ( 1 ) g 1 (z)p n,n (z) Q 2n 1 (z) = O z n+1, ( 1 ) g 2 (z)p n,n (z) R 2n 1 (z) = O z n+1.

57 Hermite-Padé approximation Let g 1 (z) = b1 Hermite-Padé approximation to (g 1, g 2 ): Then a 1 dµ 1 (x) b2 z x, g dµ 2 (x) 2(z) = a 2 z x. ( 1 ) g 1 (z)p n,n (z) Q 2n 1 (z) = O z n+1, ( 1 ) g 2 (z)p n,n (z) R 2n 1 (z) = O z n+1. Q 2n 1 (z) P n,n (z) = 2n j=1 λ (1) j,2n z x j,2n, R 2n 1 (z) P n,n (z) = 2n j=1 λ (2) j,2n z x j,2n.

58 Hermite-Padé approximation 2n j=1 2n j=1 λ (1) j,2n f (x j,2n) = 1 f (z) Q 2n 1(z) 2πi Γ P n,n (z) dz, λ (2) j,2n f (x j,2n) = 1 f (z) R 2n 1(z) 2πi Γ P n,n (z) dz, where Γ is a closed contour around [a 1, b 1 ] [a 2, b 2 ].

59 Convergence for analytic functions Theorem If f is analytic in a domain Ω that contains C 1 γ = {z C : 2U(z; ν 1 ) + U(z; ν 2 ) l 1 > γ} with γ < 0, then lim sup n 2n j=1 λ (1) b1 j,2n f (x j,2n) a 1 f (x) dµ 1 (x) 1/n e γ.

60 Convergence for analytic functions Theorem If f is analytic in a domain Ω that contains C 1 γ = {z C : 2U(z; ν 1 ) + U(z; ν 2 ) l 1 > γ} with γ < 0, then lim sup n 2n j=1 λ (1) b1 j,2n f (x j,2n) a 1 f (x) dµ 1 (x) 1/n e γ. If f is analytic in a domain Ω that contains C 2 γ = {z C : U(z; ν 1 ) + 2U(z; ν 2 ) l 2 > γ} with γ < 0, then lim sup n 2n j=1 λ (2) b2 j,2n f (x j,2n) a 2 f (x) dµ 2 (x) 1/n e γ.

61 Conclusion Two quadrature rules with 2n nodes are exact for polynomials of degree 3n 1

62 Conclusion Two quadrature rules with 2n nodes are exact for polynomials of degree 3n 1 Convergence for continuous functions if the intervals are of the same size

63 Conclusion Two quadrature rules with 2n nodes are exact for polynomials of degree 3n 1 Convergence for continuous functions if the intervals are of the same size Problems if the intervals are not of the same size: no convergence on the largest interval

64 Conclusion Two quadrature rules with 2n nodes are exact for polynomials of degree 3n 1 Convergence for continuous functions if the intervals are of the same size Problems if the intervals are not of the same size: no convergence on the largest interval Quadrature weights are not all positive and some can grow exponentially

65 Conclusion Two quadrature rules with 2n nodes are exact for polynomials of degree 3n 1 Convergence for continuous functions if the intervals are of the same size Problems if the intervals are not of the same size: no convergence on the largest interval Quadrature weights are not all positive and some can grow exponentially Not recommended for Angelesco systems with intervals of different size.

66 Conclusion Two quadrature rules with 2n nodes are exact for polynomials of degree 3n 1 Convergence for continuous functions if the intervals are of the same size Problems if the intervals are not of the same size: no convergence on the largest interval Quadrature weights are not all positive and some can grow exponentially Not recommended for Angelesco systems with intervals of different size. More useful for measures having the same support, e.g., f (x)e x2 +c 1 x dx, f (x)e x2 +c 2 x dx, f (x)e x2 +c 3 x dx where c i /2 are the wavelengths for red, blue, green and f is a light signal.

67 References C.F. Borges, On a class of Gauss-like quadrature rules, Numer. Math. 67 (1994), J. Coussement, W. Van Assche, Gaussian quadrature for multiple orthogonal polynomials, J. Comput. Appl. Math. 178 (2005), no. 1 2, D.S. Lubinsky. W. Van Assche, Simultaneous Gaussian quadrature for Angelesco systems, (manuscript) U. Fidalgo Prieto, J. Illán, G. López Lagomasino, Hermite-Padé approximation and simultaneous quadrature formulas, J. Approx. Theory 126 (2004), no. 2,