1 Edward Waring Lagrange n {(x i, y i )} n i=1 x i p i p i (x j ) = δ ij P (x) = p i p i (x) = n y i p i (x) (1) i=1 n j=1 j i x x j x i x j (2) Runge

Size: px

Start display at page:

Download "1 Edward Waring Lagrange n {(x i, y i )} n i=1 x i p i p i (x j ) = δ ij P (x) = p i p i (x) = n y i p i (x) (1) i=1 n j=1 j i x x j x i x j (2) Runge"

そうおおかわち
5 years ago
Views:

1 Edwrd Wring Lgrnge n {(x i, y i )} n x i p i p i (x j ) = δ ij P (x) = p i p i (x) = n y i p i (x) () n j= x x j x i x j (2) Runge [] [2] = ξ 0 < ξ < ξ n = b [, b] [ξ i, ξ i ] y = /( + 25x 2 ) 5 2,, 0,, 2

2 P n p 0,..., p n p 0 (x) ξ 0 x < ξ, P (x) =. p n (x) ξ n x < ξ n * deg P := mx deg p i. 0 i n k C k (, b) k {ξ i } n ξ = (ξ,..., ξ n ) k R S (k; ξ) S (k; ξ 0,..., ξ n ) S (k; ξ) s [ξ i, ξ i ] [, b] {ξ i } n i=0 k s s(k) s (k) [0, ) H k n s (k) (x) = c k + d i H(x ξ i ), x [, b] \ {ξ i } n i=0 s(x) = k c j x j + n d i (x ξ i ) k k! +, x [, b] (3) (x ξ i ) k + (x ξ i ) k + = n + k {c i } k, {d i} n n + k { H(x ξi ) k = 0 ( mx{0, x ξi } ) k k =, 2,... S (k; ξ) S (k; ξ) R (3) S (k; ξ) B. S (k; ξ 0,..., ξ n ) 0 p < q n [ξ p, ξ q ] S (k; ξ 0,..., ξ n ) s q s(x) = d i (x ξ i ) k +, x [ξ p, ξ q ] i=p * 0 2

3 d i q d i (x ξ i ) k = 0, i=p x ξ q q d i ξ j i i=p = 0, j = 0,,..., k (4) n f n x,..., x n {(x i, f(x i ))} n Legendre P P f n n {x i } n P = f x l = n x l i n j= x n δ l,n = n x l i x x j x i x j, l = 0,,..., n n j= (5) q = p + k + (4) d i d i = p+k+ j=p B k p B k p (x) := p+k+ i=p p+k+ j=p x i x j, l = 0,,..., n (5) ξ i ξ j, i = p, p +,..., p + k + ξ j ξ i (x ξ i ) k +, x R, p = 0,,..., n k (6) B p [ξ p, ξ p+k+ ] 2 B, B 2, B 3.2 B B B S (k; ξ 0,..., ξ n ) S (k; ξ 0,..., ξ n ) n + k B n k 2k ξ k, ξ k+,..., ξ, ξ n+, ξ n+2,..., ξ n+k i < j ξ i < ξ j (7) 3

4 B B 2 B 3 2 B n + k B {B k p } n p= k S (k; ξ 0,..., ξ n ) {B k p } n p= k B. 0 p < q n [ξ p, ξ q ] s S (k; ξ 0,..., ξ n ) s r r q (p + k + ) B B k p (ξ p, ξ p+k+ ) (ξ p, ξ p+ ) B k p B k p (ξ p, ξ p+k+ ) {B k p } n p= k s = n p= k s [, b] 0 λ p 0 B k p s (, ξ k ] 0 [, b] 0. (, b] 0 s [ξ k, ξ k+ ] λ p B k p s(x) = λ k B k k, x [ξ k, ξ k+ ] λ k = 0 [ξ k+, ξ k+2 ] s(x) = λ k B k k + λ k+ B k k+, x [ξ k+, ξ k+2 ] λ k = 0 λ k+ = 0 λ p 0 4

5 .2 {ξ j } n+k j= k (7) {Bk p } n+k p= k (6) B {Bp k } n+k p= k S (k; ξ 0,..., ξ n ) B B (6) x (3) B B B.3 Cox-de Boor k {ξ j } p+k+ j=p B (7) Bp k (x) = (x ξ p)bp k (x) + (ξ p+k+ x)bp+ k (x) (8) ξ p+k+ ξ p B 0 p [ξ p, ξ p+ ) B k (k ) Bp k {Bk j } Bp k, Bp+ k.4 k {ξ j } p+k+ j=p d dx Bk p (x) = (7) B k p k ( B k p ξ p+k+ ξ p k = x B B k p = (k + )B k p (x) B k p+ (x)) (9) 0 { B p k } S k s { B p k }.4 s k { B p }.5 k (7) h s = j β j B k j hs = j β j Bk j β j = β j β j n h n s (n) = j n β j Bk n j 5

6 .3 {(x i, y i )} n+k S (k; ξ 0,..., ξ n ) x i x < x 2 < < x n+k b (0) S (k; ξ 0,..., ξ n ) s(x i ) = y i, i =,..., n + k s S (k; ξ 0,..., ξ n ) S (k; ξ 0,..., ξ n ) B 2k.6 Schoenberg-Whitney (0) {(x i, y i )} n+k n p= k λ p B k p (x i ) = y i, i =, 2,..., n + k () {B k j k (x j)} n+k j= 0 {λ p} n p= k i j b ij = B k j k (x i) B B 0 b ij 0 x i / (ξ j k, ξ j ) x i ξ j k p j k, x x i B p (x) = 0 i i, j j b i j = 0 x i ξ j i i, j j b i j = 0 j b jj = 0 B, t B ( ) X 0 Y Z X j Z (n + k j) det X det Z X j 0 det X = 0 B 0 B ker B 0 s s(x) = n p= k λ p B k p (x) s(x i ) = 0, i =,..., n + k s 0 ξ k ξ p < ξ q ξ n+k [ξ p, ξ p ], [ξ q, ξ q+ ] s 0 (ξ p, ξ q ) s B 0 j x j (ξ j k, ξ j ) {x j } q j=p+k+ (ξ p, ξ q ) s (ξ p, ξ q ) q p k. s 0.2 λ p 0 6

7 [, b] {x i } n i=0 [, b] f {(x i, f(x i ))} n i=0 Lgrnge x n [x 0, x,..., x n ]f (),(2) [x 0, x,..., x n ]f f [x 0, x,..., x n ]f = n f(x k ) k=0 n j k x k x j (2) f(ξ) = ( ) k+ (x ξ) k + B (6) B k p (x) = [ξ p, ξ p+,..., ξ p+k+ ] { ( ) k+ (x ) k +}. B.7 [, b] {x i } k+ i=0 f Ck+ [, b] B k B [x 0, x,..., x n ]f = k! k+ B k (x) = (x x i ) k + i=0 B k (x)f (k+) (x)dx k+ x j x i (x x i ) k +f (k+) (x)dx = k l=0 ( ) l k! (k l)! (b x i) k l f k l (b) + ( ) k+ k!f(x i ) B k (5) = k+ = 0 k i=0 l=0 l=0 ( ) l k! (k l)! (b x i) k l f k l (b) i=0 k+ x j x i k k+ ( ) l k! (k l)! f k+ k l (b) (b x i ) k l x j x i 7

8 k+ k+ B k (x)f (k+) (x)dx = ( ) k+ k! f(x i ) k+ = k! i=0 f(x i ) i=0 k+ = k![x 0, x,..., x n ]f x i x j x j x i.4 [, b] x < x 2 < < x m {f(x i )} m C2 [, b] s s(x i ) = f(x i ), i =,..., m (3) m = 2 m > 2 (3).7 B p ( s (x) ) 2 dx (4) [x p, x p+, x p+2 ]s = [x p, x p+, x p+2 ]f, p =,..., m 2 (5) B p(x)s (x)dx = [x p, x p+, x p+2 ]f, p =,..., m 2 p+2 p+2 Bp(x) = (x x i ) + x j x i i=p j=p B p(x)u(x)dx = [x p, x p+, x p+2 ]f, p =,..., m 2 (6) 8

9 [, b] u E(u) = ( u(x) ) 2dx (7) u {B p} m 2 p= [, b] {B p} m 2 p= u v α u + αv (6) u v E(u + αv) < E(u) α u {Bp} m 2 p= u = λ j Bj (6) ( b m 2 ) Bpx λ j Bj (x) dx = [x p, x p+, x p+2 ]f, p =,..., m 2 j= ( b m 2 ) Bpx µ j Bj (x) dx = 0, p =,..., m 2 j= ( b m 2 2 µ j Bj (x)) dx = 0.2 µ j = 0 (6) u j= s u 2 s(x ), s(x 2 ) f(x ), f(x 2 ) (5) (3) s 2 s 3 3 s 2 x, x m 0 [, b] 3 [, x ], [x m, b].8 [, b] (2k + ) s (j) (x ) = s (j) (x m ) = 0, j = k +, k + 2,..., 2k (8) (2k + ) x, x m [, x ], [x m, b] k.9 (2k + ) S N (2k + ; x,..., x m ).8 m k + S N (2k + ; x,..., x m ) m.5 m k 9

10 .8 [, b] x < x 2 < < x m k k m [, b] f s(x i ) = f(x i ), i =,..., m (9) s S N (2k + ; x,..., x m ) < x [, x ] [x, x 2 ] x m < b = x, x m = b S (2k + ; x,..., x m ) (2k + m) 2k (8) S N (2k + ; x,..., x m ) m l := dim S N (2k + ; x,..., x m ).6 l = m s S N (2k + ; x,..., x m ) s(x i ) = 0, i =,..., m, s S N (2k + ; x,..., x m ) (20) s = 0 (8) s (2k+) (x i, x i+ ) (20) E ( s (k+)) = xm x m = ( ) k ( s (k+) (x) ) 2 dx m = ( ) k = 0 xi+ x i s (x) s (2k+) (x)dx s (2k+) (x i +) ( s(x i+ ) s(x i ) ) x i + (x i, x i+ ) s (k+) 0 s k k m (20) s s (9) C k+ [, b] ( s (k+) (x) ) 2 dx (9) C k+ [, b] s E ( s (k+)) = E ( s (k+)) + E ( s (k+) s (k+)) + 2 ( s (k+) (x) s (k+) (x) ) s (k+) (x)dx.8 0 E ( s (k+)) = E ( s (k+)) + E ( s (k+) s (k+)) E ( s (k+)) 0

11 s (k+) s (k+) = 0 s s k (9) s s m k m s s B 2 P {(x i, y i )} n x < < x n b s(x i ) = y i, i =,..., n s (2k + ) n ( P (s) = yi s(x i ) ) ( 2 2dx + λ s (x)) (k+) s λ k = 3 0 P λ P f C k+ [, b] {(x i, f(x i ))} n S N s.8 s.9 s P (s) P (f) P (s) 3 2

12 [] M.J.D. Powell. Approximtion Theory nd Methods. Cmbridge University Press, 98. [2].., 3,