72 5 f (x) f Tylor f (x) f (x) = f (x) + 2 f (x) + 2 3! f (x) + (5.) = f (x) + O() = f (x) 2 f (x) + 2 3! f (x) (5.2) = f (x) + O() δ f 2 = ( f (x) +
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1 7 5 Derivtives nd integrls re lso very useful s topics for tecing bout numericl computtion nd nlysis. Tey re esily understood, one cn present intuitive or pictoril motivtions, te lgebr is usully not too messy, nd yet teir clcultion is not trivil. J.R.Rice, Numericl Metods, Softwre, nd Anlysis, 2nd Ed., (Acdemic Press) ( ) (Ordinry) (Prtil) (Differentil Eqution, DE) (Integrl Eqution) 5. f (x) (forwrd, bckwrd nd centrl difference) [ ] f (x) = f (x + ) f (x) [ ] f (x) = f (x) f (x ) [ ] δ f (x) = f (x + ) f (x ) [ ] f (x) f (x) [ ] f (x) f (x) [ ] f (x) δ f (x) 2
2 72 5 f (x) f Tylor f (x) f (x) = f (x) + 2 f (x) + 2 3! f (x) + (5.) = f (x) + O() = f (x) 2 f (x) + 2 3! f (x) (5.2) = f (x) + O() δ f 2 = ( f (x) + f (x) ) 2 = f (x) + 2 3! f (x) + = f (x) + O( 2 ) 5.. f (x) = sin (cos x) x = π/4 IEEE754 f (π/4) 5. f (x) (5.) (5.2) ( f (x) f (x) ) f (x ) 2 f (x) + f (x + ) = = f (x) + O( 2 ) (5.3) f (x) = sin(cos x) f (π/4) f (π/4)/, δ f (π/4)/
3 相 対 誤 差 0.00 前 進 差 分 商 後 退 差 分 商 e-06 e-09 中 心 差 分 商 e-2 e : 5.2 f (x) x = f () Stirling [26] [3 ] f () = { 2 f ( + ) } f ( ) f (3) () 2 + (5.4) { 2 f ( + 2) + 23 f ( + ) 23 f ( ) + 2 } f ( 2) [5 ] f () = + 30 f (5) () 4 + (5.5) [7 ] f () = { 60 f ( + 3) 3 20 f ( + 2) f ( + ) 3 f ( ) f ( 2) } f ( 3) f (7) () 6 + (5.6) order , 7 6
4 74 5 f (x) = cos (sin x) f (x) x = π/4 f (π/4) = , 2,..., 2 0 (TE) (RE) 5.2 ( ) = 2 x 0 00 IEEE754.0E-0 Reltive Error.0E-05.0E-09.0E-3 3 Points, TE. 3 Points, RE 5 Points, TE 5 Points, RE 7 Points, TE 7 Points, RE.0E-7.0E ^(-x) : Stepsize 5.2: x = π/4 (TE) (RE) ( ) 3, 5, 7 2, 4, 6 IEEE754 IEEE = = = 2 0 IEEE
5 E-0.0E-05 Reltive Error.0E-09.0E-3 3 Points 5 Points 7 Points.0E-7.0E ^(-x) : Stepsize 5.3: x = π/4 f (π/4) = [3] x = [ 2π, 2π]
6 : 7 7 f (π/4) e e e e e e e e e e e Newton-Cotes Riemnn f (x)dx (5.7) Newton-Cotes (x i, f (x i )) Lgrnge [x 0, x k ] k ( ) k + (x 0, f (x 0 )),..., (x k, f (x k )) k p k (x) p k (x) f (x) x 0 x x 2 x k 5.4: k f (x) p k (x) xk x 0 f (x)dx xk x 0 p k (x)dx (5.8) p k (x) Lgrnge (4.5) (5.8)
7 5.3. Newton-Cotes 77 xk x 0 p k (x)dx = xk x 0 k j=0 k ( = ψ (x j=0 j ) k = ψ (x j ) = j=0 ψ(x) (x x j )ψ (x j ) f (x j)dx xk x 0 xk x 0 ) ψ(x) dx f (x j ) x x j k l=0, l j k (cd j ) f (x j ) = c j=0 (x x l ) dx f (x j) k d j f (x j ) (5.9) k = 2, 3,..., 7 c, d 0,..., d k 5.2 j=0 () 2 (trpezoidl rule) x = (b )/n n [x i, x i+ ] xi+ f (x)dx x i f (x)dx 2 ( f (x i) + f (x i+ )) (5.0) 2 ( f (x i) + f (x i+ )) = 2 f (x 0) + 2 f (x i ) + f (x n ) (5.) i= Simpson /3 3 2 Lgrnge Simpson /3 2n xi+2 f (x)dx x i f (x)dx 3 ( f (x i) + 4 f (x i+ ) + f (x i+2 )) (5.2) 2 3 { f (x i) + 4 f (x i+ ) + f (x i+2 )} = 3 f (x 0) + 4 f (x 2i ) + 2 f (x 2i ) + f (x 2n ) i= i= (5.3)
8 : Newton-Cotes (Abrmowitz ) 2 () c = /2 d 0 = d = 3 (Simpson /3 ) c = /3 d 0 = d = 4 d 2 = 4 (Simpson 3/8 ) c = 3/8 d 0 = d = 3 d 2 = 3 d 3 = 5 c = 2/45 d 0 = 7 d = 32 d 2 = 2 d 3 = 32 d 4 = 7 6 c = 5/288 d 0 = 9 d = 75 d 2 = 50 d 3 = 50 d 4 = 75 d 5 = 9 7 c = /40 d 0 = 4 d = 26 d 2 = 27 d 3 = 272 d 4 = 27 d 5 = 26 d 6 = 4
9 5.4. Guss (.3) Simpson /3 4 = (4/5 0)/4 = /5 5.4 Guss Guss p n (x) w(x) f (x)dx w i f (α i ) (5.4) i= w(x) w(x)dx < + α i (i =, 2,..., n) p n (x) = 0 Guss 5.4. [, b] p 0 (x) = µ 0 ( ), p (x),..., p n (x),... λ i > 0 (i = j) w(x)p i (x)p j (x)dx = 0 (i j) (5.5) {p n (x)} p i+ (x) = ( x)p i (x) 4 p i (x) (5.6) 5.3 p 0 (x) = µ 0, p (x) = µ x + r 0 p 2 (x) = µ 2 x 2 + r (x)(2 ), p 3 (x) = µ 3 x 3 + r 2 (x)(3 ),..., p n (x) = µ n x n + r n (x)(n ),... p n (x) n. n q n (x) {p n (x)} q n (x) = c i p i (x) 2 M.Abrmowitz nd I.A. Stegun Hndbook of Mtemtic Functions (Dover)
10 : p 0 (x) p (x) w(x) [, b] λ i Legendre P i (x) Cebycev T i (x) Lguerre L i (x) Hermite H i (x) x i + 0 2i + i [, ] x 0 2 x 2 [, ] 2 2i + π/2 (λ 0 = π) x i + 2i + i exp( x) [0, ) 2x 0 2 2i exp( x 2 ) (, ) 2 i i! π c i = w(x)p i (x)q n (x)dx λ i 2. p n (x) (n ) α, α 2,..., α n α i α j (i j) α i (, b) 3. Cristoffel-Drboux p n (x) = µ n (x α )(x α 2 ) (x α n ) x y p i (x)p i (y) = µ n p n (x)p n (y) p n (x)p n (y) λ i λ n µ n x y (p i (x)) 2 x = y y x = µ n ( p n (x)p n λ i λ n µ (x) + p n (x)p n(x)) n (5.7) (α, f (α )), (α 2, f (α 2 )),..., (α n, f (α n )) Lgrnge f n (x) (4.5) f n (x) = k= p n (x) (x α k )p n(α k ) f (α k) (5.8) Cristoffel-Drboux (5.7) y = α k, x = α k p i (x)p i (α k ) λ i = µ n λ n µ n p n (x)p n (α k ) x α k (p i (α k )) 2 λ i = µ n λ n µ n (p n (α k )p n(α k ))
11 5.4. Guss 8 w k f n (x) (p i (α k )) 2 = = µ n p n (α k )p w k λ i λ n µ n(α k ) (5.9) n p n (x) (x α k )p n(α k ) = w p i (x)p i (α k ) k λ i f n (x) = w k k= f (x) p i (x)p i (α k ) λ i f (α k) (5.20) Guss Guss (5.4) f (x) f n (x) w(x) f (x)dx w(x) f n (x)dx (5.20) w(x) f n (x)dx = = p i (x)p i (α k ) w(x) w k λ k= i f (α k) dx ( w k p i (α k ) f (α k ) λ i k= ) w(x)p i (x)dx (5.2) {p n (x)} (5.5) (5.2) w(x)p i (x)dx = λ 0 b (i = 0) w(x)p i (x)p 0 (x)dx = µ 0 µ 0 0 (i 0) w(x) f n (x)dx = w k f (α k ) (5.22) Guss w k p n (x) α, α 2,..., α n Newton-Cotes p n (x) 5.4 Legendre P i (x) Guss (Guss-Legendre ) P i (x) α,..., α i w i w(x) = f (x)dx k= w k f (α k ) k=
12 82 5 [, b] x = b 2 t + b + 2 f (t)dt x 2 dx = 2 Simpson /3 Guss-Legendre IEEE754 Guss-Legendre(2 ) Simpson(/3 ) Guss-Legendre(3 ) f (x) 2n g 2n (x) n n (x), q n (x) g 2n (x) = n (x)p n (x) + q n (x) p n (x) α i (i =, 2,..., n) g 2n (α i ) = q n (α i ) (i =, 2,..., n) q n (x) (α i, q n (α i )) (i =, 2,..., n) n (x) p 0 (x), p (x),..., p n (x) w(x)g 2n (x)dx = = 0 + w(x) n (x)p n (x)dx + = w k g 2n (α k ) k= w(x)q n (x)dx = w(x)q n (x)dx w k q n (α k ) Guss 2n f (x) 2n n 2n order Guss k=
13 5.4. Guss : Guss-Legendre
14 π 0 sin x dx Guss-Legendre 2, 3, () f (x) (b) 5. f (π/4)/ δ f (π/4)/ f (π/4) log x dx () (b) 4 (c) 4 Simpson /3 (d) Guss-Legendre 4 3. f (x), f (4) (x) 4. (5.9) Simpson /3 c, d 0, d, d 2 5. f (x)dx 2 ( f (x 0)+2 f (x )+ +2 f (x n )+ f (x n ))+ 24 ( f (x 0 )+ f (x )+ f (x n ) f (x n +))
2009 IA I 22, 23, 24, 25, 26, a h f(x) x x a h
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..1 f(x) n = 1 b n = 1 f f(x) cos nx dx, n =, 1,,... f(x) sin nx dx, n =1,, 3,... f(x) = + ( n cos nx + b n sin nx) n=1 1 1 5 1.1........................... 5 1.......................... 14 1.3...........................
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II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
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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,,
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() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >
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2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
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