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
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 + ) = 2 5.. = f (x) + O( 2 ) (5.3) f (x) = sin(cos x) f (π/4) f (π/4)/, δ f (π/4)/ 5. 0 3
5.2. 73 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 0 0 0-2 0-4 0-6 0-8 0-0 5.: 5.2 f (x) x = f () Stirling [26] [3 ] f () = { 2 f ( + ) } f ( ) 2 + 6 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) + 3 4 f ( + ) 3 f ( ) 4 3 + 20 f ( 2) } f ( 3) 60 + 40 f (7) () 6 + (5.6) order 3 2 5 4, 7 6
74 5 f (x) = cos (sin x) f (x) x = π/4 f (π/4) = 4.593626849327 0 2 0, 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-2 0 2 4 6 8 0 2^(-x) : Stepsize 5.2: x = π/4 (TE) (RE) ( ) 3, 5, 7 2, 4, 6 IEEE754 IEEE754 5.2 3 = 2 0 2 0 7 = 2 6 5 = 2 0 IEEE754 5.3
5.2. 75.0E-0.0E-05 Reltive Error.0E-09.0E-3 3 Points 5 Points 7 Points.0E-7.0E-2 0 2 4 6 8 0 2^(-x) : Stepsize 5.3: x = π/4 f (π/4) 7 5. 5.3 = 2 6 2 6 4.593626849327 4.593626849328 4.59362684933 0 2 [3] x = [ 2π, 2π]
76 5 5.: 7 7 f (π/4) 2 0 4.568860826503527e 0 2 4.59327000065456403e 0 2 2 4.5936279303235640e 0 2 3 4.5936267730507094e 0 2 4 4.5936268484669853e 0 2 5 4.59362684930946064e 0 2 6 4.59362684932753673e 0 2 7 4.593626849328092e 0 2 8 4.5936268493279397e 0 2 9 4.59362684932760557e 0 2 0 4.5936268493283272e 0 5.3 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)
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)
78 5 5.2: 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
5.4. Guss 79 5.3. (.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) = ( 2 + 3 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)
80 5 5.3: p 0 (x) p (x) 2 3 4 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 ))
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) 5.4.2 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=
82 5 [, b] x = b 2 t + b + 2 f (t)dt 5.4. 2 x 2 dx = 2 Simpson /3 Guss-Legendre IEEE754 Guss-Legendre(2 ) Simpson(/3 ) Guss-Legendre(3 ) 6.25000000000000000 0 4.9704420834380 0 5.04629629629629539 0 4.99874023683547497 0 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=
5.4. Guss 83 5.4: Guss-Legendre 2 0.577350269896257645094878050 +0.577350269896257645094878050 3 0.77459666924483377035853079956 0.555555555555555555555555555555 0 0.888888888888888888888888888888 +0.77459666924483377035853079956 0.555555555555555555555555555555 4 0.86363594052575223946488892 0.3478548453745385737306394922 0.3399804358485626480266575903 0.652455486254642626936050778 +0.3399804358485626480266575903 0.652455486254642626936050778 +0.86363594052575223946488892 0.3478548453745385737306394922 5 0.90679845938663992797626878299 0.236926885056890875426404079 0.53846930056830903634420700 0.478628670499366468042954835 0 0.568888888888888888888888888888 +0.53846930056830903634420700 0.478628670499366468042954835 +0.90679845938663992797626878299 0.236926885056890875426404079 6 0.93246954203520278230554493 0.7324492379703450402964272 0.66209386466264536639959509 0.3607657304843860756983353837 0.2386986093969086305072680 0.4679393457269047389870343989 +0.2386986093969086305072680 0.4679393457269047389870343989 +0.66209386466264536639959509 0.3607657304843860756983353837 +0.93246954203520278230554493 0.7324492379703450402964272 7 0.949079234275852452689684047 0.29484966688696932706432679 0.745385599394439863864773280 0.279705394892766679046777423 0.40584553773976690660642076 0.388300505058944950369775488 0 0.47959836734693877550204086 +0.40584553773976690660642076 0.388300505058944950369775488 +0.745385599394439863864773280 0.279705394892766679046777423 +0.949079234275852452689684047 0.29484966688696932706432679
84 5 5.4. π 0 sin x dx Guss-Legendre 2, 3, 4. 5.. () f (x) (b) 5. f (π/4)/ δ f (π/4)/ f (π/4) 0 5 2. 2 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 +))