Microsoft Word - 03-数値計算の基礎.docx
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- しほこ にいだ
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1 δx f x 0 + δ x n=0 a n = f ( n) ( x 0 ) n δx n f x x=0 sin x = x x3 3 + x5 5 x x ( ) = a n δ x n ( ) = sin x ak = (-mod(k,2))**(k/2) / fact_k 10
2 11
3 I = f x dx a ΔS = f ( x)h I = f a h I = h b ( ) ( ) + f ( a + h) f ( a + ih) f ( b h) N 1 i=1 ΔS = 1 2 ( ) f a + ih f ( x) + f ( x + h) h I = 1 2 f a ( ) + 2 f ( a + h ) f ( a + ih ) f ( b h) + f ( b) h 12
4 積分法 地球惑星内部物理学演習 B 資料３ 端以外の係数が２になっている まとめると I= N 1 1 f a + 2 f ( a + ih ) + f ( b ) h ( ) 2 (16) i =1 である 台形法は２次の打ち切り誤差を持つ め足し合わせる y y=f(x) 似 積分 f(x) a f(x+h) h b x てを２次以上の多項式で近似 2.3 シンプソン法 h N /2 近接するいくつかの区間を１まとめにして より高次の次数をもつ多項式で関数を近似 する方法である 精度は補間する関数の次数で決まる 例えば 区間[a,b]を偶数個ｎに 等分し ２つまとめにすると ３点を用いて２次多項式を当てはめることができる こ f (b ) h 2 f ( a + 2ih ) + のとき 小区間を２つ合わせた区間の面積は =1 ΔS = 1 f ( x ) + 4 f ( x + h ) + f ( x + 2h ) h 3 (17) である このとき 区間[a, b]全体では 1 I = [ f ( a ) + 4 f ( a + h ) + 2 f ( a + 2h ) f ( a + 2ih ) + 4 f ( a + {2i + 1} h ) f ( b 2h ) + 4 f ( b h ) + f ( b )]h (18) となる 係数が １ ４ ２ ４ ２ ２ ４ １となっている ２は小区間を２つ にまとめたΔS のつなぎ目である まとめると I= N /2 1 N /2 1 f a + 4 f a + 2i + 1 h + 2 f ( a + 2ih ) + f ( b ) h ( ) { } ( ) 3 i =1 i =1 となる ２次式で近似する方法は３次の打ち切り誤差を持つ 13 (19)
5 f x x 2/ π ( ) ( ) = e x2 = exp x 2 erf x ( ) = 2 π 0 x ( ) exp ζ 2 dζ x= 2/ π x= ( )2 1 f (x) = 2πσ exp x µ 2 2σ 2 x 2 14
6 f x x = g x ( ) = 0 ( ) x n+1 = g x n ( ) n x 0 g(x) x 1 = g x 0 ( ) x n+1 = x n x = x n+1 y y=x g(x) g(x 3 ) g(x 2 ) g(x 1 ) y=g(x) x 1 x 2 x 3... x x 15
7 ϕ x g x ( ) 0 ( ) = x ϕ ( x) f x x = g x ( ) x ϕ ( x) = g x x = g x 1 f x ( ) x n+1 = x n f x n ( ) ( ) = x f ( x) f ( x) ( ) ( ) f ( ) x n x y y=f(x) y=f (x 1 )(x-x 2 ) x x 3 x 2 x 1 x 16
8 t f ( x) = x esin x 2πt = 0 T e x t T t x x e = a2 b 2 a 2 (41) ab E E E x e x t 17
9 f95 taylor.f t_sine.f ****** Calculate sine by Talor-series expansion ****** program taylor implicit none integer i, n, ns real(8):: dx, xmax, pi real(8), allocatable:: x(:),s1(:),s2(:) real(8):: T_sine character(20):: fmt1 pi = 4.0d0 * atan(1.0d0) fmt1 = '(3f12.7)' write (6,*) 'Input number of samples for 1 period' read (5,*) ns write (6,*) 'Your input for number of samples = ',ns dx = 2.0d0 * pi / dfloat(ns) allocate ( x(0:ns),s1(0:ns),s2(0:ns) ) write (6,*) 'Input maximum order of Taylor series' 18
10 read (5,*) n write (6,*) 'Your input maximum order of Taylor series = ',n write (6,*) write (6,*) ' x sin(x) T_sine(x)' open (10,file='mysine.dat') do i = 0,ns x(i) = dx * dble(i) s1(i) = dsin(x(i)) s2(i) = T_sine(x(i),n) write ( 6,fmt1) x(i), s1(i), s2(i) write (10,fmt1) x(i), s1(i), s2(i) end do stop end program taylor ****** function T_sine ***************************************** function T_sine(x,n) implicit none integer:: k, n real(8):: pi real(8):: x, xp, fact_k, ak, fk real(8):: T_sine_k real(8):: T_sine real(8),parameter:: eps = 1.0d-10 19
11 pi = 4.0d0 * atan(1.0d0) initalize: f(x=0) = sin(0), 0 = 1, x**0 = 1.0 T_sine_k = 0.0d0 fact_k = 1.0d0 xp = 1.0d0 take summation sigma_( a(i)*x**i ) for i = 1 to n do k = 1,n fact_k = fact_k * dfloat(k) calculate factorial real(k) ak = (-mod(k,2))**(k/2) / fact_k coefficient of the series xp = xp * x fk = ak * xp calculate xp = x**k k-th order term T_sine_k = T_sine_k + fk calculate summation end do T_sine = T_sine_k return end function T_sine 20
12 real(8), allocatable:: x(:),s1(:),s2(:) allocate ( x(0:ns),s1(0:ns),s2(0:ns) ) real(8):: T_sine T_sine T_sine s2(i) = T_sine(x(i),n) T_sine write (10,500) x(i), s1(i), s2(i) real*8 function T_sine(x,n) T_sine = T_sine_k return 21
13 plot 'mysine.dat' using 1:2 plot 'mysine.dat' using 1:2, 'mysine.dat' using 1:3 plot 'mysine.dat' using 1:2 replot 'mysine.dat' using 1:3 replot 22
14 ** Error function ** Numerical integration by trapezoid method program trapezoid_i implicit none integer, parameter:: n = 1000 integer:: i real(8):: f(0:n) real(8):: x,x1,x2,dx real(8):: s1,sall real(8):: erf real(8):: pi,rootpi write (6,*) 'x1 =' read (5,*) x1 x1 = 0.0d0 write (6,*) 'x =' read (5,*) x2 dx = ( x2-x1 ) / dfloat( n ) do i = 0,n x = x1 + dx * dfloat(i) f(i) = exp( - x**2 ) end do s1 = 0.0d0 do i = 1,n-1 s1 = s d0*f(i) end do 23
15 sall = dx * 0.5d0 * ( f(0) + s1 + f(n) ) pi = 4.0d0 * atan( 1.0d0 ) rootpi = sqrt( pi ) erf = 2.0d0 / rootpi * sall write (6,*) 'Integral F = ',sall write (6,*) 'erf(',x2,')= ',erf stop end program trapezoid_i integer, parameter:: n = 1000n real(8):: f(0:n) f(i) = exp( - x**2 ) sall s1 = s d0*f(i) s 1 = n 1 i=1 ( ) f x i s1 24
16 Solve 2nd-order algebraic equation ax^2+bx+c=0 by a Newton method program newton implicit none real(4), parameter:: eps=1.0e-6 real(4):: a,b,c,x,fx,dfdx,xini,corr integer:: itr,nitr write (6,*) 'input a,b,c (use space between numbers)' read (5,*) a,b,c write (6,*) 'input inital guess' read (5,*) xini write (6,*) 'input maximum iteration' read (5,*) nitr x = xini Iteration do itr = 1,nitr fx = a*x**2 + b*x + c dfdx = 2.0 * a * x + b corr = fx / dfdx write (6,*) itr,x,fx,dfdx,corr x = x - corr If ( abs(corr/x).lt. eps ) Exit Exit loop if convergent end do write (6,*) 'solution x = ',x stop 25
17 end program newton dfdx = 2.0 * a * x + b x = x - corr If ( abs(corr/x).lt. eps ) Exit 26
Microsoft Word - 資料 (テイラー級数と数値積分).docx
δx δx n x=0 sin x = x x3 3 + x5 5 x7 7 +... x ak = (-mod(k,2))**(k/2) / fact_k ( ) = a n δ x n f x 0 + δ x a n = f ( n) ( x 0 ) n f ( x) = sin x n=0 58 I = b a ( ) f x dx ΔS = f ( x)h I = f a h h I = h
Microsoft Word - 資料 docx
y = Asin 2πt T t t = t i i 1 n+1 i i+1 Δt t t i = Δt i 1 ( ) y i = Asin 2πt i T 29 (x, y) t ( ) x = Asin 2πmt y = Asin( 2πnt + δ ) m, n δ (x, y) m, n 30 L A x y A L x 31 plot sine curve program sine implicit
情報活用資料
y = Asin 2πt T t t = t i i 1 n+1 i i+1 Δt t t i = Δt i 1 ( ) y i = Asin 2πt i T 21 (x, y) t ( ) x = Asin 2πmt y = Asin( 2πnt + δ ) m, n δ (x, y) m, n 22 L A x y A L x 23 ls -l gnuplot gnuplot> plot "sine.dat"
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情報活用資料-03-20150604
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神戸市立工業高等専門学校電気工学科 / 電子工学科専門科目 数値解析 2017.6.2 演習 2 山浦剛 ([email protected]) 講義資料ページ h t t p://clim ate.aic s. riken. jp/m embers/yamaura/num erical_analysis. html 曲線の推定 N 次多項式ラグランジュ補間 y = p N x = σ N x x
数学の基礎訓練I
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1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1)
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情報処理概論（第二日目）
情報処理概論 工学部物質科学工学科応用化学コース機能物質化学クラス 第 8 回 2005 年 6 月 9 日 前回の演習の解答例 多項式の計算 ( 前半 ): program poly implicit none integer, parameter :: number = 5 real(8), dimension(0:number) :: a real(8) :: x, total integer
9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y
5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x
x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)
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1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
最小二乗フィット、カイ二乗フィット、gnuplot
数値計算法 009 5/7 林田清 ( 大阪大学大学院理学研究科 ) 最尤法 (Maxmum Lkelhood Method) 回の ( 独立な ) 測定 xで, x,..., x 1 母集団が平均値 μgauss) 標準偏差 の正規 ( 分布の場合 1 回の測定で xから( xの間の値を観測する確率は + dx) dq = Pdx 1 1 x µ P exp π µ は不可知 推定値をとする µ
( 28 ) ( ) ( ) 0 This note is c 2016, 2017 by Setsuo Taniguchi. It may be used for personal or classroom purposes, but not for commercial purp
( 28) ( ) ( 28 9 22 ) 0 This ote is c 2016, 2017 by Setsuo Taiguchi. It may be used for persoal or classroom purposes, but ot for commercial purposes. i (http://www.stat.go.jp/teacher/c2epi1.htm ) = statistics
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A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
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4 : 1 f(x) Z b a dxf(x) (1) ( Double Exponential method=de ) 1 DE N = n T n h h =(b a)=n T n = b a f(a) +f(b) n f + f(a + j b a n )g n j=1 = b a f(a) +f(b) n f + f(a +j b a )g; n n+1 j=1 T n+1 = b a f(a)
1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
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I [email protected] 27 6 A A. /a δx = lim a + a exp π x2 a 2 = lim a + a = lim a + a exp a 2 π 2 x 2 + a 2 2 x a x = lim a + a Sic a x = lim a + a Rect a Gaussia Loretzia Bilateral expoetial Normalized
Ｃ言語による数値計算プログラミング演習
8. 数値積分 任意の区間 [,b] における f() の定積分 b () I = f ( ) d の値は, つぎのように n 点の関数値の和により近似的に与えられる () In = Ak f ( k) n k = このとき, k を分点,A k を重みという 8. ニュートン コーツ ( 複合型 ) 積分公式 積分区間 [,b] を等分割して n 個の分点をとり, 被積分関数 f() を n- 次ラグランジュ補間多項式で近似して得られる積分公式を
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Excel ではじめる数値解析 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
Excel ではじめる数値解析 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009631 このサンプルページの内容は, 初版 1 刷発行時のものです. Excel URL http://www.morikita.co.jp/books/mid/009631 i Microsoft Windows
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