num3.dvi

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2 ,, ( ) Taylor. ( 1) i )x 2i+1 sinx = (2i+1)! i=0 S=0.0D0 T=X; /* */ DO 100 I=1,N S=S+T /* */ T=-T*X*X/(I+I+2)/(I+I+3) /* */ 100 CONTINUE. S=S+(-1)**I*X**(2*i+1)/KAIJO(2*I+1). o Q log(1+x), S=S+T T=-T*X*I/(I+1) S=S+T/I, T=-T*X 1

3 2 ( 1) n 1 log2 = n n=1 n sinx sinx = x x3 3! + x5 x2n+1 + +( 1)N 5! (2N +1)! +R N R N x2n+3 (2N +3)! sin20, sinx sin(20 2π 3) o Q o e x Taylor

4 i 20 2i+1 (2i+1)! i 20 2i+1 (2i+1)!

5 C f(x) x f(x) f df =. dx. df =. dx. f(x+h) f(x) h f(x) f(x h) h f ( x) df =. dx. f(x+h) f(x h) f ( x h) 2h x h x x+ h 4 f ( x+ h) Taylor

6 5 f(x+h) = f(x)+hf (x)+ h2 2! f (x)+ h3 3! f (x)+ h4 4! f(4) (x)+ f(x h) = f(x) hf (x)+ h2 2! f (x) h3 3! f (x)+ h4 4! f(4) (x)+ f(x+h) f(x) = f (x)+ h h 2 f (x)+ f(x) f(x h) = f (x) h h 2 f (x)+ O(h) f(x+h) f(x h) = f (x)+ h2 2h 6 f (x)+ O(h 2 ) f f

7 C num3-1a.c #include<stdio.h> #include<stdlib.h> #include<math.h> double f(double x) { return sin(x); } double fordf(double a,double h) { return (f(a+h)-f(a))/h; } double backdf(double a,double h) { return (f(a)-f(a-h))/h; } double cntrdf(double a,double h) { return (f(a+h)-f(a-h))/h/2; } int main(void) { int i,n=10; double a,h=(double)1/10; printf("derivative of sin(x); give point : "); scanf("%lf",&a); for (i=0;i<n;i++){ printf("h = %18.15lf;frwd%18.15lf\n",h,fordf(a,h)); printf("back%18.15lf;",backdf(a,h)); printf("cntr%18.15lf\n",cntrdf(a,h)); h=h/10; } printf("true value : %22.15lf\n",cos(a)); return 0; } 6

8 1 x = 1 sin 7 h forward backward true value : O(h)...

9 2 h center true value : O(h 2 ) 8

10 9 f(x+h) f(x) f(x+h) f(x)+ε h h f(x+h) f(x) h = f (x)+o(h)+o ( ε) h h h = ε, h h = ε = 10 8 h ε double f(x) = h ε h f(x+h) f(x h) 2h f (x) = O(h 2 )+O ( ε) h h 2 = ε, h h = 3 ε = 10 5 h 2 = ε 2/3 = 10 10

11 f(x+h) = f(x)+hf (x)+ h2 2! f (x)+ h3 3! f (x)+ h4 4! f(4) (x)+ f(x h) = f(x) hf (x)+ h2 2! f (x) h3 3! f (x)+ h4 4! f(4) (x)+ f(x+h)+f(x h) 2f(x) h 2 = f (x)+ h2 12 f(4) (x)+ f(x+h)+f(x h) 2f(x) h 2 = f (x)+o(h 2 )+O ( ε h 2 ) h 2 = ε h2 f(x) =.. 1 h = ε 1/4 = 10 4 h 2 = ε 1/2 =

12 FORTRAN (F77) num3-2.f 11 PROGRAM DF2SIN DOUBLE PRECISION DF2,X,H INTRINSIC DSIN H=1.0D-1 WRITE(*,*) Give X : READ(*,*)X DO 100 I=1,10 WRITE(*,200) H =,H, ; 2nd Center =,DF2(DSIN,X,H) H=H/ CONTINUE WRITE(*,300) True value :,-DSIN(X) 200 FORMAT(1H,A4,F13.10,A15,F22.16) 300 FORMAT(1H,A13,F22.16) END DOUBLE PRECISION FUNCTION DF2(F,X,H) DOUBLE PRECISION X,H,F EXTERNAL F DF2=(F(X+H)+F(X-H)-F(X)*2)/H/H RETURN END

13 FORTRAN END 12 PROGRAM MAINYO... END FUNCTION KANSU(HIKISU,...)... RETURN <- END SUBROUTINE SUB(HIKISU,...)... RETURN <- END C FORTRAN C COMMON FORTRAN sin exp DSIN DEXP EXTERNAL INTRINSIC

14 sinx x = 1 13 h 2nd center true value : h 2 = ε h2 f(x) =.. 1 h = ε 1/4 = 10 4 h 2 = ε 1/2 = 10 8.

15 14 Taylor f(x+h) = f(x)+f (x)h+ f (x+θh) h 2, 2! f(x+h) f(x) f (x) M 2 h h 2 M 2 = sup f (x) ( ) f(x) = x 2 x = 1 (best possible i.e. ) f(x+h) = f(x)+f (x)h+ f (x) h 2 + f(3) (x+θ 1 h) h 3, 2! 3! f(x h) = f(x) f (x)h+ f (x) h 2 f(3) (x θ 2 h) h 3, 2! 3! f(x+h) f(x h) f (x) 2h = 1 f (3) (x+θ 1 h)+f (3) (x θ 2 h) h 2 M 3 2 3! 6 h2, M 3 = sup f (3) (x) f(x) = x 3 x = 1

16 f(x+h) = f(x)+f (x)h+ f (x) 2! f(x h) = f(x) f (x)h+ f (x) h 2 + f (x) 3! h 2 f (x) 2! 3! f(x+h)+f(x h) 2f(x) f (x) h 2 h 3 + f(4) (x+θ 1 h) h 4 4! h 3 + f(4) (x θ 2 h) h 4 4! = f(4) (x+θ 1 h)+f (4) (x θ 2 h) h 2 M 4 4! 12 h2 f(x) = x 4 x = 1 15

17 h f n = f (n) (x) f(x+2h) = f 0 +2f 1 h+ 4f 2 2! h2 + 8f 3 3! h3 + 16f 4 4! f(x+h) = f 0 +f 1 h+ f 2 2! h2 + f 3 3! h3 + f 4 4! h4 + f 5 5! h5 + f(x h) = f 0 f 1 h+ f 2 2! h2 f 3 3! h3 + f 4 4! h4 f 5 5! h5 + h f 5 h 5 + 5! f(x 2h) = f 0 2f 1 h+ 4f 2 2! h2 8f 3 3! h3 + 16f 4 h 4 32f 5 h 5 + 4! 5! a{f(x+2h) f(x 2h)}+b{f(x+h) f(x h)} f 0, f 2, f 3, f 4 a,b f 0, f 2, f 4 8 f 1 4a+2b = 1, f 3 3 a+ 1 3 b = 0 a = 1 12, b = 2 3 f(x+2h)+f(x 2h)+8f(x+h) 8f(x h) i.e. 12h = f (x) f(5) (x) h

18 FORTRAN (F77) num3-3.f 17 PROGRAM DFSINR DOUBLE PRECISION DFR,X,H INTRINSIC DSIN H=1.0D-1 WRITE(*,*) Give X : READ(*,*)X DO 100 I=1,10 WRITE(*,200) H =,H, ; 4th Center =,DFR(DSIN,X,H) H=H/ CONTINUE WRITE(*,300) True value :,DCOS(X) 200 FORMAT(1H,A4,F13.10,A15,F22.16) 300 FORMAT(1H,A13,F22.16) END DOUBLE PRECISION FUNCTION DFR(F,X,H) DOUBLE PRECISION X,H,F EXTERNAL F DFR=(-F(X+H*2)+F(X-H*2)+F(X+H)*8-F(X-H)*8)/H/12 RETURN END

19 h 4-th center True value : h 4 = ε h h = ε1/5 = 10 3 h 4 = ε 4/5 =

20 19 (1) num3-1.c num3-1a.c,.,. (2) logx. (3) num3-2.f,. (4) logx. (5) num3-3.f,.., C. ( )

21 3.1 f(x) h f (a) 3.2 f(x) 3.3 f (x) f(x), f(x+h), f(x+2h) O(h 2 ) ( ) 3.4 f(x) = Arcsinx, C Arcsinx C asin(x), FORTRAN ASIN(X) ( ) DASIN(X) ( ). 3.5 a 1, a 2, a 3 f (a 2 ) h = a 3 a ( ) 3.6 h a{f(x+2h)+f(x 2h)}+b{f(x+h)+f(x h)}+cf(x) Taylor 20

num2.dvi

num2.dvi kanenko@mbk.nifty.com http://kanenko.a.la9.jp/ 16 32...... h 0 h = ε () 0 ( ) 0 1 IEEE754 (ieee754.c Kerosoft Ltd.!) 1 2 : OS! : WindowsXP ( ) : X Window xcalc.. (,.) C double 10,??? 3 :, ( ) : BASIC,