C 2 / 21 1 y = x 1.1 lagrange.c 1 / Laglange / 2 #include <stdio.h> 3 #include <math.h> 4 int main() 5 { 6 float x[10], y[10]; 7 float xx, pn, p; 8 in

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1 C 1 / 21 C 2005 A * * * * *GnuPlot C * *Gnuplot *Gnuplot

2 C 2 / 21 1 y = x 1.1 lagrange.c 1 / Laglange / 2 #include <stdio.h> 3 #include <math.h> 4 int main() 5 { 6 float x[10], y[10]; 7 float xx, pn, p; 8 int i, j; 9 10 / / 11 for( i = 0; i 9; i++ ){ 12 xx = ( float )i; 13 x[i] = xx; 14 y[i] = sqrt( xx ); 15 } xx = 4.4; 18 pn = 0; 19 for( i = 0; i 9; i++ ){ 20 p = y[i]; 21 for( j = 1; j 9; j++ ){ 22 if( i j ){ 23 p = p ( xx x[j] ) / ( x[i] x[j] ); 24 } 25 } 26 pn = pn + p; 27 } 28 printf( " %f %f\n", sqrt( xx ), pn ); 29 return 0; 2

3 C 3 / } : y = x : x = : %f %-20.15f x. 1.2 * newtonint.c 1 / / 2 #include <stdio.h> 3 #include <math.h> 4 #define N 9 5 int main() 6 { 7 float x[15], w[15]; 8 float xx, kotae; 9 int i, j; / / 12 for( i = 1; i N; i++ ){ 13 xx = ( float )i; 14 x[i] = xx; 15 w[i] = sqrt( xx ); 16 } for( i = 1; i N; i++ ){ 19 for( j = N; j > i; j ){ 20 w[j] = ( w[j] w[j 1] ) / ( x[j] x[j i] ); 21 } 22 } 23 3

4 C 4 / xx = 4.4; 25 kotae = w[n]; 26 for( i = N 1; i 1; i ){ 27 kotae = kotae ( xx x[i] ) + w[i]; 28 } 29 printf( " %f %f\n", kotae, sqrt( xx )); 30 return 0; 31 } : y = x : x = : x x 3 dx chuteni.c 1 / / 2 #include <stdio.h> 3 #include <math.h> 4 #define N float f( float x ) 7 { 8 return x x x; 9 } int main() 12 { 13 float x, y, sum, f0 = 0., fn = 1., step; sum = 0.; 16 step = ( fn f0 ) / (2 N); 4

5 C 5 / for( x = f0 + step; x < fn; x = x + 2 step ) sum = sum + f( x ); 18 printf( "%f\n", 2 step sum ); 19 return 0; 20 } 4 : N : f(x) = x 3 13 : f0 fn 16 : step 2N : x 3 dx daikei.c 1 / / 2 #include <stdio.h> 3 #include <math.h> 4 #define N float f( float x ) 7 { 8 return x x x; 9 } int main() 12 { 13 float x, y, sum, f0 = 0., fn = 1., step; sum = 0.; 16 step = ( fn f0 ) / N; 17 for( x = f0 + step; x < fn; x = x + step ) sum = sum + f( x ); 18 printf( "%f\n", step / 2. ( f( f0 ) + f( fn ) + 2. sum )); 19 return 0; 5

6 C 6 / } 4 : N : f(x) = x 3 13 : f0 fn 16 : step N : x 3 dx symp.c 1 / / 2 #include <stdio.h> 3 #include <math.h> 4 #define N float f( float x ) 7 { 8 return x x x; 9 } int main() 12 { 13 float x, y, sum, f0 = 0., fn = 1., step; 14 int i; sum = f( f0 ) + f( fn ); 17 step = ( fn f0 ) / N; 18 i = 1; 19 for( x = f0 + step; x < fn; x = x + step ){ 20 if( i % 2 == 0 ){ 21 sum = sum + 2. f( x ); 22 }else{ 6

7 C 7 / sum = sum + 4. f( x ); 24 } 25 i++; 26 } 27 printf( "%f\n", step / 3. sum ); 28 return 0; 29 } 4 : N : f(x) = x 3 13 : f0 fn 17 : step N : 2.4 * dx π 1 + x2 numint.c 1 / / 2 #include <stdio.h> 3 #include <stdlib.h> 4 5 double f(double x) 6 { 7 return 4 / (1 + x x); 8 } 9 10 int main() 11 { 12 int i, n, nmax = 32; 13 double a = 0, b = 1, h, trapezoid, midpoint, simpson; printf(" n: Simpson\n"); 7

8 C 8 / h = b a; trapezoid = h (f(a) + f(b)) / 2; 17 for (n = 1; n nmax; n = 2) { 18 midpoint = 0; 19 for (i = 1; i n; i++) midpoint += f(a + h (i 0.5)); 20 midpoint = h; 21 simpson = (trapezoid + 2 midpoint) / 3; 22 printf("%5d % f % f % f \n", 23 n, trapezoid, midpoint, simpson); 24 h / 2; trapezoid = (trapezoid + midpoint) / 2; 25 } 26 return EXIT SUCCESS; 27 } 5 8 : f(x) = 4 1+x 2 12 : nmax=32 13 : a b 16 : h : h h 2 h 4, h 8, h 16, h nmax= equation1.c 1 #include <stdio.h> 2 #include <math.h> 3 4 main() 5 { 8

9 C 9 / 21 6 double a, b, c, d; 7 double x1, x2; 8 a=1;b=2;c=1; 9 d = b b 4.0 a c; 10 if (d 0.0) { 11 x1 = ( b+sqrt(d))/(2.0 a); 12 x2 = ( b sqrt(d))/(2.0 a); 13 printf("x1=%-27.20f (%-27.20f)\n", x1, a x1 x1+b x1+c); 14 printf("x2=%-27.20f (%-27.20f)\n", x2, a x2 x2+b x2+c); 15 } 16 else { 17 printf("x1=n.a.\n"); 18 printf("x2=n.a.\n"); 19 } 20 exit(0); 21 } 6 : a,b,c 9 : d : 13 : x 1 ax bx 1 + c 0 14 : 13 x : N.A. a, b, c equation2.c 1 #include <stdio.h> 2 #include <math.h> 3 4 main() 5 { 6 double a, b, c, d; 9

10 C 10 / 21 7 double x1, x2; 8 9 a=1;b=2;c=1; 10 d = b b 4.0 a c; 11 if (d 0.0) { 12 x1 = (fabs(b)+sqrt(d))/(2.0 a); 13 x1 = (b<0.0)?x1: x1; 14 x2 = c/(a x1);/ / 15 printf("x1=%-27.20f (%-27.20f)\n", x1, a x1 x1+b x1+c); 16 printf("x2=%-27.20f (%-27.20f)\n", x2, a x2 x2+b x2+c); 17 } 18 else { 19 printf("x1=n.a.\n"); 20 printf("x2=n.a.\n"); 21 } 22 exit(0); 23 } equation1.c : x 1 = b + d,b 0 x 1 2a x 1 14 : x 2 a, b, c bisect.c 1 / / 2 #include <stdio.h> 3 #include <math.h> 4 #define e #define kukana 1. 6 #define kukanb 2. 10

11 C 11 / float f( float x ) 10 { 11 return x x 2.; 12 } int main() 15 { 16 float xa, xb,xc,y; 17 int i; 18 xa = kukana; 19 xb = kukanb; 20 for( i = 1; i 50; i++ ){ 21 xc = ( xa + xb ) / 2.; 22 y=f(xc); 23 printf( "%d, %f\n", i,xc ); 24 if( fabs( y ) < e ){ 25 return 0; 26 } 27 if( f( xa ) f( xc ) < 0 ){ 28 xb = xc; 29 }else{ 30 xa = xc; 31 } 32 } 33 printf( "\n" ); 34 return 0; 35 } 4 6 : e= ,kukana=1,kukanb=2 kukana kukanb : f(x) = x : xa,xb : 2 11

12 C 12 / : y = f(xc) e 33 : 50 y = f(xc) e * 2 x 2 2 regula.c 1 / / 2 #include <stdio.h> 3 #include <math.h> 4 #define e #define kukana 1. 6 #define kukanb float f( float x ) 9 { 10 return( x x 2. ); 11 } int main() 14 { 15 float xa, xb, xc, y; 16 int i; xa = kukana; 19 xb = kukanb; 20 for( i = 1; i 50; i++ ){ 21 xc = ( xa f( xb ) xb f( xa )) / ( f( xb ) f( xa )); 22 y = f( xc ); 23 printf( " %d, %f\n", i,xc ); 24 if( fabs( y ) < e ){ 25 return 0; 12

13 C 13 / } 27 if( f( xa ) f( xc ) < 0 ){ 28 xb = xc; 29 }else{ 30 xa = xc; 31 } 32 } 33 printf( "\n" ); 34 return 0; 35 } 2 21 (xa, f(xa)), (xb, f(xb)) x Newton x newton.c 1 / / 2 / 0 / 3 #include <stdio.h> 4 #include <math.h> 5 #define e #define initial float f( float x ) 9 { 10 return x x 2; 11 } 12 float diff( float x ) 13

14 C 14 / { 14 return 2. x; 15 } 16 int main() 17 { 18 float xa, xb; 19 int i; xa = initial; 22 for( i = 1; i 50; i++ ){ 23 xb = xa f(xa) /diff(xa); 24 printf( "%d, %f\n", i, xb ); 25 if( fabs( f(xb)) < e ){ 26 return 0; 27 } 28 xa = xb; 29 } 30 printf( " \n" ); 31 return 0; 32 } 6 : initial = 2 x 8 10 : f(x) = x : f (x) 21 : xa = initial 23 : xb = xa f(xa) f (xa) 28 : xa x xb *GnuPlot GnuPlot GnuPlot data newtong.c 1 / / 2 / 0 / 3 / gnuplot / 14

15 C 15 / 21 4 #include <stdio.h> 5 #include <math.h> 6 #define e #define initial float f( float x ) 10 { 11 return x x 2; 12 } 13 float diff( float x ) 14 { 15 return 2. x; 16 } int main() 20 { 21 float xa, xb; 22 int i; xa = initial; 25 for( i = 1; i 50; i++ ){ 26 printf( "%f %f\n",xa,f(xa)); 27 xb = xa f(xa) /diff(xa); 28 printf( "%f 0.0\n", xb); 29 printf("\n"); 30 if( fabs( f(xb)) < e ){ 31 return 0; 32 } 33 xa = xb; 34 } 35 return 0; 36 } 15

16 C 16 / 21 4 f(x) α (f(α) = α) α = e α α 4.1 iteration.c 1 / / 2 #include <stdio.h> 3 #include <math.h> 4 #define begin 0.5 / / 5 #define eps / / 6 7 float f( float x ) 8 { 9 return exp( x); 10 } int main() 13 { 14 float xa, xb; 15 int i; xa = begin; 18 for( i = 1; i 50; i++ ) 19 { 20 xb = f(xa); 21 printf( "%f\n", xb ); 22 if( fabs(xa xb) < eps ) 23 { 24 printf( "%f\n",xb ); 25 return 0; 26 } 27 xa = xb; 16

17 C 17 / } 29 printf( " \n" ); 30 return 0; 31 } 4 : begin = 0.5 x 7 10 : f(x) = e x 17 : xa = begin x : xb = f(xa) xa xb < eps 29 : 50 xa xb eps 4.2 *GnuPlot iterationg.c 1 / / 2 / gnuplot / 3 #include <stdio.h> 4 #include <math.h> 5 #define begin 1.0 / / 6 #define eps / / 7 8 float f( float x ) 9 { 10 return exp( x); 11 } int main() 14 { 15 float xa, xb; 16 int i; xa = begin; 19 for( i = 1; i 50; i++ ) 20 { 21 xb = f(xa); 22 printf( "%f, %f\n", xa,xa ); 17

18 C 18 / printf( "%f, %f\n", xa,xb ); 24 if( fabs(xa xb) < eps ) 25 { 26 return 0; 27 } 28 xa = xb; 29 } 30 return 0; 31 } C Unix > C sample.c > gcc -lm sample.c >./a.out mule 5.2 * >./a.out > test.dat test.dat 5.3 *Gnuplot gnuplot gnuplot > gnuplot test.dat gnuplot > plot test.dat with lines test.dat gnuplot > plot test.dat with dots test.dat 18

19 C 19 / 21 gnuplot view.gp gnuplot > load view.gp view.gp gnuplot 5.4 *Gnuplot newtong.c > gcc -lm newtong.c >./a.out > newton.dat newton.dat gnuplot > plot x*x -2 y = x 2 2 gnuplot > replot 0 y = 0 x gnuplot > replot newton.dat with lines y = x iterationg.c > gcc -lm iterationg.c >./a.out > iteration.dat 19

20 C 20 / 21 iteration.dat gnuplot > plot exp(-x) y = e x gnuplot > replot x y = x gnuplot > replot iteration.dat with lines 20

21 C 21 / f(x) = x [0, 10] x = 0, 1, 2,, 10 2 x = 9 : lagrange.c 6.2 f(x) = e x2 2 f(x)dx 8 8 f(x)dx 2π C 2π sqrt(8*atan(1)) exp( x 2 ) C exp(-x*x/2) : daikei.c 6.3 f(x) = x 3 1 f(x) = x = 0.1 : bisect.c,newton.c 6.4 m = x E sin x f(x) = m + E sin x x = f(x) m = 0.8, E = 0.2 : [0, 1] sin x C sin(x) : iteration.c 21

y = x 4 y = x 8 3 y = x 4 y = x 3. 4 f(x) = x y = f(x) 4 x =,, 3, 4, 5 5 f(x) f() = f() = 3 f(3) = 3 4 f(4) = 4 3 S S = f() + f() + f(3) + f(4) () 4

Simpson H4 BioS. Simpson 3 3 0 x. β α (β α)3 (x α)(x β)dx = () * * x * * ɛ δ y = x 4 y = x 8 3 y = x 4 y = x 3. 4 f(x) = x y = f(x) 4 x =,, 3, 4, 5 5 f(x) f() = f() = 3 f(3) = 3 4 f(4) = 4 *3 S S = f()