範例EX4-1 利用梯行法計算 e^x , x=0 to 1 的積分值

梯形數值積分法

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若 f(x) 在 [a,b] 中連續，則可積。 梯形數值積分法 (trapezoidal rule) 是求定積分之近似值的一種方法， 其公式如下

其中 n 表示將 [a,b] 等分成 n 段，令 

而且 x₀ = a、 x_n = b，一般而言 x_k = a + k h。

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計算 

#include <stdio.h>
#include <math.h>

/* 示範梯形數值積分法  (trapeziodal.c) */
#define STEP 10

double f(double);

main() {
    int n, k, i;
    double h, T, a, b, x;
    T = 0;
    a = 0;
    b = 1;
    for (n=1, i=0; i < STEP; ++i) {
        n = 2*n;
        h = (b-a)/n;
        x = a;
        T = f(x)/2;
        for (k=1; k<n ; ++k) {
           x = x + h;
           T = T + f(x);
        }
        T = T + f(b)/2;
        T = h*T;
        printf("T_%-5d = %17.14f.\n", n, T);
    }
}

double f(double x) {
    return exp(x);
}

T_2     =  1.75393109246483.                                                                                                                   

T_4     =  1.72722190455752.                                                                                                                   

T_8     =  1.72051859216430.                                                                                                                   

T_16    =  1.71884112857999.                                                                                                                   

T_32    =  1.71842166031633.                                                                                                                   

T_64    =  1.71831678685009.                                                                                                                   

T_128   =  1.71829056808348.                                                                                                                   

T_256   =  1.71828401336682.                                                                                                                   

T_512   =  1.71828237468609.                                                                                                                   

T_1024  =  1.71828196501581.                                                                                                                   

                                                                                                                                               

                                                                                                                                               

...Program finished with exit code 0                                                                                                           

Press ENTER to exit console.