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# 台形則（数値積分）

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# 台形則とは

• 数値積分の解法の一つ
• 関数$f(x)$において、微小区間$[x_0,x_1]$内での関数値を一次方程式で近似する

# 算法

$$\int_{x_0}^{x_1}f(x)dx=\frac{h}{2}(y_0+y_1)$$
となるので、区間$[a,b]$内の積分は各微小区間内での積分値の和で表せるから、
$$\int_a^bf(x)dx=\frac{h}{2}(y_0+y_1)+\frac{h}{2}(y_1+y_2)+\cdots+\frac{h}{2}(y_{n-1}+y_n)$$$$=\frac{h}{2}(y_0+2y_1+2y_2+\cdots+2y_{n-1}+y_n)$$

# サンプルコード

$f(x)=\sqrt{1-x^2}$において、区間$[0,1]$の定積分の値を求めるプログラム。

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

double f (double x) {
return sqrt(1-x*x);
}

/* 台形則（区間[a,b]をn分割） */
double trapezoidal_rule (double a, double b, int n) {
double h;
int i;
double value=0;

h = (b - a) / n;   // 区間幅の計算
for (i = 0; i <= n; i++) {
if (i == 0 || i == n) value += f(a + i*h);
else value += 2 * f(a + i*h);
}
value = value*h/2;
return value;
}

int main (void) {
printf("Analytical solution: %f\n", M_PI/4);
printf("Numerical solution : %f\n", trapezoidal_rule(0, 1, 4));
return 0;
}


Analytical solution: 0.785398
Numerical solution : 0.748927


# 特徴

• 全区間の誤差 $$E=\frac{h^2(b-a)}{12}f^{\prime\prime}(\xi) \ \ , \ \ (a<\xi<b)$$
• 分割数を倍にすると誤差は$1/4$に減少
• でも、区間数を増やしすぎると丸め誤差が増加する
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