trapeze积分算法

Trapezoidal rule is a numerical integration method that approximates the definite integral of a function by dividing the area under the curve into trapezoids and summing their areas. The formula for the trapezoidal rule is:

∫ a^b f(x) dx ≈ (b-a)/2 * [f(a) + f(b)]

where a and b are the limits of integration, and f(x) is the integrand.

The trapezoidal rule approximates the area under the curve by approximating it with a series of trapezoids. The area of each trapezoid is given by the average of the heights of its two bases, multiplied by the length of its two bases.

To use the trapezoidal rule, the interval [a,b] is divided into n subintervals of equal width h=(b-a)/n. The formula for the trapezoidal rule can then be written as:

∫ a^b f(x) dx ≈ h/2 * [f(a) + 2f(a+h) + 2f(a+2h) + ... + 2f(b-h) + f(b)]

where h = (b-a)/n, and the terms 2f(a+h), 2f(a+2h), ..., 2f(b-h) represent twice the value of the function at the midpoints of the subintervals.

The trapezoidal rule is a simple and easy-to-use method for numerical integration, but it has limitations. It can only provide an approximation of the integral, and the accuracy of the approximation depends on the number of subintervals used. In general, the more subintervals used, the more accurate the approximation. However, using too many subintervals can lead to numerical instability and round-off errors.