In many cases, it is more accurate than the trapezoidal rule and others.

In C++, one can implement the trapezoidal rule as follows.

For definiteness, the trapezoidal rule was used but this could have been any other method.

This method is known as the "trapezoidal rule" for differential equations.

The trapezoidal rule often converges very quickly for periodic functions.

The trapezoidal rule is a collocation method (as discussed in that article).

The trapezoidal rule is easily implemented in a spreadsheet.

This includes the left-half plane, so the trapezoidal rule is A-stable.

Thus, the trapezoidal rule is a second-order method.

In fact, the region of absolute stability for the trapezoidal rule is precisely the left-half plane.

Another example for an implicit Runge-Kutta method is the trapezoidal rule.

This is called the trapezoidal rule.

The (composite) trapezoidal rule can be implemented in Python as follows:

It explains the superior performance of the trapezoidal rule on smooth periodic functions and is used in certain extrapolation methods.

Now combine both formulas and use that and to get the trapezoidal rule for solving ordinary differential equations.

Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, which can be analyzed in various ways.

The trapezoidal rule has the smallest error constant amongst the A-stable linear multistep methods of order 2.

The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result:

For various classes of functions that are not twice-differentiable, the trapezoidal rule has sharper bounds than Simpson's rule.

Due to the dependence on the length of 'x' in the trapezoidal rule, the area estimation is highly dependent on the blood/plasma sampling schedule.

(see trapezoidal rule).

An example of a second-order A-stable method is the trapezoidal rule mentioned above, which can also be considered as a linear multistep method.

An alternate method is computing lower and upper bounds with the trapezoidal rule; a mesh of progressively finer sizes allows for arbitrary accuracy.

The second Dahlquist barrier states that the trapezoidal rule is the most accurate amongst the A-stable linear multistep methods.

However for various classes of rougher functions (ones with weaker smoothness conditions), the trapezoidal rule has faster convergence in general than Simpson's rule.