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ISBN 0-521-00794-1. ... It follows from the above formulas for the errors of the midpoint and trapezoidal rule that ... Averaging of Simpson's 1/3 rule ...
The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable ...
This is called the midpoint rule or rectangle rule ... of which the rectangle rule and the trapezoidal rule are examples. Simpson's rule, ... ISBN 978-0-521-88068-8;
A simple predictor–corrector method (known as Heun's method) can be constructed from the Euler method (an explicit method) and the trapezoidal rule (an implicit method). Consider the differential equation ′ = (,), =, and denote the step size by .
The zeroeth extrapolation, R(n, 0), is equivalent to the trapezoidal rule with 2 n + 1 points; the first extrapolation, R(n, 1), is equivalent to Simpson's rule with 2 n + 1 points. The second extrapolation, R(n, 2), is equivalent to Boole's rule with 2 n + 1 points. The further extrapolations differ from Newton-Cotes formulas.
Adaptive Simpson's method, also called adaptive Simpson's rule, is a method of numerical integration proposed by G.F. Kuncir in 1962. [1] It is probably the first recursive adaptive algorithm for numerical integration to appear in print, [ 2 ] although more modern adaptive methods based on Gauss–Kronrod quadrature and Clenshaw–Curtis ...
Trapezoidal rule — second-order method, based on (piecewise) linear approximation; Simpson's rule — fourth-order method, based on (piecewise) quadratic approximation Adaptive Simpson's method; Boole's rule — sixth-order method, based on the values at five equidistant points; Newton–Cotes formulas — generalizes the above methods
It is assumed that the value of a function f defined on [,] is known at + equally spaced points: < < <.There are two classes of Newton–Cotes quadrature: they are called "closed" when = and =, i.e. they use the function values at the interval endpoints, and "open" when > and <, i.e. they do not use the function values at the endpoints.