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The midpoint method computes + so that the red chord is approximately parallel to the tangent line at the midpoint (the green line). In numerical analysis , a branch of applied mathematics , the midpoint method is a one-step method for numerically solving the differential equation ,
If = (+) / for all i, the method is the midpoint rule [2] [3] and gives a middle Riemann sum. If f ( x i ∗ ) = sup f ( [ x i − 1 , x i ] ) {\displaystyle f(x_{i}^{*})=\sup f([x_{i-1},x_{i}])} (that is, the supremum of f {\textstyle f} over [ x i − 1 , x i ] {\displaystyle [x_{i-1},x_{i}]} ), the method is the upper rule and gives an upper ...
In numerical analysis, Romberg's method [1] is used to estimate the definite integral by applying Richardson extrapolation [2] repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array.
A quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral.
A loose rule of thumb dictates that stiff differential equations require the use of implicit schemes, whereas non-stiff problems can be solved more efficiently with explicit schemes. The so-called general linear methods (GLMs) are a generalization of the above two large classes of methods.
The Gauss–Legendre method of order two is the implicit midpoint rule. Its Butcher tableau is: 1/2: ... The method of order 2 is just an implicit midpoint method.
The implicit midpoint method is of second order. It is the simplest method in the class of collocation methods known as the Gauss-Legendre methods . It is a symplectic integrator .
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.