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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 ,
Euler method and midpoint method, related methods for solving differential equations; Lebesgue integration; Riemann integral, limit of Riemann sums as the partition becomes infinitely fine; Simpson's rule, a powerful numerical method more powerful than basic Riemann sums or even the Trapezoidal rule
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 rule has similar geometric properties. To summarize: the pendulum example shows that, besides the explicit and implicit Euler methods not being good choices of method to solve the problem, the symplectic Euler method and implicit midpoint rule agree well with the exact flow of the system, with the midpoint rule agreeing ...
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 .
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 .
The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb. [2] "Quadrature" is a historical mathematical term that means calculating area. Quadrature problems have served as one of the main sources of mathematical analysis.