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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
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 .
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.
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 .
Gauss–Kronrod formulas are extensions of the Gauss quadrature formulas generated by adding + points to an -point rule in such a way that the resulting rule is exact for polynomials of degree less than or equal to + (Laurie (1997, p. 1133); the corresponding Gauss rule is of order ).
The modified midpoint method by itself is a second-order method, ... (1983), "Smoothing the extrapolated midpoint rule", Numerische Mathematik, 41 (2): 165 ...