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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 ,
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 .
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.
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 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 ...
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; Trapezoidal rule, numerical method based on the average of the left and right Riemann sum
The Gauss–Legendre method of order two is the implicit midpoint rule. Its Butcher tableau is: 1/2: 1/2 1 The Gauss–Legendre method of order four has Butcher ...