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Trapezoidal rule. The function f (x) (in blue) is approximated by a linear function (in red). In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule) [a] is a technique for numerical integration, i.e., approximating the definite integral: The trapezoidal rule works by approximating the region under the graph of ...
Trapezoidal rule (differential equations) In numerical analysis and scientific computing, the trapezoidal rule is a numerical method to solve ordinary differential equations derived from the trapezoidal rule for computing integrals. The trapezoidal rule is an implicit second-order method, which can be considered as both a Runge–Kutta method ...
Heun's method. In mathematics and computational science, Heun's method may refer to the improved[1] or modified Euler's method (that is, the explicit trapezoidal rule[2]), or a similar two-stage Runge–Kutta method. It is named after Karl Heun and is a numerical procedure for solving ordinary differential equations (ODEs) with a given initial ...
Upper and lower methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively. The values of the sums converge as the subintervals halve from top-left to bottom-right. In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum.
Another example for an implicit Runge–Kutta method is the trapezoidal rule. Its Butcher tableau is: The trapezoidal rule is a collocation method (as discussed in that article). All collocation methods are implicit Runge–Kutta methods, but not all implicit Runge–Kutta methods are collocation methods.
Romberg's method. 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. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally ...
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 . First, the predictor step: starting from the current value , calculate an initial guess value via the Euler ...
The Newmark-beta method is a method of numerical integration used to solve certain differential equations. It is widely used in numerical evaluation of the dynamic response of structures and solids such as in finite element analysis to model dynamic systems. The method is named after Nathan M. Newmark, [1] former Professor of Civil Engineering ...