Search results
Results from the WOW.Com Content Network
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 the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area.
In fact, the region of absolute stability for the trapezoidal rule is precisely the left-half plane. This means that if the trapezoidal rule is applied to the linear test equation y' = λy, the numerical solution decays to zero if and only if the exact solution does. However, the decay of the numerical solution can be many orders of magnitude ...
The ancient Babylonians used the trapezoidal rule to integrate the motion of Jupiter along the ecliptic. [3] Antique method to find the Geometric mean. For a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side = (the Geometric mean of a and b).
While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.
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 value.
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly.
You can find instant answers on our AOL Mail help page. Should you need additional assistance we have experts available around the clock at 800-730-2563.
Consider , the exact solution to a differential equation in an appropriate normed space (, | | | |). Consider a numerical approximation u h {\displaystyle u_{h}} , where h {\displaystyle h} is a parameter characterizing the approximation, such as the step size in a finite difference scheme or the diameter of the cells in a finite element method .