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The next, "corrector" step refines the initial approximation by using the predicted value of the function and another method to interpolate that unknown function's value at the same subsequent point. Predictor–corrector methods for solving ODEs
where is a function : [,), and the initial condition is a given vector. First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted ...
The solution is the weighted average of six increments, where each increment is the product of the size of the interval, , and an estimated slope specified by function f on the right-hand side of the differential equation.
Let us now apply Euler's method again with a different step size to generate a second approximation to y(t n+1). We get a second solution, which we label with a (). Take the new step size to be one half of the original step size, and apply two steps of Euler's method. This second solution is presumably more accurate.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
This differs from the (standard, or forward) Euler method in that the function is evaluated at the end point of the step, instead of the starting point. The backward Euler method is an implicit method , meaning that the formula for the backward Euler method has y n + 1 {\displaystyle y_{n+1}} on both sides, so when applying the backward Euler ...
Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt ...
The function r is called the stability function. [31] It follows from the formula that r is the quotient of two polynomials of degree s if the method has s stages. Explicit methods have a strictly lower triangular matrix A , which implies that det( I − zA ) = 1 and that the stability function is a polynomial.