三、 荣格库塔(Runge-Kutta Methods--RK) 我们可以对modified Euler method进行演化,如下图所示: 欧拉法是用该点以及该点的斜率来预测下一点的值,改良欧拉法则用了n 点与n+1点的斜率,并给每个斜率1/2的权重来预测n+1点的值。欧拉法往后推演,就是荣格库塔法。如下图所示,荣格库塔法在n与n+1点之间取了多...
这通常需要求解一个线性方程。- 改良欧拉法（Modified Euler method）: 使用中间点的斜率来减少预测误差。欧拉法虽然简单，但在实际应用中可能会因为预测误差随步长增加而累积，导致精度下降。因此，对于更精确的计算，通常会使用更高阶的数值方法，如龙格-库塔法（Runge-Kutta methods）。
Euler's methods for differential equations were the first methods to be discovered. They are still of more than historical interest, because their study opens the door for understanding more modern methods and existence results. For complicated problems, often of very high dimension, they are even...
荣格库塔（Runge-Kutta Methods--RK）法进一步提升了欧拉法的精度。该方法通过在n与n+1点之间选取多个点，给各点的斜率配以权重，从而预测n+1点的值。这种方法从欧拉法出发，不断演化，最终形成了丰富的RK法家族。常用的荣格库塔方法包括一阶RK（即欧拉法）、二阶RK（包括modified Euler method或midp...
1.4.3 Errors in Numerical Methods 1.4.4 The Big Three 前一节讨论的斜率场的几何概念与近似微分方程解的基本数值方法密切相关。对于一个初值问题,我们可以通过先画出 t−y 平面上的斜率场,然后从初值 (t0,y0) 开始,画出与斜率场相切的图形来大致了解它的解。在本节中,我们将描述一个使这一想法自动化的...
implictEulermethodsforthe nonlinear systems ofneutral delayintegro-differential equations(NDIDES).The Lipschitznumberofthis paper isafunctionofthevariablet,notconstant.For nonllnearneutral delayintegro-differentialequations,ifthe Lipschitz numbermeet certain ...
方法(Methods) # .copy ( euler : Euler ) : Euler 将euler 的属性拷贝到当前对象。 # .clone () : Euler 返回一个与当前参数相同的新欧拉角。 # .equals ( euler : Euler ) : Boolean 检查euler 是否与当前对象相同。 # .fromArray ( array : Array ) : Euler 长度为3或4的一个 array 。array[...
Inthispaper,onthebaseofthebackgroundofSDEs(Firstly)we introduce several numerical methods which are commonly used and the exponential Runge-Kutta method in ordinary differential equations,whose one order schema is exponential Euler method.We have proved the convergence order is 0.5 when exponential Euler...
ODE1 implements Euler's method. It provides an introduction to numerical methods for ODEs and to the MATLAB suite of ODE solvers. Exponential growth and compound interest are used as examples. Related MATLAB code files can be downloaded from MATLAB Central ...
Strong Predictor-Corrector Euler Methods for Stochastic Differential Equations This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The propose... N Bruti-Liberati,E Platen - 《Stochastics & Dynamics》 被引量:...