本文对缩张算法进行了多处改进，其中最主要的是与基于数值微分的改良高斯-牛顿法相结合，形成了曲线与曲面拟合的改进缩张算法。新算法不需提供非线性方程的偏导数，但可利用近似偏导数指导搜索方向，提高搜索效率。改进缩张算法减少了复杂非线性方程拟合的运算负荷，提高了利用度点反馈调节搜索步长和区域的敏感度，增强了跳出局部最优陷阱的能力。新算法可在较大初值范围内实现全局最优拟合。本文以3个不同类型的实例验证了新算法的拟合功效。编制了可动态显示拟合过程的Matlab程序，可供曲线和曲面拟合应用者使用。
The improved contraction-expansion algorithm was established mainly by combining previous C-E algorithm with Levenberg-Marquardt method based on numerical derivative technique. The new algorithm could use differential information to guide searching direction without providing derivatives, so as to simplify fitting process and raise searching efficiency. The new C-E algorithm greatly reduced computing load for those multiple parameters situation and strengthened the ability to jump out off pitfalls. The feedback system was more sensitive than previous one in using scatter-spread of the spring points to adjust search step length. Improved effects were obtained in fitting various nonlinear equations with new algorithm. Some examples were given to demonstrate the power in curve and surface fitting.
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