单摆周期近似公式的数值优化

Approximate Simple Pendulum Period Formulae: Optimization in a Numerical Way

运用MATLAB中的最小二乘曲线拟合方法,在[0,π/2]及[0,π]区间对现有单摆周期近似公式分别进行了数值修正,得到与原公式具有相同的形式但不同系数的修正公式。对原公式及修正后公式的方差和及相对误差进行了比较。结果表明,原近似公式的精确度得到了显著的改进。此外,作者发现,原公式及修正公式在[0,π/2]区间的近似程度要优于[0,π]区间,并且,在[0,π/2]区间具有较高精确度的公式在[0,π]区间未必同样精确,从而揭示出了这些公式近似程度对讨论区间的依赖性。

Approximate formulae of period of a simple pendulum with oscillation amplitude in and were optimized in a numerical way with the function of Lsqcurvefit in MATLAB.New formulae with the same forms as but different coefficients from the original ones were obtained correspondingly.The sums of squared difference and relative error of each original formula and its corresponding revised one with the exact value of the period of a simple pendulum were compared.The results show that every original formula has been remarkably improved.Furthermore,it is found that a formula（no matter it is an original one or a revised one） performs better inthan in.Besides,a formula that performs relatively better in is not necessarily better in,showing the dependence of the precision of a formula on the interval.