高阶微分方程整数型通用算法的实际实现与误差界
Practical implementation and error bounds of integer-type general algorithm for higher order differential equations
AI总结：
提出一种仅使用整数四则运算求解Fuchs型高阶线性常微分方程有限范数解的算法，并给出误差上界和数值实验验证。
摘要：
在我们之前的论文中，我们提出了一种算法，用于获得Fuchs型高阶线性常微分方程 [\sum_m p_m (x) (d/dx)^m] f(x) = 0（其中p_m是系数为有理数的多项式）的有限范数解，该算法仅使用整数的四则运算，并证明了其有效性。对于任何非负整数k，数学上保证该方法能在某些条件下产生所有满足 \int |f(x)|^2 (x^2+1)^k dx < \infty 的解。我们在实际程序中实现了该算法。其中使用的整数型准正交化可以抑制计算爆炸。此外，我们给出了误差的上限。我们还给出了一些数值实验的结果，并将其与相应的精确解析解进行比较，结果表明所提出的算法能够成功获得高精度的解（仅使用整数的算术运算）。
英文摘要：
In our preceding paper, we have proposed an algorithm for obtaining finite-norm solutions of higher-order linear ordinary differential equations of the Fuchsian type [\sum_m p_m (x) (d/dx)^m] f(x) = 0 (where p_m is a polynomial with rational-number-valued coefficients), by using only the four arithmetical operations on integers, and we proved its validity. For any nonnegative integer k, it is guaranteed mathematically that this method can produce all the solutions satisfying \int |f(x)|^2 (x^2+1)^k dx < \infty, under some conditions. We materialize this algorithm in practical procedures. An interger-type quasi-orthogonalization used there can suppress the explosion of calculations. Moreover, we give an upper limit of the errors. We also give some results of numerical experiments and compare them with the corresponding exact analytical solutions, which show that the proposed algorithm is successful in yielding solutions with high accuracy (using only arithmetical operations on integers).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
