Lagrange插值方法的一致收敛性及算法研究

Uniform Convergence and Algorithm of Lagrange Interpolation Method

为解决Lagrange插值多项式并非对任意的连续函数都一致收敛的问题,通过一点修正法的Bernstein型算子与两点修正法的三角插值多项式算子、代数多项式算子改善其收敛性。利用一点修正法构造了一个代数多项式算子F_n(f;x)(即Bernstein型算子),利用两点修正法构造了三角插值多项式算子T_n (f;r,x)(r为自然数)、代数多项式算子G_(n,R) (f;x)(R为自然数),通过C语言编写具体程序,分析其逼近效果的优劣。结果表明:与Lagrange插值多项式算子相比,采用一点修正法与两点修正法优化的Bernstein型算子、三角插值多项式算子与代数多项式算子的运行速度更快,误差值更小,收敛结果与精确解更逼近。

To solve the problem that Lagrange interpolation polynomials do not uniformly converge for any continuous function,the convergence is improved by the Bernstein-type operator with the one-point correction method and the trigonometric interpolation polynomial operators and algebraic polynomial operator with the two-point correction method.The one-point correction method is used to construct an algebraic polynomial operator F_(n)(f;x)(Bernstein-type operator),two-point correction method is used to construct trigonometric interpolation polynomial operator T_(n)(f;r,x)(r is a natural number),and algebraic polynomial operator G_(n,R)(f;x)(R is a natural number),and a specific program is written in C language to analyze the superiority and inferiority of its approximation effects.The results show that compared with Lagrange interpolation polynomial operators,the Bernstein-type operator,trigonometric interpolation polynomial operator and algebraic polynomial operator optimized by the one-point correction method and the two-point correction method have faster running speed,smaller error values,and closer convergence results to the exact solution.