中文摘要：

Lanczos方法是求解对称不定线性方程组的有效方法之一,本文利用Lanczos算法求解位场的向下延拓的方程组,而后利用数值计算检验该算法,发现其延拓结果的均方误差与拟合数据的平均残差范数并非单调递减,并且迭代次数较多的结果是一个不可预测、不确定、随机性的输出.为获稳定近似解,采用Lanczos算法求解与位场向下延拓方程组等价的法方程组,实现了位场向下延拓的法方程Lanczos方法,而后再进行数值计算检验,并将本文提出的位场向下延拓方法与Lanczos方法进行比较,结果表明,位场向下延拓的法方程Lanczos方法是一种抑制噪声能力较强,下延稳定的下延方法,且延拓结果具有均方误差与拟合数据的平均残差范数单调递减的良好特性.

英文摘要：

Lanczos method is an efficient and effective method for solving symmetric indefinite linear equations. However, when utilizing the Lanczos algorithm to solve the potential field downward continuation equations, we found both the mean square fitting error and the average residual norm are not generally monotonically decreasing, but unpredictable and random. In this paper, we proposed a new Lanczos-based field downward continuation method to achive more stable approximate solution. Specifically, We solve the normal equations equalent to the original one, instead of solving the original equations directly. In the numerical examples, we compared our method and traditional Lanczos one. The numerical experiments shows that our method is stable for downward continuation and insensitive to data noise. Besides, the mean square fitting error and the average residual norm are now generally monotonical decreasing as the increase of the iteration times, which altogether demonstrate the proposed method is effective and practical