A comparison of several methods for inverting large symmetric positive definite matrices
A comparison of several methods for inverting large symmetric positive definite matrices
复制标题
几种大型对称正定矩阵求逆方法的比较
DOI：
10.1090/s0025-5718-1964-0166914-1
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发表时间：
1964
影响因子：
2
通讯作者：
M. P. Lietzke
中科院分区：
文献类型：
--
作者：
M. H. Lietzke;R. Stoughton;M. P. Lietzke
参数个数在计算过程中舍入误差的积累，造成显著性的损失，是最严重的限制之一。由于在高速计算机上用最小二乘法对大量参数及其方差进行评价涉及对称正定矩阵的求逆，因此选择一种使舍入误差累积的影响最小化的求逆方案变得很重要。如果存储器空间有限，无法使用双精度运算，则尤其如此。本文比较了求这类矩阵逆的几种直接方法。已尝试考虑的条件和顺序的矩阵被逆的计算逆的精确逆的接近度的影响。比较的矩阵求逆方法有Gauss-Jordan [1]、Choleski [2]、同余变换[3]和秩零化[4]方案。为了公平地比较这些方法，每种方法都用IBM 7090 FORTRAN II（第2版）编程，只使用单精度算术（精确到大约8位小数）。然后使用双精度算术（精确到约16位小数）计算误差指标，以便后一种计算不是限制因素。在下面的讨论中，矩阵的符号将加下划线，而其他符号将不加下划线。
number of parameters the accumulation of round-off error during the course of the computation, resulting in a loss of significance, is one of the most serious restrictions. Since the evaluation of a large number of parameters and their variances by a least squares procedure on a high speed computer involves the inversion of a symmetric, positive-definite matrix it becomes important to choose an inversion scheme in which the effects of the accumulation of round-off error are minimized. This is especially true if limited memory space for storage precludes the use of double precision arithmetic. In this paper a comparison of several direct methods for inverting such matrices is given. An attempt has been made to consider the effects of both condition and order of the matrix to be inverted on the closeness of the computed inverse to the exact inverse. The matrix inversion methods compared are the Gauss-Jordan [1], Choleski [2], congruent transformation [3], and rank annihilation [4] schemes. To give a fair comparison of the methods each was programmed in IBM 7090 FORTRAN II (Version 2) using only single precision arithmetic (good to about 8 decimal digits). The error indicators were then computed using double precision arithmetic (good to about 16 decimal digits) so that the latter calculation was not a limiting factor. In the following discussion the symbols for matrices will be underlined while other symbols will not be.