Method for processing zero denominator condition of matrix inversion lemma, and method for recursively solving inverse matrix level by level

Abstract

The invention discloses a processing method for overcoming a zero denominator condition of a matrix inversion lemma through limit operation, and a method for recursively solving an inverse matrix level by level. The method comprises the main steps of firstly, decomposing any n-order invertible matrix K into a sum of a diagonal element matrix diag(K) and n rank-1 matrixes, defined in the specification, wherein ui is an ith column of the matrix (ki,i is set to be 0), vi is an n-dimensional row vector (the rest are equal to 0 except vi(i) is equal to 1); secondly, according to a recursion formula defined in the specification, performing calculation, wherein i is greater than or equal to 1 and less than or equal to n, C0 is equal to diag(K)<-1>, and Cn is equal to K<-1>; in the calculation, when an element ki,i of diag(K) is equal to 0, assuming a formula defined in the specification firstly, performing recursion twice, and defining C2=lima->infinity(C2); and in the calculation, when the condition of 1+vi.Ci-1.ui=0 occurs, assuming a formula defined in the specification firstly, performing recursion and defining Ci+1=lima->infinity(Ci+1). According to the recursion method, inverse of a Hilbert matrix can be accurately solved; a Gauss-Jordan elimination process can be realized by left multiplication of an elimination matrix and an original matrix; and in combination with a condensable selection freedom degree of elementary matrix transformation, the elimination matrix is used for preprocessing a finite element stiffness matrix, so that a condition number can be greatly reduced.

Description

technical field [0001] The invention relates to a matrix recursion inversion method, which uses limit processing to overcome the zero denominator of the matrix inversion lemma, and the recursion formula can efficiently and accurately solve the inverse matrix of a Hilbert matrix (a typical seriously ill-conditioned matrix). Background technique [0002] Matrix inversion methods include adjoint matrix method, matrix decomposition method, block matrix method, elementary transformation method, edge addition method, iterative method for approximate inverse matrix, etc.; [0003] The calculation amount of the adjoint matrix method is too large, and generally cannot be used for the actual calculation of the inversion of a high-order number matrix; [0004] Common matrix decomposition methods include LU decomposition, QR decomposition, singular value decomposition, etc. Since the inversion of triangular matrix and unitary matrix is ​​relatively simple, the inverse of the original ma...

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Problems solved by technology

[0006] The edge-adding method is based on the matrix inversion lemma, which can be calculated step by step, but it cannot be calculated when the main element is zero or there is a zero denominator in the calculation;

[0007] The iterative method of finding the approximate inverse matrix of a matrix is ​​invalid for ill-conditioned matrices with large condition numbers;

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