CN105630741A - Improved module for solving inverse matrixes of matrixes according to bit replacement method - Google Patents

Abstract

The invention discloses an improved module for solving inverse matrixes of matrixes according to a bit replacement method. The module comprises a boundary element obtaining unit, an inner element obtaining unit, a lower triangular matrix obtaining unit, an upper triangular matrix obtaining unit and an inverse matrix obtaining unit, wherein the boundary element obtaining unit is used for obtaining the boundary elements of a reduced coefficient matrix of a matrix to be inversed; the inner element obtaining unit is used for obtaining the inner elements of the reduced coefficient matrix of the matrix to be inversed; the lower triangular matrix obtaining unit is used for obtaining an inverse matrix of a lower triangular matrix obtained through decomposing the matrix to be inversed; the upper triangular matrix obtaining unit is used for obtaining an inverse matrix of an upper triangular matrix obtained through decomposing the matrix to be inversed; and the inverse matrix obtaining unit is used for obtaining an inverse matrix of the matrix to be inversed. The module disclosed in the invention is capable of reducing the complexity of the matrix inversion computation, compressing the storage space and improving the parallelizability of the inversion computation so as to rapidly and efficiently complete the matrix inversion computation.

Description

The step-by-step Shift Method of a kind of improvement seeks matrix inversion matrix module

Technical field

The present invention relates to matrix operations, particularly relate to a kind of novel plug seeking matrix inversion matrix based on the step-by-step Shift Method improved.

Background technology

Matrix operations is the basic problem of science and engineering calculation, and matrix operations is to describe the indispensable instrument of mathematical relationship in many engineering problems, has and can be attributed to matrix operations greatly in scientific algorithm. Matrix operations includes matrix multiplication, matrix decomposition, matrix inversion etc., and the difficult point of matrix operations is in that matrix inversion. All it is widely used in many signal processing, image procossing and communication aspects in matrix inversion operation at present, there is important engineering significance. These engineering fields often have high-throughput, requirement of real-time, and how efficent use of resources realizes quickly extensive matrix inversion operation becomes emphasis and the difficult point of design.

The common low order matrix inversion algorithms of current matrix mainly has: definition method, the adjoint matrix tactical deployment of troops, elementary row rank transformation method, equivalent standard type method, Hamilton-Caley method, block matrix method, additionally, have Nonlinear descriptor systems to invert fast algorithm, Toeplitz matrix inversion fast algorithm etc. for Special matrix conventional method of inverting. The above algorithm proposes from mathematical angle, inverts for high accuracy extensive in engineer applied, no special matrix, and conventional Gauss-Jordan elimination approach, matrix decomposition method carry out matrix inversion operation.

In any of the above inversion technique to no special matrix, the adjoint matrix tactical deployment of troops needs to ask substantial amounts of determinant, and each determinant nearly all to calculate all of matrix element, computationally intensive, and the demand of memory space is also very big; Elementary transform method needs increase order of matrix number and relate to the matrixing operation of complexity, and memory space requires and computation complexity is all very high; And matrix inversion algorithm conventional in engineering, Gauss-Jordan elimination approach is owing to having used substantial amounts of sorting operation, and amount of calculation is bigger, it is achieved process not easily adopts parallel computation, and real-time is not strong; And matrix decomposition is inverted the shortcoming of rule customer service said method, it is triangular matrix or the product of triangular matrix and Special matrix by matrix decomposition to be inverted, utilizes triangular matrix to invert simple feature, improve efficiency and the concurrency of inversion operation. At present, conventional matrix decomposition inversion algorithms has QR decomposition method and LU factorization, wherein QR decompose comprise a large amount of evolution, square, the complex calculation such as division, LU decomposes then to be needed to use the process such as unit, iteration that disappears, both decomposes inversion algorithms all needs first to be decomposed into by original matrix two triangular matrixes, again triangular matrix is inverted, product finally by triangle inverse matrix obtains final inverse matrix result, operand is bigger, there is substantial amounts of iterative process centre, operation time is longer, and degree of parallelism is not high, and the memory space needed is also bigger.

To sum up, current existing matrix inversion techniques still has certain limitation in engineer applied, and key factor is some deficiency following:

First, many dependence recursion alternative manners, serial requires higher, carries out concurrent operation difficulty bigger, it is difficult to meet requirement of real-time in engineer applied.

Second, computational complexity is higher, and operand is relatively big, calculates the time longer.

3rd, computational space complexity is higher, and the memory space taken is relatively big, not high at concrete engineering application resource utilization.

Summary of the invention

The present invention is for avoiding the deficiencies in the prior art part, the step-by-step Shift Method proposing a kind of new improvement seeks matrix inversion matrix norm block, to reducing computational complexity, compresses memory space, improve inversion operation can concurrency, and then faster, be efficiently completed matrix inversion operation.

The present invention solves that technical problem adopts the following technical scheme that

The step-by-step Shift Method of a kind of improvement of the present invention asks the feature of matrix inversion matrix norm block to include: boundary element acquiring unit, inner element acquiring unit, lower triangular matrix acquiring unit, upper triangular matrix acquiring unit and inverse matrix acquiring unit;

Described boundary element acquiring unit is used for obtaining matrix to be inverted $A=\left(\begin{array}{cccccc}{a}_{11}& a_{12}& ...& a_{1i}& ...& a_{1M}\\ a_{21}& a_{22}& ...& a_{2i}& ...& a_{2M}\\ .& .& & .& & .\\ .& .& ...& .& ...& .\\ .& .& & .& & .\\ a_{j1}& a_{j2}& ...& a_{ji}& ...& a_{jM}\\ .& .& & .& & .\\ .& .& ...& .& ...& .\\ .& .& & .& & .\\ a_{M1}& a_{M2}& ...& a_{Mi}& ...& a_{MM}\end{array}\right)$ The boundary element of reduction coefficient matrix N; Described matrix A to be inverted is that to meet each rank the Principal Minor Sequence be not the M rank square formation of 0; a_jiRepresent jth row the i-th column element; I, j=1,2,3 ..., M;

Described inner element acquiring unit is for obtaining the inner element of the reduction coefficient matrix N of matrix A to be inverted; Thus obtaining reduction coefficient matrix N;

Described lower triangular matrix acquiring unit is for obtaining the inverse matrix of matrix A triangle decomposition gained lower triangular matrix to be inverted;

Described upper triangular matrix acquiring unit is for obtaining the inverse matrix of matrix A triangle decomposition gained upper triangular matrix to be inverted;

Described inverse matrix acquiring unit is for obtaining the inverse matrix A of matrix A to be inverted^-1��

The step-by-step Shift Method of improvement of the present invention asks the feature of matrix inversion matrix norm block to lie also in:

Described boundary element acquiring unit utilizes formula (1) to obtain the boundary element n of reduction coefficient matrix N according to described matrix A to be inverted_1i��0And n_j1��0:

$\left\{\begin{array}{c}{n}_{1i\·0}=a_{1i}\\ n_{j1\·0}=a_{j1}\end{array}\right.---\left(1\right)$

Described inner element acquiring unit utilizes formula (2) to obtain the diagonal element n of reduction coefficient matrix N_{ii��(i-1)}:

$n_{ii\·(i-1)}=a_{ii}-\underset{k=1}{\overset{i-1}{\Σ}}\frac{n_{ik\·(k-1)}\·n_{ki\·(k-1)}}{n_{kk\·(k-1)}}---\left(2\right)$

In formula (2), k=2,3 ... i-1;

Described inner element acquiring unit utilizes formula (3) to obtain the lower triangle element n of reduction coefficient matrix N_{ji��(i-1)}:

$n_{ji\&CenterDot;(i-1)}=\left\{\begin{array}{c}{a}_{ji}-\underset{k=1}{\overset{i-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}},n_{ii\&CenterDot;(i-1)}>0\\ -(a_{ji}-\underset{k=1}{\overset{i-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}}),n_{ii\&CenterDot;(i-1)}<0\end{array}\right.---\left(3\right)$

In formula (3), i=2,3 ..., M-1; J=i+1, i+2 ..., M;

Described inner element acquiring unit utilizes formula (4) to obtain the upper triangle element n of reduction coefficient matrix N_{ji��(j-1)}:

$n_{ji\&CenterDot;(j-1)}=\left\{\begin{array}{c}{a}_{ji}-\underset{k=1}{\overset{j-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}},n_{jj\&CenterDot;(j-1)}>0\\ -(a_{ji}-\underset{k=1}{\overset{j-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}}),n_{jj\&CenterDot;(j-1)}<0\end{array}\right.---\left(4\right)$

In formula (4), i=j+1, j+2 ..., M; J=2,3 ..., M-1;

Thus obtaining reduction coefficient matrix N it is: $A=\left(\begin{array}{cccccc}{n}_{11\&CenterDot;0}& ...& n_{1(i-1)\&CenterDot;0}& n_{1i\&CenterDot;0}& ...& n_{1M\&CenterDot;0}\\ n_{21\&CenterDot;0}& ...& n_{2(i-1)\&CenterDot;1}& n_{2i\&CenterDot;1}& ...& n_{2M\&CenterDot;1}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ n_{j1\&CenterDot;0}& ...& n_{j(i-1)\&CenterDot;(i-2)}& n_{ji\&CenterDot;(i-1)}& ...& n_{jM\&CenterDot;(j-1)}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ n_{M1\&CenterDot;0}& ...& n_{M(i-1)\&CenterDot;(i-2)}& n_{Mi\&CenterDot;(i-1)}& ...& n_{MM\&CenterDot;M-1}\end{array}\right)$

Described lower triangular matrix acquiring unit, according to described reduction coefficient matrix N, utilizes formula (5) to obtain lower triangular matrix

$\begin{array}{c}\left\{\begin{array}{c}{l}_{ii}=\frac{1}{\sqrt{n_{ii\&CenterDot;(i-1)}}},n_{ii\&CenterDot;(i-1)}>0;i=1,2,...,M\\ l_{ii}=-\frac{1}{\sqrt{\left|n_{ii\&CenterDot;(i-1)}\right|}},n_{ii\&CenterDot;(i-1)}<0;i=1,2,...,M\end{array}\right.\\ l_{ji}=-l_{jj}\&CenterDot;\underset{k=i}{\overset{j-1}{\&Sigma;}}{n}_{jk\&CenterDot;(k-1)}\&CenterDot;l_{kk}\&CenterDot;l_{ki},i=1,2,...,M-1;j=i+1,i+2,...,M\end{array}---\left(5\right)$

Described upper triangular matrix acquiring unit, according to described reduction coefficient matrix N, utilizes formula (6) to obtain upper triangular matrix

$\begin{array}{c}\left\{\begin{array}{c}{r}_{ii}=\frac{1}{\sqrt{n_{ii\&CenterDot;(i-1)}}},n_{ii\&CenterDot;(i-1)}>0;i=1,2,...,M\\ r_{ii}=\frac{1}{\sqrt{\left|n_{ii\&CenterDot;(i-1)}\right|}},n_{ii\&CenterDot;(i-1)}<0;i=1,2,...,M\end{array}\right.\\ r_{ji}=-r_{ii}\&CenterDot;\underset{k=j}{\overset{i-1}{\&Sigma;}}{n}_{ki\&CenterDot;(k-1)}\&CenterDot;r_{kk}\&CenterDot;r_{jk},j=1,2,...,M-1;i=j+1,j+2,...,M;\end{array}---\left(6\right)$

Described inverse matrix acquiring unit utilizes formula (7) to obtain the inverse matrix A of described matrix A to be inverted^-1:

A^-1=R^-1��L^-1(7)��

Compared with the prior art, the present invention has the beneficial effect that:

1, the matrix inversion module that the present invention proposes, in whole inversion process, only in inner element acquiring unit, lower triangular matrix acquiring unit and in upper triangular matrix acquiring unit, need the diagonal element of reduction coefficient matrix does evolution or inverse (division), the remainder of whole module of inverting pertains only to simple multiply-add operation process, avoid substantial amounts of evolution in prior art, square, ask the computing such as norm, division, greatly simplifie calculating process.

2, the matrix inversion module that the present invention proposes, by obtaining in boundary element acquiring unit and inner element acquiring unit and creating reduction coefficient matrix, decompose serial iteration process, make while obtaining reduction coefficient matrix element, the reduction coefficient matrix element that can be obtained by prime inside upper triangular matrix acquiring unit and lower triangular matrix acquiring unit, on Parallel implementation, lower triangular matrix element, overcome existing matrix decomposition inversion techniques, owing to adopting iteration serial computing to calculate one by one, what lower triangular matrix element caused can the not strong problem of concurrency.

3, the matrix decomposition module that the present invention proposes, it is equally based on the algorithm idea that matrix decomposition is inverted, but carry out triangle decomposition without first treating finding the inverse matrix, obtain two triangular matrixes successively, seek the inverse matrix of the two triangular matrix again, but the inverse matrix of the two triangular matrix can be directly sought by reduction coefficient matrix, and inner element obtain that unit is internal can executed in parallel, upper triangular matrix acquiring unit and lower triangular matrix acquiring unit can executed in parallel, whole inside modules calculating process is very simple, thus operation time is very short, solve in existing matrix inversion techniques, the problem that operation time complexity is higher.

4, the matrix inversion module that the present invention proposes, replaces algorithm based on the step-by-step improved so that whole module is except the memory space inputted shared by matrix to be inverted, it is not necessary to take memory space extra in a large number. Owing to existing Matrix Technology to take substantial amounts of memory space, cause, in ultra-large matrix inversion engineer applied, there is certain limitation, the invention solves this problem.

Detailed description of the invention

In the present embodiment, a kind of step-by-step Shift Method based on improvement asks matrix inversion matrix norm block to include: boundary element acquiring unit, inner element acquiring unit, lower triangular matrix acquiring unit, upper triangular matrix acquiring unit and inverse matrix acquiring unit; Its solution throughway is the matrix to be inverted that 1 basis is given, seeks the boundary element of its reduction coefficient matrix; 2 boundary elements according to matrix to be inverted He its reduction coefficient matrix, seek the inner element of its reduction coefficient matrix; Matrix to be inverted is considered as the product of two triangular matrixes by 3, according to reduction coefficient matrix, solves the inverse matrix of the two triangular matrix; Two triangle inverse matrixs are carried out matrix multiplication by 4, obtain inverse of a matrix matrix to be inverted, thus completing inverting of whole matrix. Specifically,

Boundary element acquiring unit is used for obtaining matrix to be inverted $A=\left(\begin{array}{cccccc}{a}_{11}& a_{12}& ...& a_{1i}& ...& a_{1M}\\ a_{21}& a_{22}& ...& a_{2i}& ...& a_{2M}\\ .& .& & .& & .\\ .& .& ...& .& ...& .\\ .& .& & .& & .\\ a_{j1}& a_{j2}& ...& a_{ji}& ...& a_{jM}\\ .& .& & .& & .\\ .& .& ...& .& ...& .\\ .& .& & .& & .\\ a_{M1}& a_{M2}& ...& a_{Mi}& ...& a_{MM}\end{array}\right)$ The boundary element of reduction coefficient matrix N; Matrix A to be inverted is that to meet each rank the Principal Minor Sequence be not the M rank square formation of 0; a_jiRepresent jth row the i-th column element; I, j=1,2,3 ..., M; In the present embodiment, the matrix A to be inverted that employing Matlab creates is each rank the Principal Minor Sequence randomly generated is not the 8 rank square formations of 0:

$A=\left(\begin{array}{cccccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}& a_{16}& a_{17}& a_{18}\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}& a_{26}& a_{27}& a_{28}\\ a_{31}& a_{32}& a_{33}& a_{34}& a_{35}& a_{36}& a_{37}& a_{38}\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}& a_{46}& a_{47}& a_{48}\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}& a_{56}& a_{57}& a_{58}\\ a_{61}& a_{62}& a_{63}& a_{64}& a_{65}& a_{66}& a_{67}& a_{68}\\ a_{71}& a_{72}& a_{73}& a_{74}& a_{75}& a_{76}& a_{77}& a_{78}\\ a_{81}& a_{82}& a_{83}& a_{84}& a_{85}& a_{86}& a_{87}& a_{88}\end{array}\right)$

$=\left(\begin{array}{cccccccc}-2& 2& -2& 7& -2& -1& 6& 4\\ 4& 2& -5& 8& -6& -7& 5& -3\\ 0& -3& 0& 2& -2& 0& -2& -6\\ 6& -4& -2& 0& 3& 2& 4& 1\\ -4& 5& -3& 0& -2& 6& -4& 2\\ -3& -3& 0& 0& -3& -3& -5& -2\\ -7& -5& -4& -1& 4& -4& 3& 5\\ -6& 0& -1& 0& 7& -4& -4& 1\end{array}\right)$

Specifically, boundary element acquiring unit utilizes formula (1) to obtain the boundary element n of reduction coefficient matrix N according to matrix A to be inverted_1i��0And n_j1��0:

$\left\{\begin{array}{c}{n}_{1i\&CenterDot;0}=a_{1i}\\ n_{j1\&CenterDot;0}=a_{j1}\end{array}\right.---\left(1\right)$

In the present embodiment, boundary element acquiring unit obtains the boundary element of reduction coefficient matrix N according to formula (1), as shown in formula (1.1).

$\begin{array}{c}N=\left(\begin{array}{cccccccc}{n}_{11\&CenterDot;0}& n_{12\&CenterDot;0}& n_{13\&CenterDot;0}& n_{14\&CenterDot;0}& n_{15\&CenterDot;0}& n_{16\&CenterDot;0}& n_{17\&CenterDot;0}& n_{18\&CenterDot;0}\\ n_{21\&CenterDot;0}& n_{22\&CenterDot;1}& n_{23\&CenterDot;1}& n_{24\&CenterDot;1}& n_{25\&CenterDot;1}& n_{26\&CenterDot;1}& n_{27\&CenterDot;1}& n_{28\&CenterDot;1}\\ n_{31\&CenterDot;0}& n_{32\&CenterDot;1}& n_{33\&CenterDot;2}& n_{34\&CenterDot;2}& n_{35\&CenterDot;2}& n_{36\&CenterDot;2}& n_{37\&CenterDot;2}& n_{38\&CenterDot;2}\\ n_{41\&CenterDot;0}& n_{42\&CenterDot;1}& n_{43\&CenterDot;2}& n_{44\&CenterDot;3}& n_{45\&CenterDot;3}& n_{46\&CenterDot;3}& n_{47\&CenterDot;3}& n_{48\&CenterDot;3}\\ n_{51\&CenterDot;0}& n_{52\&CenterDot;1}& n_{53\&CenterDot;2}& n_{54\&CenterDot;3}& n_{55\&CenterDot;4}& n_{56\&CenterDot;4}& n_{57\&CenterDot;4}& n_{58\&CenterDot;4}\\ n_{61\&CenterDot;0}& n_{62\&CenterDot;1}& n_{63\&CenterDot;2}& n_{64\&CenterDot;3}& n_{65\&CenterDot;4}& n_{66\&CenterDot;5}& n_{67\&CenterDot;5}& n_{68\&CenterDot;5}\\ n_{71\&CenterDot;0}& n_{72\&CenterDot;1}& n_{73\&CenterDot;2}& n_{74\&CenterDot;3}& n_{75\&CenterDot;4}& n_{76\&CenterDot;5}& n_{77\&CenterDot;6}& n_{78\&CenterDot;6}\\ n_{81\&CenterDot;0}& n_{82\&CenterDot;1}& n_{83\&CenterDot;2}& n_{84\&CenterDot;3}& n_{85\&CenterDot;4}& n_{86\&CenterDot;5}& n_{87\&CenterDot;6}& n_{88\&CenterDot;7}\end{array}\right)\\ =\left(\begin{array}{cccccccc}-2& -2& 2& -7& 2& 1& -6& -4\\ -4& n_{22\&CenterDot;1}& n_{23\&CenterDot;1}& n_{24\&CenterDot;1}& n_{25\&CenterDot;1}& n_{26\&CenterDot;1}& n_{27\&CenterDot;1}& n_{28\&CenterDot;1}\\ 0& n_{32\&CenterDot;1}& n_{33\&CenterDot;2}& n_{34\&CenterDot;2}& n_{35\&CenterDot;2}& n_{36\&CenterDot;2}& n_{37\&CenterDot;2}& n_{38\&CenterDot;2}\\ -6& n_{42\&CenterDot;1}& n_{43\&CenterDot;2}& n_{44\&CenterDot;3}& n_{45\&CenterDot;3}& n_{46\&CenterDot;3}& n_{47\&CenterDot;3}& n_{48\&CenterDot;3}\\ 4& n_{52\&CenterDot;1}& n_{53\&CenterDot;2}& n_{54\&CenterDot;3}& n_{55\&CenterDot;4}& n_{56\&CenterDot;4}& n_{57\&CenterDot;4}& n_{58\&CenterDot;4}\\ 3& n_{62\&CenterDot;1}& n_{63\&CenterDot;2}& n_{64\&CenterDot;3}& n_{65\&CenterDot;4}& n_{66\&CenterDot;5}& n_{67\&CenterDot;5}& n_{68\&CenterDot;5}\\ 7& n_{72\&CenterDot;1}& n_{73\&CenterDot;2}& n_{74\&CenterDot;3}& n_{75\&CenterDot;4}& n_{76\&CenterDot;5}& n_{77\&CenterDot;6}& n_{78\&CenterDot;6}\\ 6& n_{82\&CenterDot;1}& n_{83\&CenterDot;2}& n_{84\&CenterDot;3}& n_{85\&CenterDot;4}& n_{86\&CenterDot;5}& n_{87\&CenterDot;6}& n_{88\&CenterDot;7}\end{array}\right)\end{array}---\left(1.1\right)$

Inner element acquiring unit is for obtaining the inner element of the reduction coefficient matrix N of matrix A to be inverted; Thus obtaining reduction coefficient matrix N; Specifically,

Inner element acquiring unit obtains the diagonal element n of reduction coefficient matrix N first with formula (2)_{ii��(i-1)}:

$n_{ii\&CenterDot;(i-1)}=a_{ii}-\underset{k=1}{\overset{i-1}{\&Sigma;}}\frac{n_{ik\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}}---\left(2\right)$

In formula (2), k=2,3 ... i-1;

In the present embodiment, inner element acquiring unit utilizes formula (2) to obtain the diagonal element n of reduction coefficient matrix N_{ii��(i-1)}, as shown in formula (2.1):

$\begin{array}{c}N=\left(\begin{array}{cccccccc}-2& -2& 2& -7& 2& 1& -6& -4\\ -4& n_{22\&CenterDot;1}& n_{23\&CenterDot;1}& n_{24\&CenterDot;1}& n_{25\&CenterDot;1}& n_{26\&CenterDot;1}& n_{27\&CenterDot;1}& n_{28\&CenterDot;1}\\ 0& n_{32\&CenterDot;1}& n_{33\&CenterDot;2}& n_{34\&CenterDot;2}& n_{35\&CenterDot;2}& n_{36\&CenterDot;2}& n_{37\&CenterDot;2}& n_{38\&CenterDot;2}\\ -6& n_{42\&CenterDot;1}& n_{43\&CenterDot;2}& n_{44\&CenterDot;3}& n_{45\&CenterDot;3}& n_{46\&CenterDot;3}& n_{47\&CenterDot;3}& n_{48\&CenterDot;3}\\ 4& n_{52\&CenterDot;1}& n_{53\&CenterDot;2}& n_{54\&CenterDot;3}& n_{55\&CenterDot;4}& n_{56\&CenterDot;4}& n_{57\&CenterDot;4}& n_{58\&CenterDot;4}\\ 3& n_{62\&CenterDot;1}& n_{63\&CenterDot;2}& n_{64\&CenterDot;3}& n_{65\&CenterDot;4}& n_{66\&CenterDot;5}& n_{67\&CenterDot;5}& n_{68\&CenterDot;5}\\ 7& n_{72\&CenterDot;1}& n_{73\&CenterDot;2}& n_{74\&CenterDot;3}& n_{75\&CenterDot;4}& n_{76\&CenterDot;5}& n_{77\&CenterDot;6}& n_{78\&CenterDot;6}\\ 6& n_{82\&CenterDot;1}& n_{83\&CenterDot;2}& n_{84\&CenterDot;3}& n_{85\&CenterDot;4}& n_{86\&CenterDot;5}& n_{87\&CenterDot;6}& n_{88\&CenterDot;7}\end{array}\right)\\ =\left(\begin{array}{cccccccc}-2& -2& 2& -7& 2& 1& -6& -4\\ -4& 6& n_{23\&CenterDot;1}& n_{24\&CenterDot;1}& n_{25\&CenterDot;1}& n_{26\&CenterDot;1}& n_{27\&CenterDot;1}& n_{28\&CenterDot;1}\\ 0& n_{32\&CenterDot;1}& -4.5& n_{34\&CenterDot;2}& n_{35\&CenterDot;2}& n_{36\&CenterDot;2}& n_{37\&CenterDot;2}& n_{38\&CenterDot;2}\\ -6& n_{42\&CenterDot;1}& n_{43\&CenterDot;2}& -0.7778& n_{45\&CenterDot;3}& n_{46\&CenterDot;3}& n_{47\&CenterDot;3}& n_{48\&CenterDot;3}\\ 4& n_{52\&CenterDot;1}& n_{53\&CenterDot;2}& n_{54\&CenterDot;3}& -109.1429& n_{56\&CenterDot;4}& n_{57\&CenterDot;4}& n_{58\&CenterDot;4}\\ 3& n_{62\&CenterDot;1}& n_{63\&CenterDot;2}& n_{64\&CenterDot;3}& n_{65\&CenterDot;4}& -7.9771& n_{67\&CenterDot;5}& n_{68\&CenterDot;5}\\ 7& n_{72\&CenterDot;1}& n_{73\&CenterDot;2}& n_{74\&CenterDot;3}& n_{75\&CenterDot;4}& n_{76\&CenterDot;5}& -2.3182& n_{78\&CenterDot;6}\\ 6& n_{82\&CenterDot;1}& n_{83\&CenterDot;2}& n_{84\&CenterDot;3}& n_{85\&CenterDot;4}& n_{86\&CenterDot;5}& n_{87\&CenterDot;6}& 152.2675\end{array}\right)\end{array}---\left(2.1\right)$

Inner element acquiring unit recycling formula (3) obtains the lower triangle element n of reduction coefficient matrix N_{ji��(i-1)}:

$n_{ji\&CenterDot;(i-1)}=\left\{\begin{array}{c}{a}_{ji}-\underset{k=1}{\overset{i-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}},n_{ii\&CenterDot;(i-1)}>0\\ -(a_{ji}-\underset{k=1}{\overset{i-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}}),n_{ii\&CenterDot;(i-1)}<0\end{array}\right.---\left(3\right)$

In formula (3), i=2,3 ..., M-1; J=i+1, i+2 ..., M;

In the present embodiment, inner element acquiring unit utilizes formula (3) to obtain the lower triangle element n of reduction coefficient matrix N_{ii��(i-1)}, as shown in formula (3.1):

$\begin{array}{c}N=\left(\begin{array}{cccccccc}-2& -2& 2& -7& 2& 1& -6& -4\\ -4& 6& n_{23\&CenterDot;1}& n_{24\&CenterDot;1}& n_{25\&CenterDot;1}& n_{26\&CenterDot;1}& n_{27\&CenterDot;1}& n_{28\&CenterDot;1}\\ 0& n_{32\&CenterDot;1}& -4.5& n_{34\&CenterDot;2}& n_{35\&CenterDot;2}& n_{36\&CenterDot;2}& n_{37\&CenterDot;2}& n_{38\&CenterDot;2}\\ -6& n_{42\&CenterDot;1}& n_{43\&CenterDot;2}& -0.7778& n_{45\&CenterDot;3}& n_{46\&CenterDot;3}& n_{47\&CenterDot;3}& n_{48\&CenterDot;3}\\ 4& n_{52\&CenterDot;1}& n_{53\&CenterDot;2}& n_{54\&CenterDot;3}& -109.1429& n_{56\&CenterDot;4}& n_{57\&CenterDot;4}& n_{58\&CenterDot;4}\\ 3& n_{62\&CenterDot;1}& n_{63\&CenterDot;2}& n_{64\&CenterDot;3}& n_{65\&CenterDot;4}& -7.9771& n_{67\&CenterDot;5}& n_{68\&CenterDot;5}\\ 7& n_{72\&CenterDot;1}& n_{73\&CenterDot;2}& n_{74\&CenterDot;3}& n_{75\&CenterDot;4}& n_{76\&CenterDot;5}& -2.3182& n_{78\&CenterDot;6}\\ 6& n_{82\&CenterDot;1}& n_{83\&CenterDot;2}& n_{84\&CenterDot;3}& n_{85\&CenterDot;4}& n_{86\&CenterDot;5}& n_{87\&CenterDot;6}& 152.2675\end{array}\right)\\ =\left(\begin{array}{cccccccc}-2& -2& 2& -7& 2& 1& -6& -4\\ -4& 6& n_{23\&CenterDot;1}& n_{24\&CenterDot;1}& n_{25\&CenterDot;1}& n_{26\&CenterDot;1}& n_{27\&CenterDot;1}& n_{28\&CenterDot;1}\\ 0& -3& -4.5& n_{34\&CenterDot;2}& n_{35\&CenterDot;2}& n_{36\&CenterDot;2}& n_{37\&CenterDot;2}& n_{38\&CenterDot;2}\\ -6& 2& 5& -0.7778& n_{45\&CenterDot;3}& n_{46\&CenterDot;3}& n_{47\&CenterDot;3}& n_{48\&CenterDot;3}\\ 4& 1& -2.5& 10.4444& -109.1429& n_{56\&CenterDot;4}& n_{57\&CenterDot;4}& n_{58\&CenterDot;4}\\ 3& -6& 6& 5.8333& 61.5& -7.9771& n_{67\&CenterDot;5}& n_{68\&CenterDot;5}\\ 7& -12& 15& 24.8333& 244.6429& 31.9902& -2.3182& n_{78\&CenterDot;6}\\ 6& -6& 4& 10.5556& 100.8571& 20.6047& 16.3905& 152.2675\end{array}\right)\end{array}---\left(3.1\right)$

Inner element acquiring unit finally utilizes formula (4) to obtain the upper triangle element n of reduction coefficient matrix N_{ji��(j-1)}:

$n_{ji\&CenterDot;(j-1)}=\left\{\begin{array}{c}{a}_{ji}-\underset{k=1}{\overset{j-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}},n_{jj\&CenterDot;(j-1)}>0\\ -(a_{ji}-\underset{k=1}{\overset{j-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}}),n_{jj\&CenterDot;(j-1)}<0\end{array}\right.---\left(4\right)$

In formula (4), i=j+1, j+2 ..., M; J=2,3 ..., M-1;

In the present embodiment, inner element acquiring unit utilizes formula (4) to obtain the upper triangle element n of reduction coefficient matrix N_{ii��(i-1)}, as shown in formula (4.1):

$N=\left(\begin{array}{cccccccc}-2& -2& 2& -7& 2& 1& -6& -4\\ -4& 6& n_{23\&CenterDot;1}& n_{24\&CenterDot;1}& n_{25\&CenterDot;1}& n_{26\&CenterDot;1}& n_{27\&CenterDot;1}& n_{28\&CenterDot;1}\\ 0& -3& -4.5& n_{34\&CenterDot;2}& n_{35\&CenterDot;2}& n_{36\&CenterDot;2}& n_{37\&CenterDot;2}& n_{38\&CenterDot;2}\\ -6& 2& 5& -0.7778& n_{45\&CenterDot;3}& n_{46\&CenterDot;3}& n_{47\&CenterDot;3}& n_{48\&CenterDot;3}\\ 4& 1& -2.5& 10.4444& -109.1429& n_{56\&CenterDot;4}& n_{57\&CenterDot;4}& n_{58\&CenterDot;4}\\ 3& -6& 6& 5.8333& 61.5& -7.9771& n_{67\&CenterDot;5}& n_{68\&CenterDot;5}\\ 7& -12& 15& 24.8333& 244.6429& 31.9902& -2.3182& n_{78\&CenterDot;6}\\ 6& -6& 4& 10.5556& 100.8571& 20.6047& 16.3905& 152.2675\end{array}\right)$

$=\left(\begin{array}{cccccccc}-2& -2& 2& -7& 2& 1& -6& -4\\ -4& 6& -9& 22& -10& -9& 17& 5\\ 0& -3& -4.5& -13& 7& 4.5& -6.5& 3.5\\ -6& 2& 5& -0.7778& -8.1111& -7& -9.1111& -15.2222\\ 4& 1& -2.5& 10.4444& -109.1429& 87& 137.5714& 213.1905\\ 3& -6& 6& 5.8333& 61.5& -7.9771& -3.5190& -7.6289\\ 7& -12& 15& 24.8333& 244.6429& 31.9902& -2.3182& 26.0863\\ 6& -6& 4& 10.5556& 100.8571& 20.6047& 16.3905& 152.2675\end{array}\right)---\left(4.1\right)$

Thus obtaining reduction coefficient matrix N it is: $N=\left(\begin{array}{cccccc}{n}_{11\&CenterDot;0}& ...& n_{1(i-1)\&CenterDot;0}& n_{1i\&CenterDot;0}& ...& n_{1M\&CenterDot;0}\\ n_{21\&CenterDot;0}& ...& n_{2(i-1)\&CenterDot;1}& n_{2i\&CenterDot;1}& ...& n_{2M\&CenterDot;1}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ n_{j1\&CenterDot;0}& ...& n_{j(i-1)\&CenterDot;(i-2)}& n_{ji\&CenterDot;(i-1)}& ...& n_{jM\&CenterDot;(j-1)}\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ .& .& .& .& .& .\\ n_{M1\&CenterDot;0}& ...& n_{M(i-1)\&CenterDot;(i-2)}& n_{Mi\&CenterDot;(i-1)}& ...& n_{MM\&CenterDot;M-1}\end{array}\right)$

In the present embodiment, it is thus achieved that reduction coefficient matrix N be:

$N=\left(\begin{array}{cccccccc}-2& -2& 2& -7& 2& 1& -6& -4\\ -4& 6& -9& 22& -10& -9& 17& 5\\ 0& -3& -4.5& -13& 7& 4.5& -6.5& 3.5\\ -6& 2& 5& -0.7778& -8.1111& -7& -9.1111& -15.2222\\ 4& 1& -2.5& 10.4444& -109.1429& 87& 137.5714& 213.1905\\ 3& -6& 6& 5.8333& 61.5& -7.9771& -3.5190& -7.6289\\ 7& -12& 15& 24.8333& 244.6429& 31.9902& -2.3182& 26.0863\\ 6& -6& 4& 10.5556& 100.8571& 20.6047& 16.3905& 152.2675\end{array}\right)$

Lower triangular matrix acquiring unit decomposes the inverse matrix of gained lower triangular matrix for obtaining matrix A to be inverted;

Specifically, lower triangular matrix acquiring unit, according to reduction coefficient matrix N, utilizes formula (5) to obtain lower triangular matrix

$\begin{array}{c}\left\{\begin{array}{c}{l}_{ii}=\frac{1}{\sqrt{n_{ii\&CenterDot;(i-1)}}},n_{ii\&CenterDot;(i-1)}>0;i=1,2,...,M\\ l_{ii}=-\frac{1}{\sqrt{\left|n_{ii\&CenterDot;(i-1)}\right|}},n_{ii\&CenterDot;(i-1)}<0;i=1,2,...,M\end{array}\right.\\ l_{ji}=-l_{jj}\&CenterDot;\underset{k=i}{\overset{j-1}{\&Sigma;}}{n}_{jk\&CenterDot;(k-1)}\&CenterDot;l_{kk}\&CenterDot;l_{ki},i=1,2,...,M-1;j=i+1,i+2,...,M\end{array}---\left(5\right)$

In the present embodiment, lower triangular matrix acquiring unit, according to formula (4.1), utilizes formula (5) to obtain lower triangular matrix L^-1, as shown in formula (5.1):

$L^{-1}=\left(\begin{array}{cccccccc}-0.7071& 0& 0& 0& 0& 0& 0& 0\\ 0.8165& 0.4082& 0& 0& 0& 0& 0& 0\\ -0.4714& -0.2375& -0.4714& 0& 0& 0& 0& 0\\ -1.3895& 1.0079& 1.2599& -1.1339& 0& 0& 0& 0\\ 1.7412& -1.1532& -1.4814& 1.2854& -0.0957& 0& 0& 0\\ -0.0885& -0.0748& 0.6092& -0.0236& 0.1995& -0.3541& 0& 0\\ 1.37& -0.5661& -2.859& 1.3768& -0.012& 2.6339& -0.6568& 0\\ 1.0078& -0.4177& -2.1430& 1.093& -0.0326& 2.0885& -0.573& 0.081\end{array}\right)---\left(5.1\right)$

Upper triangular matrix acquiring unit decomposes the inverse matrix of gained upper triangular matrix for obtaining matrix A to be inverted;

Specifically, upper triangular matrix acquiring unit, according to reduction coefficient matrix N, utilizes formula (6) to obtain upper triangular matrix

$\begin{array}{c}\left\{\begin{array}{c}{r}_{ii}=\frac{1}{\sqrt{n_{ii\&CenterDot;(i-1)}}},n_{ii\&CenterDot;(i-1)}>0;i=1,2,...,M\\ r_{ii}=\frac{1}{\sqrt{\left|n_{ii\&CenterDot;(i-1)}\right|}},n_{ii\&CenterDot;(i-1)}<0;i=1,2,...,M\end{array}\right.\\ r_{ji}=-r_{ii}\&CenterDot;\underset{k=j}{\overset{i-1}{\&Sigma;}}{n}_{ki\&CenterDot;(k-1)}\&CenterDot;r_{kk}\&CenterDot;r_{jk},j=1,2,...,M-1;i=j+1,j+2,...,M;\end{array}---\left(6\right)$

In the present embodiment, upper triangular matrix acquiring unit, according to formula (4.1), utilizes formula (6) to obtain upper triangular matrix R^-1, as shown in formula (6.1):

$R^{-1}=\left(\begin{array}{cccccccc}0.7071& 0.4082& 0.2357& 1.4489& 1.2649& 0.5139& -0.0999& 0.2503\\ 0& 0.4082& 0.7071& 0.7559& 0.6017& 0.3504& -0.2257& 0.2903\\ 0& 0& 0.4714& 3.2757& 2.7349& 0.7878& 0.1667& -0.0628\\ 0& 0& 0& 1.1339& 0.9982& 0.2433& -0.7405& 1.0167\\ 0& 0& 0& 0& 0.0957& -0.2822& -1.0588& 1.2501\\ 0& 0& 0& 0& 0& 0.3541& 0.2879& -0.3248\\ 0& 0& 0& 0& 0& 0& 0.6568& -0.9119\\ 0& 0& 0& 0& 0& 0& 0& 0.081\end{array}\right)---\left(6.1\right)$

Inverse matrix acquiring unit for obtain matrix A to be inverted inverse matrix A^-1;

Specifically, inverse matrix acquiring unit utilizes formula (7) to obtain the inverse matrix A of matrix A to be inverted^-1:

A^-1=R^-1��L^-1(7)

According to the L that triangular matrix acquiring unit obtains^-1Matrix and R^-1Matrix, utilizes formula (7) to obtain the inverse matrix A of matrix A^-1, as shown in formula (7.1). Thus, complete whole inversion operation.

$\begin{array}{c}{A}^{-1}=R^{-1}\&CenterDot;L^{-1}\\ =\left(\begin{array}{cccccccc}-0.0139& 0.0259& -0.0939& 0.1067& -0.0081& 0.0757& -0.0778& 0.0203\\ -0.0476& 0.0483& -0.0358& -0.0854& 0.0245& -0.1121& -0.0181& 0.0235\\ 0.0953& -0.0904& -0.0086& -0.0567& -0.1086& 0.0289& -0.0735& -0.0051\\ 0.1553& -0.0320& 0.0363& 0.0833& -0.0050& 0.0868& -0.0962& 0.0824\\ 0.0009& -0.0120& 0.0345& 0.0382& -0.0120& -0.0781& -0.0209& 0.1013\\ 0.0383& -0.0549& 0.0834& 0.0356& 0.0566& -0.0405& -0.0042& -0.0263\\ -0.0192& 0.0091& 0.0765& -0.0924& -0.0376& -0.1747& 0.0912& -0.0739\\ 0.0817& -0.0338& -0.1737& 0.0886& 0.0026& 0.1693& -0.0464& 0.0066\end{array}\right)\end{array}---\left(7.1\right)$

In order to verify the effect of matrix inversion module proposed in this patent, randomly select the sample matrix that many group exponent number M are different, matrix element span is different, input this novel matrix as sample matrix to be inverted and invert module carries out matrix inversion experiment. Performance for the matrix inversion module that objective appraisal this patent proposes, the result and the Matlab matrix inversion function simulation results that obtain the module of inverting adopting this patent to propose contrast, employing formula (8) obtains maximum absolute error, in formula (8), matrix I is the operation result matrix utilizing Matlab matrix inversion system function to invert. Result under different experimental conditions has carried out contrast evaluation and test, and concrete outcome is as shown in table 1 below:

��=Max (| I-C |), C=R^-1��L^-1(8)

The different matrix decomposition experimental error result of table 1

In table 1, experiment has randomly selected 8 rank, 64 rank, three kinds of 512 rank scale sample matrix, every kind of scale matrix elemental range respectively (-1,1), (-20,20), (-1000,1000), every kind of condition random is chosen sample matrix four groups different and is tested, from maximum error result data in table, adopt the result of invert result and the emulation of Matlab system function of the module acquisition of inverting of this patent proposition closely, absolute error and relative error are all less, have higher operational precision.

In addition, from embodiment calculating process, the module of inverting that this patent proposes, when inverting, first replaced the original matrix element of corresponding position by reduction coefficient matrix element, then replaced the reduction coefficient matrix element on corresponding position by triangle inverse matrix element, last by triangle inverse matrix multiplied result replacement triangle inverse matrix element, whole process does not need additionally to take memory space, and in module each unit perform process can degree of parallelism higher, high degree reduce operation time. It is hereby understood that the matrix inversion module not only operational precision that this patent proposes is higher, operation use time is shorter, and computational complexity is low, can degree of parallelism is high, save memory space, there is extraordinary theory and engineer applied be worth.

Claims (2)

1. the step-by-step Shift Method improved seeks a matrix inversion matrix norm block, and its feature includes: boundary element acquiring unit, inner element acquiring unit, lower triangular matrix acquiring unit, upper triangular matrix acquiring unit and inverse matrix acquiring unit;

Described boundary element acquiring unit is used for obtaining matrix to be inverted $A=\left(\begin{array}{cccccc}{a}_{11}& a_{12}& ...& a_{1i}& ...& a_{1M}\\ a_{21}& a_{22}& ...& a_{2i}& ...& a_{2M}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}\\ ...\\ \end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}\\ ...\\ \end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ a_{j1}& a_{j2}& ...& a_{ji}& ...& a_{jM}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}\\ ...\\ \end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}\\ ...\\ \end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ a_{M1}& a_{M2}& ...& a_{Mi}& ...& a_{MM}\end{array}\right)$ The boundary element of reduction coefficient matrix N; Described matrix A to be inverted is that to meet each rank the Principal Minor Sequence be not the M rank square formation of 0; a_jiRepresent jth row the i-th column element; I, j=1,2,3 ..., M;

Described inner element acquiring unit is for obtaining the inner element of the reduction coefficient matrix N of matrix A to be inverted; Thus obtaining reduction coefficient matrix N;

Described lower triangular matrix acquiring unit is for obtaining the inverse matrix of matrix A triangle decomposition gained lower triangular matrix to be inverted;

Described upper triangular matrix acquiring unit is for obtaining the inverse matrix of matrix A triangle decomposition gained upper triangular matrix to be inverted;

Described inverse matrix acquiring unit is for obtaining the inverse matrix A of matrix A to be inverted^-1��

2. the step-by-step Shift Method of improvement according to claim 1 seeks matrix inversion matrix norm block, it is characterized in that:

Described boundary element acquiring unit utilizes formula (1) to obtain the boundary element n of reduction coefficient matrix N according to described matrix A to be inverted_1i��0And n_j1��0:

$\left\{\begin{array}{c}{n}_{1i\&CenterDot;0}=a_{1i}\\ n_{j1\&CenterDot;0}=a_{j1}\end{array}\right.---\left(1\right)$

Described inner element acquiring unit utilizes formula (2) to obtain the diagonal element n of reduction coefficient matrix N_{ii��(i-1)}:

$n_{ii\&CenterDot;(i-1)}=a_{ii}-\underset{k=1}{\overset{i-1}{\&Sigma;}}\frac{n_{ik\&CenterDot;(k-1)}\&CenterDot;n_{ki\&CenterDot;(k-1)}}{n_{kk\&CenterDot;(k-1)}}---\left(2\right)$

In formula (2), k=2,3 ... i-1;

Described inner element acquiring unit utilizes formula (3) to obtain the lower triangle element n of reduction coefficient matrix N_{ji��(i-1)}:

$n_{ji\&CenterDot;\left(i-1\right)}=\left\{\begin{array}{c}{a}_{ji}-\underset{k=1}{\overset{i-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;\left(k-1\right)}\&CenterDot;n_{ki\&CenterDot;\left(k-1\right)}}{n_{kk\&CenterDot;\left(k-1\right)}},n_{ii\&CenterDot;\left(i-1\right)}>0\\ -\left(a_{ji}-\underset{k=1}{\overset{i-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;\left(k-1\right)}\&CenterDot;n_{ki\&CenterDot;\left(k-1\right)}}{n_{kk\&CenterDot;\left(k-1\right)}}\right),n_{ii\&CenterDot;\left(i-1\right)}<0\end{array}\right.---\left(3\right)$

In formula (3), i=2,3 ..., M-1; J=i+1, i+2 ..., M;

Described inner element acquiring unit utilizes formula (4) to obtain the upper triangle element n of reduction coefficient matrix N_{ji��(j-1)}:

$n_{ji\&CenterDot;\left(j-1\right)}=\left\{\begin{array}{c}{a}_{ji}-\underset{k=1}{\overset{j-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;\left(k-1\right)}\&CenterDot;n_{ki\&CenterDot;\left(k-1\right)}}{n_{kk\&CenterDot;\left(k-1\right)}},n_{jj\&CenterDot;\left(j-1\right)}>0\\ -\left(a_{ji}-\underset{k=1}{\overset{j-1}{\&Sigma;}}\frac{n_{jk\&CenterDot;\left(k-1\right)}\&CenterDot;n_{ki\&CenterDot;\left(k-1\right)}}{n_{kk\&CenterDot;\left(k-1\right)}}\right),n_{jj\&CenterDot;\left(j-1\right)}<0\end{array}\right.---\left(4\right)$

In formula (4), i=j+1, j+2 ..., M; J=2,3 ..., M-1;

Thus obtaining reduction coefficient matrix N it is: $N=\left(\begin{array}{cccccc}{n}_{11\&CenterDot;0}& ...& n_{1\left(i-1\right)\&CenterDot;0}& n_{1i\&CenterDot;0}& ...& n_{1M\&CenterDot;0}\\ n_{21\&CenterDot;0}& ...& n_{2\left(i-1\right)\&CenterDot;1}& n_{2i\&CenterDot;1}& ...& n_{2M\&CenterDot;1}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ n_{j1\&CenterDot;0}& ...& n_{j\left(i-1\right)\&CenterDot;\left(i-2\right)}& n_{ji\&CenterDot;\left(i-1\right)}& ...& n_{jM\&CenterDot;\left(j-1\right)}\\ \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}& \begin{array}{c}.\\ .\\ .\end{array}\\ n_{M1\&CenterDot;0}& ...& n_{M\left(i-1\right)\&CenterDot;\left(i-2\right)}& n_{Mi\&CenterDot;\left(i-1\right)}& ...& n_{MM\&CenterDot;M-1}\end{array}\right)$

Described lower triangular matrix acquiring unit, according to described reduction coefficient matrix N, utilizes formula (5) to obtain lower triangular matrix

$\left\{\begin{array}{ccc}{l}_{ii}=\frac{1}{\sqrt{n_{ii\&CenterDot;\left(i-1\right)}}},& n_{ii\&CenterDot;\left(i-1\right)}>0;& i=1,2,...,M\\ l_{ii}=-\frac{1}{\sqrt{\left|n_{ii\&CenterDot;\left(i-1\right)}\right|}},& n_{ii\&CenterDot;\left(i-1\right)}<0;& i=1,2,...,M\end{array}\right.---\left(5\right)$

$l_{ji}=-l_{jj}\&CenterDot;\underset{k=i}{\overset{j-1}{\&Sigma;}}{n}_{jk\&CenterDot;(k-1)}\&CenterDot;l_{kk}\&CenterDot;l_{ki},i=1,2,...,M-1;j=i+1,i+2,...,M$

Described upper triangular matrix acquiring unit, according to described reduction coefficient matrix N, utilizes formula (6) to obtain upper triangular matrix

$\left\{\begin{array}{ccc}{r}_{ii}=\frac{1}{\sqrt{n_{ii\&CenterDot;\left(i-1\right)}}},& n_{ii\&CenterDot;\left(i-1\right)}>0;& i=1,2,...,M\\ r_{ii}=\frac{1}{\sqrt{\left|n_{ii\&CenterDot;\left(i-1\right)}\right|}},& n_{ii\&CenterDot;\left(i-1\right)}<0;& i=1,2,...,M\end{array}\right.---\left(6\right)$

$r_{ji}=-r_{ii}\&CenterDot;\underset{k=j}{\overset{i-1}{\&Sigma;}}{n}_{ki\&CenterDot;(k-1)}\&CenterDot;r_{kk}\&CenterDot;r_{jk},j=1,2,...,M-1;i=j+1,j+2,...,M;$

Described inverse matrix acquiring unit utilizes formula (7) to obtain the inverse matrix A of described matrix A to be inverted^-1:

A^-1=R^-1��L^-1(7)��