CN106980602A - The processing of matrix inversion lemma zero mother's situation and a kind of method for solving inverse matrix that increases by degrees - Google Patents
The processing of matrix inversion lemma zero mother's situation and a kind of method for solving inverse matrix that increases by degrees Download PDFInfo
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- CN106980602A CN106980602A CN201710167646.2A CN201710167646A CN106980602A CN 106980602 A CN106980602 A CN 106980602A CN 201710167646 A CN201710167646 A CN 201710167646A CN 106980602 A CN106980602 A CN 106980602A
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- G—PHYSICS
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- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
Abstract
By taking Limit Operation overcome the processing method of the female situation of matrix inversion lemma zero the invention discloses a kind of, and give a kind of method of Recursive Solution matrix inversion.Key step includes：The first step, the sum that it is decomposed into diagonal element matrix diag (K) and n rank-one matrix by any n ranks invertible matrix K, i.e.,： Wherein uiArranged for matrix i-th and (make ki,i=0), vi(v is removed for n dimension row vectorsi(i) 0) remaining be outside=1；Second step, according to recurrence formulaCalculate, wherein 1≤i≤n, C0=diag (K)‑1, C can be obtainedn=K‑1.As a certain element k of diag (K) in calculatingi,iWhen=0, can first it assumeRecursion makes C afterwards twice2=lima→∞(C2)；Occurs 1+v in calculatingi·Ci‑1·uiWhen=0, can first it assumeRecursion simultaneously makes Ci+1=lima→∞(Ci+1).The recurrence method can accurately solve the inverse of Hilbert matrix, it can realize that Gauss about works as forward steps using elimination matrix premultiplication original matrix, associate(d) matrix elementary transformation can condense the selected free degree, and its conditional number can be greatly reduced with elimination matrix pretreatment finite element matrix.
Description
Technical field
The present invention relates to matrix recursion inversion technique, using taking limit process to overcome the female situation of matrix inversion lemma zero,
Recurrence formula can solve the inverse matrix of Hilbert matrix (typical Very Ill-conditioned matrix) with efficiently and accurately.
Background technology
Matrix inversion technique includes the adjoint matrix tactical deployment of troops, matrix decomposition method, block matrix method, elementary transform method, edged method, asked
Iterative method of approximation inverse matrix etc.；
Adjoint matrix tactical deployment of troops amount of calculation is excessive, fundamentally cannot be used the actual calculating that high-order matrix number is inverted；
Common matrix decomposition method includes LU decomposition, QR decomposition, singular value decomposition etc., because triangular matrix, unitary matrice are asked
It is inverse relatively easy, matrix multiplication is carried out after inverting again and can obtain the inverse of original matrix.
In LU factorization, pivoting operation must be carried out when host element is zero, in addition, if main in actual calculating process
Member is close to zero, it is necessary to carry out pivoting operation, to avoid the problems such as precision is reduced or can not calculated；
Edged method be based on matrix inversion lemma, can step-by-step calculation, but when host element be zero or calculate in occur zero mother
During situation, it is impossible to calculated；
Ask the iterative method of the approximation inverse matrix of matrix invalid to big conditional number ill-condition matrix；
The content of the invention
The present invention is based on matrix inversion lemma, overcomes zero host element in calculating process and zero female using limit process is taken
Phenomenon, the finding the inverse matrix that increases by degrees it is inverse, compared with seeking the iterative method of approximate matrix inverse matrix, available for the ill square of big conditional number
The recursion of battle array is inverted, and key step includes：
The sum that it is decomposed into diagonal element matrix diag (K) and n rank-one matrix by any n ranks invertible matrix K, i.e.,
According to recurrence formula Calculate elimination matrixs at different levels；
Increase by degrees solution, you can obtains Cn=K-1。
If diagonal element matrix diag (K) a certain element ki,iWhen=0, can first it assumeSolve C0；Recursion two
C is obtained after secondary2Expression formula, order, subsequent stages elimination matrix can normal recursion；
Occurs 1+v in calculating processi·Ci-1·uiWhen=0, it is female that zero occurs in recurrence formula, it is impossible to directly calculates, now,
Can first it assume
Recursion obtains C afterwards twicei+1Expression formula, take the limit, subsequent stages elimination matrix
Normal recursion.
Brief description of the drawings
Fig. 1 is calculation flow chart；
Embodiment
With reference to two examples, matrix recursion inversion process is shown in detail, example 1 is directed to zero host element, and example 2 is directed to
Occurs the female situation of zero in calculating process.
Example 1. is setRecursive Solution elimination matrixs at different levels.
1. C is calculated0：
Due toMiddle K1,1=0, it is impossible to directly invert, order
2. C is calculated1：
u1=[0,2,3,4]T,v1=[1,0,0,0], 1+v1·C0·u1=1,
3. C is calculated2：
u2=[2,0,5,6]T,v2=[0,1,0,0],
Take the limit
4. C is calculated3：
u3=[3,5,0,7]T,v3=[0,0,1,0],
5. C is calculated4：
u4=[4,6,7,0]T,v4=[0,0,0,1],
Checking is apparent from the correctness and C of elimination matrixs at different levels4=K-1。
Example 2. is setRecursive Solution elimination matrixs at different levels.
1. C is calculated1：v1=[1,0,0,0], 1+v1·C0·u1=1,
2. C is calculated2：u2=[2,0,0,0]T, v2=[0,1,0,0], due to 1+v2·C1·u2=0, it is impossible to directly utilize formulaCalculate C2, it is assumed thatIt can obtain：
3. C is calculated3：u3=[1,0,0,0]T, v3=[0,0,1,0],
Take the limit
4. C is calculated4：u4=[2,0,0,0]T, v4=[0,0,0,1],
Checking is apparent from the correctness and C of elimination matrixs at different levels4=K-1。
Claims (5)
1. the processing of matrix inversion lemma zero mother's situation and a kind of method for solving inverse matrix that increases by degrees, it is characterised in that
Comprise the following steps：
1) sum that it is decomposed into diagonal element matrix diag (K) and n rank-one matrix by any n ranks invertible matrix K, i.e.,
2) according to recurrence formula Calculate elimination matrixs at different levels；
3) increase by degrees solution, you can obtains Cn=K-1。
The inversion technique 2. matrix according to right 1 increases by degrees, it is characterised in that step 1) comprise the following steps：
1) n ranks invertible matrix K its be decomposed into the sum of diagonal element matrix diag (K) and n rank-one matrix, if diagonal element square
Battle array diag (K) a certain element ki,iWhen=0, can first it assumeSolve C0；
2) recursion obtains C afterwards twice2Expression formula, order, subsequent stages elimination matrix can normal recursion, calculate
Process refers to example 1.
The inversion technique 3. matrix according to right 1 increases by degrees, it is characterised in that step 2) comprise the following steps：
1) occurs 1+v in calculating processi·Ci-1·uiWhen=0, it is female that zero occurs in recurrence formula, it is impossible to directly calculates, now, can
First assume
2) recursion obtains C afterwards twicei+1Expression formula, order, subsequent stages elimination matrix can pass normally
Push away, calculating process refers to example 2.
Following second of recurrence algorithm can also be used 4. increasing by degrees and inverting, it is characterised in that comprised the following steps：
1) the reversible real matrix K of n ranks is decomposed into：
2) for the sake of simplicity, it is following block form that K, which decomposes postscript,：
Wherein Kn-1For the sub- square of n-1 ranks
Battle array, unFor n-1 dimensional vectors, vnRow vector is tieed up for n-1.
3)K-1Expression formula： Wherein Cn-1=Kn-1 -1, A=1-vnCn-1unB,
4)K-1Recursion expression formula of inverting is identical in form with " edged method " finding the inverse matrix formula, but herein using the side for taking the limit
There is zero diagonal element and occur zero mother in calculating in method processing array.
5. second of recurrence algorithm according to right 4, it is characterised in that occur zero pivot or the female feelings of zero in calculating process
During condition, using claim 3) described in take limit process.
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108536651A (en) * | 2018-04-19 | 2018-09-14 | 武汉轻工大学 | The method and apparatus for generating reversible modal m matrix |
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108536651A (en) * | 2018-04-19 | 2018-09-14 | 武汉轻工大学 | The method and apparatus for generating reversible modal m matrix |
| CN108536651B (en) * | 2018-04-19 | 2022-04-05 | 武汉轻工大学 | Method and apparatus for generating reversible modulo m matrix |
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