CN105608059A

CN105608059A - Module for solving triangular matrix decomposition based on improved bitwise substitution method

Info

Publication number: CN105608059A
Application number: CN201510981481.3A
Authority: CN
Inventors: 张多利; 王浩; 宋宇鲲
Original assignee: Hefei University of Technology
Current assignee: Hefei University of Technology
Priority date: 2015-12-22
Filing date: 2015-12-22
Publication date: 2016-05-25
Also published as: JP2018506757A; JP6388713B2; WO2017107337A1

Abstract

The invention discloses a module for solving triangular matrix decomposition based on an improved bitwise substitution method, which is characterized by comprising a boundary element acquisition unit, an internal element acquisition unit, an upper triangular matrix decomposition unit and a lower triangular matrix decomposition unit, wherein the boundary element acquisition unit is used for acquiring a boundary element of a reduced coefficient matrix of a matrix to be decomposed; the internal element acquisition unit is used for acquiring an internal element of the reduced coefficient matrix of the matrix to be decomposed to acquire the reduced coefficient matrix; the upper triangular matrix decomposition unit is used for decomposing an upper triangular matrix of the matrix to be decomposed; and the lower triangular matrix decomposition unit is used for decomposing a lower triangular matrix of the matrix to be decomposed. The module disclosed by the invention can reduce the computational complexity of matrix decomposition, compress the storage space and improve the parallelizability of the decomposition operation, so as to quickly and effectively complete the matrix decomposition operation.

Description

A kind of module of asking matrix triangle decomposition based on improved position displacement method

Technical field

The present invention relates to matrix operation, relate in particular to a kind of New-type mould of asking matrix triangle decomposition based on improved position displacement methodPiece.

Background technology

Matrix operation is the basis that Science and engineering calculates. Flexibility and the high numerical value of succinct form of presentation, the computing intuitively of matrixStability makes matrix operation become key technology and the key problem of numerous engineering projects. Matrix operation comprises matrix multiplication, matrixDecomposition, matrix inversion etc. Wherein matrix decomposition is the inverse process of matrix multiplication, is the mode that solves of a kind of simplification of matrix inversion,Develop also universal. Because the matrix after decomposing has more significantly numerical characteristics or physical meaning, matrix decomposition is at numerical valueAnalyze and engineering field acquisition extensive use. For example, in environmental management risk assessment, Digital Image Processing and encryption, Fluid ComputationThe large-scale data analysis fields such as dynamics, signal processing and control, matrix decomposition algorithm or has become core support.How efficent use of resources is realized the computing of rapid large-scale matrix decomposition becomes the Focal point and difficult point of design.

Current matrix decomposes of a great variety, and in engineering application, conventional have QR decompositions, LU decomposition, a singular value decomposition etc., tiesClose concrete application and can select different decomposition methods. Wherein QR decomposes the decomposition that can be used for Arbitrary Matrix, and its essence is to appointMeaning matrix A resolves into the product of an orthogonal matrix Q and a triangular matrix R. QR decomposes classic algorithm gram-schmidtMethod, householder converter technique and givens rotary process, adopt recurrence method, and computation complexity is large, flow process control difficulty, andRow is poor. It is a kind of matrix decomposition mode for nonsingular square matrix (being that sequence of matrices principal minor is not all 0) that LU decomposes, its basisMatter is the product that matrix A is decomposed into a lower triangle battle array L and upper triangular matrix U. But LU decomposition computation complexity is high, adoptRecurrence method, serial is had relatively high expectations, and takies memory space large. Singular value decomposition is normal matrix diagonalization at the tenth of the twelve Earthly Branches in matrix analysisPopularization, a complex matrix A is decomposed into two unitary matrice U, V and a diagonal matrix S by the essence of singular value decompositionProduct. But singular value decomposition is difficult to split into uncorrelated sub-computing, and singular value decomposition concurrency is poor, and computation complexity is higher, meterCalculate efficiency, real-time poor.

To sum up, current existing matrix decomposition technology still has certain limitation in engineering application, be mainly summed up as following someNot enough:

The first, adopt recursion alternative manner, serial is had relatively high expectations, and carries out concurrent operation difficulty larger, and being difficult to meet engineering shouldUse middle requirement of real-time.

The second, computational complexity is higher, and operand is larger, and computing time is longer.

The 3rd, computational space complexity is higher, and the memory space taking is larger, 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, proposes a kind ofly new to ask matrix triangle based on improved position displacement methodThe module of decomposing, to reducing computational complexity, compression memory space, that improves decomposition operation can concurrency, and then very fastSpeed, complete matrix decomposition computing efficiently.

The present invention is that technical solution problem adopts following technical scheme:

The present invention is a kind of asks the feature of the module of matrix triangle decomposition to comprise based on improved position displacement method: boundary element obtains listUnit, inner element acquiring unit, upper triangular matrix resolving cell and lower triangular matrix resolving cell;

Described boundary element acquiring unit is used for obtaining matrix to be decomposed

A = (\begin{matrix} a_{11} & a_{12} & ... & a_{1 i} & ... & a_{1 M} \\ a_{21} & a_{22} & ... & a_{2 i} & ... & a_{2 M} \\ . & . & . & . \\ . & . & ... & . & ... & . \\ . & . & . & . \\ a_{j 1} & a_{j 2} & ... & a_{j i} & ... & a_{j M} \\ . & . & . & . \\ . & . & ... & . & ... & . \\ . & . & . & . \\ a_{M 1} & a_{M 2} & ... & a_{M i} & ... & a_{M M} \end{matrix})

ReductionThe boundary element of coefficient matrix N; Described matrix A to be decomposed is that to meet each rank the Principal Minor Sequence be not 0 M rank square formation; a_jiTableShow the capable i column element of j; I, j=1,2,3 ..., M;

Described inner element acquiring unit is used for the inner element of the reduction coefficient matrix N that obtains matrix A to be decomposed; Thereby obtainReduction coefficient matrix N;

Described upper triangular matrix resolving cell is for decomposing the upper triangular matrix of matrix A to be decomposed;

Described lower triangular matrix resolving cell is for decomposing the lower triangular matrix of matrix A to be decomposed.

Of the present inventionly ask the feature of the module of matrix triangle decomposition to be also based on improved position displacement method:

Described boundary element acquiring unit, according to matrix A to be decomposed, utilizes formula (1) to obtain the boundary element of reduction coefficient matrix Nn_1i·0And n_j1·0：

\{\begin{matrix} n_{1 i \cdot 0} = a_{1 i} \\ n_{j 1 \cdot 0} = a_{j 1} \end{matrix} - - - (1)

Described inner element acquiring unit utilizes formula (2) to obtain the diagonal element n of reduction coefficient matrix N_ii·(i-1)：

n_{i i \cdot (i - 1)} = a_{i i} - Σ_{k = 1}^{i - 1} \frac{n_{i k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}} - - - (2)

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_{j i \cdot (i - 1)} = \{\begin{matrix} a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}, n_{i i \cdot (i - 1)} > 0 \\ - (a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}), n_{i i \cdot (i - 1)} < 0 \end{matrix} - - - (3)

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_{j i \cdot (i - 1)} = \{\begin{matrix} a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}, n_{j j \cdot (j - 1)} > 0 \\ - (a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}), n_{j j \cdot (j - 1)} < 0 \end{matrix} - - - (4)

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

Thereby obtain reduction coefficient matrix N be:

N = (\begin{matrix} n_{11 \cdot 0} & ... & n_{1 (i - 1) \cdot 0} & n_{1 i \cdot 0} & ... & n_{1 M \cdot 0} \\ n_{21 \cdot 0} & ... & n_{2 (i - 1) \cdot 1} & n_{2 i \cdot 0} & ... & n_{2 M \cdot 1} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ n_{j 1 \cdot 0} & ... & n_{j (i - 1) \cdot (i - 2)} & n_{j i \cdot (i - 1)} & ... & n_{j M \cdot (j - 1)} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ n_{M 1 \cdot 0} & ... & n_{M (i - 1) \cdot (i - 2)} & n_{M i \cdot (i - 1)} & ... & n_{M M \cdot M - 1} \end{matrix})

Described lower triangular matrix resolving cell, according to described reduction coefficient matrix N, utilizes formula (5) that described matrix A to be decomposed is dividedSeparate as lower triangular matrix

l_ji＝0,j＝1,2,…,M-1；i＝j+1,j+2,…,M

\{\begin{matrix} l_{i i} = \sqrt{n_{i i \cdot (i - 1)}}, & n_{i i \cdot (i - 1)} > 0; & i = 1, 2, ..., M \\ l_{i i} = - \sqrt{| n_{i i \cdot (i - 1)} |}, & n_{i i \cdot (i - 1)} < 0; & i = 1, 2, ..., M \end{matrix}

\{\begin{matrix} l_{j i} = \frac{1}{\sqrt{n_{i i \cdot (i - 1)}}} \cdot n_{j i \cdot (i - 1)}, & i = 1, 2, ..., M - 1; & j = i + 1, i + 2, ..., M; n_{i i \cdot (i - 1)} > 0 \\ l_{j i} = \frac{1}{- \sqrt{| n_{i i \cdot (i - 1)} |}} \cdot n_{j i \cdot (i - 1)}, & i = 1, 2, ..., M - 1; & j = i + 1, i + 2, ..., M; n_{i i \cdot (i - 1)} < 0 \end{matrix} - - - (5)

Described upper triangular matrix resolving cell, according to described reduction coefficient matrix N, utilizes formula (6) that described matrix A to be decomposed is dividedSeparate as upper triangular matrix

r_ji＝0,i＝1,2,…,M-1；j＝i+1,i+2,…,M

\{\begin{matrix} r_{i i} = \sqrt{n_{i i \cdot (i - 1)}}, & n_{i i \cdot (i - 1)} > 0; & i = 1, 2, ..., M \\ r_{i i} = \sqrt{| n_{i i \cdot (i - 1)} |}, & n_{i i \cdot (i - 1)} < 0; & i = 1, 2, ..., M \end{matrix}

\{\begin{matrix} r_{j i} = \frac{1}{\sqrt{n_{j j \cdot (j - 1)}}} \cdot n_{j i \cdot (j - 1)}, & n_{j j \cdot (j - 1)} > 0; & j = 1, 2, ..., M - 1; i = j + 1, j + 2, ..., M \\ r_{j i} = \frac{1}{\sqrt{| n_{j j \cdot (j - 1)} |}} \cdot n_{j i \cdot (j - 1)}, & n_{j j \cdot (j - 1)} < 0; & j = 1, 2, ..., M - 1; i = j + 1, j + 2, ..., M \end{matrix} - - - (6) .

Compared with the prior art, beneficial effect of the present invention is embodied in:

1, the matrix decomposition module that the present invention proposes, in whole calculating process, only in inner element acquiring unit, lower threeIn angular moment battle array resolving cell and in upper triangular matrix resolving cell, need to do evolution or inverse to the diagonal element of reduction coefficient matrix(division), the remainder of whole decomposing module only relates to simple multiply-add operation process, has avoided a large amount of opening in prior artSide, square, ask the computing such as norm, division, greatly simplified calculating process.

2, the matrix decomposition module that the present invention proposes, is carrying out on the basis of matrix Algorithm for Triangular Decomposition based on former position displacement method,Its algorithm is revised and improved, produced improved Efficient Matrix Decomposition algorithm. The matrix decomposition module that the present invention proposesBased on improved position displacement matrix decomposition algorithm, not only widen the computing scope of application, and greatly simplified calculating process, makeComputational complexity is lower.

3, the matrix decomposition module that the present invention proposes, by obtaining in boundary element acquiring unit and inner element acquiring unitAnd create reduction coefficient matrix, and decompose serial iteration process, make in obtaining reduction coefficient matrix element upper triangleThe reduction coefficient matrix element that matrix decomposition unit and lower triangular matrix resolving cell inside can be obtained by prime, parallel solve,Lower triangular matrix element, has overcome in existing matrix decomposition technology, owing to adopting iteration serial computing to calculate one by one upper and lower triangleWhat matrix element caused can the not strong problem of concurrency.

4, the matrix decomposition module that the present invention proposes, based on the method for position displacement, makes whole module except input square to be decomposedOutside the shared memory space of battle array, without the memory space outside occupying volume. Existing matrix decomposition technology, a large amount of owing to takingMemory space, thus in the engineering application of ultra-large matrix decomposition, there is certain limitation, and the present invention has solved this justOne problem.

5, the matrix decomposition module that the present invention proposes, each unit internal arithmetic process computational complexity is lower, and inner element obtainsGetting inside, unit can be by upper and lower triangle executed in parallel calculating process, and upper triangular matrix resolving cell and lower triangular matrix decompose singleUnit can parallel work-flow, thereby operation time is shorter, has solved the higher problem of existing matrix decomposition technology complexity operation time.

Detailed description of the invention

In the present embodiment, a kind ofly ask the module of matrix triangle decomposition to comprise based on improved position displacement method: boundary element obtains listUnit, inner element acquiring unit, upper triangular matrix resolving cell and lower triangular matrix resolving cell; It decomposes thinking: 1According to given matrix to be decomposed, ask the boundary element of its reduction coefficient matrix; 2 according to matrix to be decomposed and its reduction coefficient matrixBoundary element, ask the inner element of its reduction coefficient matrix; 3 according to reduction coefficient matrix, is upper by matrix decomposition to be decomposedTriangular matrix and lower triangular matrix, thus the decomposition of whole matrix completed; Specifically,

Boundary element acquiring unit is used for obtaining matrix to be decomposed

A = (\begin{matrix} a_{11} & a_{12} & ... & a_{1 i} & ... & a_{1 M} \\ a_{21} & a_{22} & ... & a_{2 i} & ... & a_{2 M} \\ . & . & . & . \\ . & . & ... & . & ... & . \\ . & . & . & . \\ a_{j 1} & a_{j 2} & ... & a_{j i} & ... & a_{j M} \\ . & . & . & . \\ . & . & ... & . & ... & . \\ . & . & . & . \\ a_{M 1} & a_{M 2} & ... & a_{M i} & ... & a_{M M} \end{matrix})

Reduction coefficientThe boundary element of matrix N; Matrix A to be decomposed is that to meet each rank the Principal Minor Sequence be not 0 M rank square formation; a_jiRepresent that j is capableI column element; I, j=1,2,3 ..., M; In the present embodiment, the matrix A to be decomposed that adopts Matlab to create is random generationEach rank the Principal Minor Sequence be not 08 rank square formations:

\begin{matrix} A = (\begin{matrix} 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{matrix}) \\ = (\begin{matrix} - 7 & - 12 & - 9 & 0 & - 5 & 3 & - 8 & 1 \\ 14 & - 7 & - 14 & 12 & - 8 & 10 & 12 & 4 \\ - 7 & 2 & - 11 & - 18 & 5 & - 12 & 12 & 13 \\ - 15 & 2 & - 10 & 9 & 5 & 0 & 7 & - 4 \\ - 1 & 10 & - 19 & - 13 & 0 & 5 & 7 & 4 \\ 4 & - 15 & 7 & 12 & - 9 & - 1 & 7 & - 4 \\ 8 & 13 & 4 & - 3 & - 5 & 17 & 6 & - 16 \\ 4 & 7 & 8 & - 5 & - 1 & - 7 & 3 & 1 \end{matrix}) \end{matrix}

Specifically, boundary element acquiring unit is to utilize formula (1) to obtain the limit of reduction coefficient matrix N according to matrix A to be decomposedBound component n_1i·0And n_j1·0：

\{\begin{matrix} n_{1 i \cdot 0} = a_{1 i} \\ n_{j 1 \cdot 0} = a_{j 1} \end{matrix} - - - (1)

In the present embodiment, boundary element acquiring unit is to utilize formula (1) to obtain reduction coefficient square according to the matrix A to be decomposed of inputThe boundary element of battle array N, shown in (1.1):

\begin{matrix} N = (\begin{matrix} n_{11 \cdot 0} & n_{12 \cdot 0} & n_{13 \cdot 0} & n_{14 \cdot 0} & n_{15 \cdot 0} & n_{16 \cdot 0} & n_{17 \cdot 0} & n_{18 \cdot 0} \\ n_{21 \cdot 0} & n_{22 \cdot 1} & n_{23 \cdot 1} & n_{24 \cdot 1} & n_{25 \cdot 1} & n_{26 \cdot 1} & n_{27 \cdot 1} & n_{28 \cdot 1} \\ n_{31 \cdot 0} & n_{32 \cdot 1} & n_{33 \cdot 2} & n_{34 \cdot 2} & n_{35 \cdot 2} & n_{36 \cdot 2} & n_{37 \cdot 2} & n_{38 \cdot 2} \\ n_{41 \cdot 0} & n_{42 \cdot 1} & n_{43 \cdot 2} & n_{44 \cdot 3} & n_{45 \cdot 3} & n_{46 \cdot 3} & n_{47 \cdot 3} & n_{48 \cdot 3} \\ n_{51 \cdot 0} & n_{52 \cdot 1} & n_{53 \cdot 2} & n_{54 \cdot 3} & n_{55 \cdot 4} & n_{56 \cdot 4} & n_{57 \cdot 4} & n_{58 \cdot 4} \\ n_{61 \cdot 0} & n_{62 \cdot 1} & n_{63 \cdot 2} & n_{64 \cdot 3} & n_{65 \cdot 4} & n_{66 \cdot 5} & n_{67 \cdot 5} & n_{68 \cdot 5} \\ n_{71 \cdot 0} & n_{72 \cdot 1} & n_{73 \cdot 2} & n_{74 \cdot 3} & n_{75 \cdot 4} & n_{76 \cdot 5} & n_{77 \cdot 6} & n_{78 \cdot 6} \\ n_{81 \cdot 0} & n_{82 \cdot 1} & n_{83 \cdot 2} & n_{84 \cdot 3} & n_{85 \cdot 4} & n_{86 \cdot 5} & n_{87 \cdot 6} & n_{88 \cdot 7} \end{matrix}) \\ = (\begin{matrix} - 7 & - 12 & - 9 & 0 & - 5 & 3 & - 8 & 1 \\ 14 & n_{22 \cdot 1} & n_{23 \cdot 1} & n_{24 \cdot 1} & n_{25 \cdot 1} & n_{26 \cdot 1} & n_{27 \cdot 1} & n_{28 \cdot 1} \\ - 7 & n_{32 \cdot 1} & n_{33 \cdot 2} & n_{34 \cdot 2} & n_{35 \cdot 2} & n_{36 \cdot 2} & n_{37 \cdot 2} & n_{38 \cdot 2} \\ - 15 & n_{42 \cdot 1} & n_{43 \cdot 2} & n_{44 \cdot 3} & n_{45 \cdot 3} & n_{46 \cdot 3} & n_{47 \cdot 3} & n_{48 \cdot 3} \\ - 1 & n_{52 \cdot 1} & n_{53 \cdot 2} & n_{54 \cdot 3} & n_{55 \cdot 4} & n_{56 \cdot 4} & n_{57 \cdot 4} & n_{58 \cdot 4} \\ 4 & n_{62 \cdot 1} & n_{63 \cdot 2} & n_{64 \cdot 3} & n_{65 \cdot 4} & n_{66 \cdot 5} & n_{67 \cdot 5} & n_{68 \cdot 5} \\ 8 & n_{72 \cdot 1} & n_{73 \cdot 2} & n_{74 \cdot 3} & n_{75 \cdot 4} & n_{76 \cdot 5} & n_{77 \cdot 6} & n_{78 \cdot 6} \\ 4 & n_{82 \cdot 1} & n_{83 \cdot 2} & n_{84 \cdot 3} & n_{85 \cdot 4} & n_{86 \cdot 5} & n_{87 \cdot 6} & n_{88 \cdot 7} \end{matrix}) \end{matrix} - - - (1.1)

Inner element acquiring unit is used for the inner element of the reduction coefficient matrix N that obtains matrix A to be decomposed; Thereby acquisition reductionCoefficient matrix N;

Specifically, inner element acquiring unit first utilizes formula (2) to obtain the diagonal element n of reduction coefficient matrix N_ii·(i-1)：

n_{i i \cdot (i - 1)} = a_{i i} - Σ_{k = 1}^{i - 1} \frac{n_{i k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}} - - - (2)

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

In the present embodiment, inner element acquiring unit is utilized the diagonal element of formula (2) acquisition reduction coefficient matrix, suc as formula (2.1)Shown in:

\begin{matrix} N = (\begin{matrix} - 7 & - 12 & - 9 & 0 & - 5 & 3 & - 8 & 1 \\ 14 & n_{22 \cdot 1} & n_{23 \cdot 1} & n_{24 \cdot 1} & n_{25 \cdot 1} & n_{26 \cdot 1} & n_{27 \cdot 1} & n_{28 \cdot 1} \\ - 7 & n_{32 \cdot 1} & n_{33 \cdot 2} & n_{34 \cdot 2} & n_{35 \cdot 2} & n_{36 \cdot 2} & n_{37 \cdot 2} & n_{38 \cdot 2} \\ - 15 & n_{42 \cdot 1} & n_{43 \cdot 2} & n_{44 \cdot 3} & n_{45 \cdot 3} & n_{46 \cdot 3} & n_{47 \cdot 3} & n_{48 \cdot 3} \\ - 1 & n_{52 \cdot 1} & n_{53 \cdot 2} & n_{54 \cdot 3} & n_{55 \cdot 4} & n_{56 \cdot 4} & n_{57 \cdot 4} & n_{58 \cdot 4} \\ 4 & n_{62 \cdot 1} & n_{63 \cdot 2} & n_{64 \cdot 3} & n_{64 \cdot 4} & n_{66 \cdot 5} & n_{67 \cdot 5} & n_{68 \cdot 5} \\ 8 & n_{72 \cdot 1} & n_{73 \cdot 2} & n_{74 \cdot 3} & n_{75 \cdot 4} & n_{76 \cdot 5} & n_{77 \cdot 6} & n_{78 \cdot 6} \\ 4 & n_{82 \cdot 1} & n_{83 \cdot 2} & n_{84 \cdot 3} & n_{85 \cdot 4} & n_{86 \cdot 5} & n_{88 \cdot 6} & n_{88 \cdot 7} \end{matrix}) \\ = (\begin{matrix} - 7 & - 12 & - 9 & 0 & - 5 & 3 & - 8 & 1 \\ 14 & - 31 & n_{23 \cdot 1} & n_{24 \cdot 1} & n_{25 \cdot 1} & n_{26 \cdot 1} & n_{27 \cdot 1} & n_{28 \cdot 1} \\ - 7 & n_{32 \cdot 1} & - 16.4516 & n_{34 \cdot 2} & n_{35 \cdot 2} & n_{36 \cdot 2} & n_{37 \cdot 2} & n_{38 \cdot 2} \\ - 15 & n_{42 \cdot 1} & n_{43 \cdot 2} & 34.5024 & n_{45 \cdot 3} & n_{46 \cdot 3} & n_{47 \cdot 3} & n_{48 \cdot 3} \\ - 1 & n_{52 \cdot 1} & n_{53 \cdot 2} & n_{54 \cdot 3} & - 8.4079 & n_{56 \cdot 4} & n_{57 \cdot 4} & n_{58 \cdot 4} \\ 4 & n_{62 \cdot 1} & n_{63 \cdot 2} & n_{64 \cdot 3} & n_{65 \cdot 4} & - 9.4401 & n_{67 \cdot 5} & n_{68 \cdot 5} \\ 8 & n_{72 \cdot 1} & n_{73 \cdot 2} & n_{74 \cdot 3} & n_{75 \cdot 4} & n_{76 \cdot 5} & 22.9315 & n_{78 \cdot 6} \\ 4 & n_{82 \cdot 1} & n_{83 \cdot 2} & n_{84 \cdot 3} & n_{85 \cdot 4} & n_{86 \cdot 5} & n_{87 \cdot 6} & 1.8465 \end{matrix}) \end{matrix} - - - (2.1)

Inner element acquiring unit recycling formula (3) obtains the lower triangle element n of reduction coefficient matrix N_ji·(i-1)：

n_{j i \cdot (i - 1)} = \{\begin{matrix} a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}, n_{i i \cdot (i - 1)} > 0 \\ - (a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}), n_{i i \cdot (i - 1)} < 0 \end{matrix} - - - (3)

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

In the present embodiment, inner element acquiring unit is utilized the lower triangle element of formula (3) acquisition reduction coefficient matrix, suc as formula (3.1)Shown in:

N = (\begin{matrix} - 7 & - 12 & - 9 & 0 & - 5 & 3 & - 8 & 1 \\ 14 & - 31 & n_{23 \cdot 1} & n_{24 \cdot 1} & n_{25 \cdot 1} & n_{26 \cdot 1} & n_{27 \cdot 1} & n_{28 \cdot 1} \\ - 7 & n_{32 \cdot 1} & - 16.4516 & n_{34 \cdot 2} & n_{35 \cdot 2} & n_{36 \cdot 2} & n_{37 \cdot 2} & n_{38 \cdot 2} \\ - 15 & n_{42 \cdot 1} & n_{43 \cdot 2} & 34.5024 & n_{45 \cdot 3} & n_{46 \cdot 3} & n_{47 \cdot 3} & n_{48 \cdot 3} \\ - 1 & n_{52 \cdot 1} & n_{53 \cdot 2} & n_{54 \cdot 3} & - 8.4079 & n_{56 \cdot 4} & n_{57 \cdot 4} & n_{58 \cdot 4} \\ 4 & n_{62 \cdot 1} & n_{63 \cdot 2} & n_{64 \cdot 3} & n_{65 \cdot 4} & - 9.4401 & n_{67 \cdot 5} & n_{68 \cdot 5} \\ 8 & n_{72 \cdot 1} & n_{73 \cdot 2} & n_{74 \cdot 3} & n_{75 \cdot 4} & n_{76 \cdot 5} & 22.9315 & n_{78 \cdot 6} \\ 4 & n_{82 \cdot 1} & n_{83 \cdot 2} & n_{84 \cdot 3} & n_{85 \cdot 4} & n_{86 \cdot 5} & n_{87 \cdot 6} & 1.8465 \end{matrix})

= (\begin{matrix} - 7 & - 12 & - 9 & 0 & - 5 & 3 & - 8 & 1 \\ 14 & - 31 & n_{23 \cdot 1} & n_{24 \cdot 1} & n_{25 \cdot 1} & n_{26 \cdot 1} & n_{27 \cdot 1} & n_{28 \cdot 1} \\ - 7 & - 14 & - 16.4516 & n_{34 \cdot 2} & n_{35 \cdot 2} & n_{36 \cdot 2} & n_{37 \cdot 2} & n_{38 \cdot 2} \\ - 15 & - 27.7143 & 19.3226 & 34.5024 & n_{45 \cdot 3} & n_{46 \cdot 3} & n_{47 \cdot 3} & n_{48 \cdot 3} \\ - 1 & - 11.7143 & 29.8065 & 14.3277 & - 8.4079 & n_{56 \cdot 4} & n_{57 \cdot 4} & n_{58 \cdot 4} \\ 4 & 21.8571 & - 24.4194 & - 15.1345 & - 2.4816 & - 9.4401 & n_{67 \cdot 5} & n_{68 \cdot 5} \\ 8 & 0.7143 & 5.5484 & 0.9664 & 10.8584 & 0.5766 & 22.9315 & n_{78 \cdot 6} \\ 4 & - 0.1429 & - 2.7097 & - 7.0168 & 4.1557 & 11.7530 & - 16.0717 & 1.8465 \end{matrix}) - - - (3.1)

Inner element acquiring unit finally utilizes formula (4) to obtain the upper triangle element n of reduction coefficient matrix N_ji·(j-1)：

n_{j i \cdot (i - 1)} = \{\begin{matrix} a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}, n_{j j \cdot (j - 1)} > 0 \\ - (a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}), n_{j j \cdot (j - 1)} < 0 \end{matrix} - - - (4)

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

In the present embodiment, inner element acquiring unit is utilized the upper triangle element of formula (4) acquisition reduction coefficient matrix, suc as formula (4.1)Shown in:

\begin{matrix} N = (\begin{matrix} - 7 & - 12 & - 9 & 0 & - 5 & 3 & - 8 & 1 \\ 14 & - 31 & n_{23 \cdot 1} & n_{24 \cdot 1} & n_{25 \cdot 1} & n_{26 \cdot 1} & n_{27 \cdot 1} & n_{28 \cdot 1} \\ - 7 & - 14 & - 16.4516 & n_{34 \cdot 2} & n_{35 \cdot 2} & n_{36 \cdot 2} & n_{37 \cdot 2} & n_{38 \cdot 2} \\ - 15 & - 27.7143 & 19.3226 & 34.5024 & n_{45 \cdot 3} & n_{46 \cdot 3} & n_{47 \cdot 3} & n_{48 \cdot 3} \\ - 1 & - 11.7143 & 29.8065 & 14.3277 & - 8.4079 & n_{56 \cdot 4} & n_{57 \cdot 4} & n_{58 \cdot 4} \\ 4 & 21.8571 & - 24.4194 & - 15.1345 & - 2.4816 & - 9.4401 & n_{67 \cdot 5} & n_{68 \cdot 5} \\ 8 & 0.7143 & 5.5484 & 0.9664 & 10.8584 & 0.5766 & 22.9315 & n_{78 \cdot 6} \\ 4 & - 0.1429 & - 2.7097 & - 7.0168 & 4.1557 & 11.7530 & - 16.0717 & 1.8465 \end{matrix}) \\ = (\begin{matrix} - 7 & - 12 & - 9 & 0 & - 5 & 3 & - 8 & 1 \\ 14 & - 31 & 32 & - 12 & 18 & - 16 & 4 & - 6 \\ - 7 & - 14 & - 16.4516 & 12.5806 & - 1.8710 & 7.7742 & - 18.1935 & - 14.7079 \\ - 15 & - 27.7143 & 19.3226 & 34.5024 & - 2.5754 & 17.0064 & - 0.8017 & - 18.0555 \\ - 1 & - 11.7143 & 29.8065 & 14.3277 & - 8.4079 & - 17.6407 & 25.9982 & 13.0286 \\ 4 & 21.8571 & - 24.4194 & - 15.1345 & - 2.4816 & - 9.4401 & - 24.2288 & - 2.4098 \\ 8 & 0.7143 & 5.5484 & 0.9664 & 10.8584 & 0.5766 & 22.9315 & - 2.7720 \\ 4 & - 0.1429 & - 2.7097 & - 7.0168 & 4.1557 & 11.7530 & - 16.0717 & 1.8465 \end{matrix}) \end{matrix} - - - (4.1)

Thereby obtain reduction coefficient matrix N be:

N = (\begin{matrix} n_{11 \cdot 0} & ... & n_{1 (i - 1) \cdot 0} & n_{1 i \cdot 0} & ... & n_{1 M \cdot 0} \\ n_{21 \cdot 0} & ... & n_{2 (i - 1) \cdot 1} & n_{2 i \cdot 0} & ... & n_{2 M \cdot 1} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ n_{j 1 \cdot 0} & ... & n_{j (i - 1) \cdot (i - 2)} & n_{j i \cdot (i - 1)} & ... & n_{j M \cdot (j - 1)} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ n_{M 1 \cdot 0} & ... & n_{M (i - 1) \cdot (i - 2)} & n_{M i \cdot (i - 1)} & ... & n_{M M \cdot M - 1} \end{matrix})

In the present embodiment, the reduction coefficient matrix N of acquisition is:

N = (\begin{matrix} - 7 & - 12 & - 9 & 0 & - 5 & 3 & - 8 & 1 \\ 14 & - 31 & 32 & - 12 & 18 & - 16 & 4 & - 6 \\ - 7 & - 14 & - 16.4516 & 12.5806 & - 1.8710 & 7.7742 & - 18.1935 & - 14.7079 \\ - 15 & - 27.7143 & 19.3226 & 34.5024 & - 2.5754 & 17.0064 & - 0.8017 & - 18.0555 \\ - 1 & - 11.7143 & 29.8065 & 14.3277 & - 8.4079 & - 17.6407 & 25.9982 & 13.0286 \\ 4 & 21.8571 & - 24.4194 & - 15.1345 & - 2.4816 & - 9.4401 & - 24.2288 & - 2.4098 \\ 8 & 0.7143 & 5.5484 & 0.9664 & 10.8584 & 0.5766 & 22.9315 & - 2.7720 \\ 4 & - 0.1429 & - 2.7097 & - 7.0168 & 4.1557 & 11.7530 & - 16.0717 & 1.8465 \end{matrix})

Upper triangular matrix resolving cell is for decomposing the upper triangular matrix of matrix A to be decomposed;

Specifically, upper triangular matrix resolving cell, according to reduction coefficient matrix N, utilizes formula (5) that matrix A to be decomposed is decomposedFor lower triangular matrix

l_ji＝0,j＝1,2,…,M-1；i＝j+1,j+2,…,M

\{\begin{matrix} l_{i i} = \sqrt{n_{i i \cdot (i - 1)}}, & n_{i i \cdot (i - 1)} > 0; & i = 1, 2, ..., M \\ l_{i i} = - \sqrt{| n_{i i \cdot (i - 1)} |}, & n_{i i \cdot (i - 1)} < 0; & i = 1, 2, ..., M \end{matrix}

\{\begin{matrix} l_{j i} = \frac{1}{\sqrt{n_{i i \cdot (i - 1)}}} \cdot n_{j i \cdot (i - 1)}, & i = 1, 2, ..., M - 1; & j = i + 1, i + 2, ..., M; n_{i i \cdot (i - 1)} > 0 \\ l_{j i} = \frac{1}{- \sqrt{| n_{i i \cdot (i - 1)} |}} \cdot n_{j i \cdot (i - 1)}, & i = 1, 2, ..., M - 1; & j = i + 1, i + 2, ..., M; n_{i i \cdot (i - 1)} < 0 \end{matrix} - - - (5)

In the present embodiment, upper triangular matrix resolving cell, according to formula (4.1), utilizes formula (5) that matrix A to be decomposed is decomposed into down to threeAngle matrix L, shown in (5.1):

L = (\begin{matrix} - 2.6458 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 5.2915 & - 5.5678 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 2.6458 & 2.5145 & - 4.0561 & 0 & 0 & 0 & 0 & 0 \\ - 5.6695 & 4.9776 & - 4.7639 & 5.8740 & 0 & 0 & 0 & 0 \\ - 0.3780 & 2.1039 & - 7.3486 & 2.4392 & - 2.8996 & 0 & 0 & 0 \\ 1.5119 & - 3.9257 & 6.0205 & - 2.5765 & 0.8558 & - 3.0725 & 0 & 0 \\ 3.0237 & - 0.1283 & - 1.3679 & 0.1654 & - 3.7447 & - 0.1877 & 4.7887 & 0 \\ 1.5119 & 0.0257 & 0.6681 & - 1.1945 & - 1.4332 & - 3.8253 & - 3.3562 & 1.3589 \end{matrix}) - - - (5.1)

Lower triangular matrix resolving cell is for decomposing the lower triangular matrix of matrix A to be decomposed;

Specifically, lower triangular matrix resolving cell, according to reduction coefficient matrix N, utilizes formula (6) that matrix A to be decomposed is decomposedFor upper triangular matrix

r_ji＝0,i＝1,2,…,M-1；j＝i+1,i+2,…,M

\{\begin{matrix} r_{i i} = \sqrt{n_{i i \cdot (i - 1)}}, & n_{i i \cdot (i - 1)} > 0; & i = 1, 2, ..., M \\ r_{i i} = \sqrt{| n_{i i \cdot (i - 1)} |}, & n_{i i \cdot (i - 1)} < 0; & i = 1, 2, ..., M \end{matrix}

\{\begin{matrix} r_{j i} = \frac{1}{\sqrt{n_{j j \cdot (j - 1)}}} \cdot n_{j i \cdot (j - 1)}, & n_{j j \cdot (j - 1)} > 0; & j = 1, 2, ..., M - 1; i = j + 1, j + 2, ..., M \\ r_{j i} = \frac{1}{\sqrt{| n_{j j \cdot (j - 1)} |}} \cdot n_{j i \cdot (j - 1)}, & n_{j j \cdot (j - 1)} < 0; & j = 1, 2, ..., M - 1; i = j + 1, j + 2, ..., M \end{matrix} - - - (6)

In the present embodiment, lower triangular matrix resolving cell, according to formula (4.1), utilizes formula (6) that matrix A to be decomposed is decomposed intoTriangular matrix R, shown in (6.1):

R = (\begin{matrix} 2.6485 & 4.5356 & 3.4017 & 0 & 1.8898 & - 1.1339 & 3.0237 & - 0.3780 \\ 0 & 5.5678 & 5.7474 & - 2.1553 & 3.2329 & - 2.8737 & 0.7184 & - 1.0776 \\ 0 & 0 & 4.0561 & 3.1017 & - 0.4613 & 1.9167 & - 4.4855 & - 3.6266 \\ 0 & 0 & 0 & 5.8740 & - 0.4384 & 2.8952 & - 0.1365 & - 3.0738 \\ 0 & 0 & 0 & 0 & 2.8996 & - 6.0837 & 8.9660 & 4.4932 \\ 0 & 0 & 0 & 0 & 0 & 3.0725 & - 7.8858 & - 0.7843 \\ 0 & 0 & 0 & 0 & 0 & 0 & 4.7887 & - 0.5789 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.3589 \end{matrix}) - - - (6.1)

In order to verify the effect of the matrix decomposition module proposing in this patent, the exponent number of group more than choosing at random M differences, matrix element are gotThe matrix that value scope is different, inputs in this novel matrix triangle decomposition module and carries out matrix decomposition experiment as sample matrix to be decomposed.The performance of matrix decomposition module proposing for objective appraisal this patent, by three angular moments that adopt after this patent decomposing module is decomposedBattle array result of product and former sample matrix to be decomposed contrast, and employing formula (7) is calculated and obtained maximum absolute error ε, and to differenceResult under experiment condition is evaluated and tested, and concrete outcome is as shown in table 1 below:

ε＝Max(|A-D|),D＝L·R(7)

The different matrix decomposition experimental error of table 1 result

In table 1,8 rank, 64 rank, three kinds, 1024 rank scale sample matrix, every kind of scale matrix element have been chosen in experiment at randomScope is respectively (1,1), (20,20), (1000,1000), and every kind of condition random is chosen four groups of different sample matrix and carried outTest, from worst error result data in table, the triangular matrix phase after the matrix decomposition decomposition module that employing this patent proposesTake advantage of result and former sample matrix to be decomposed very approaching, absolute error and relative error are all less, have higher operational precision,The decomposing module that adopts this patent to propose is decomposed effectively.

In addition,, from embodiment calculating process, the decomposing module that this patent proposes, in the time decomposing, first by reduction coefficientMatrix element is replaced the original matrix element of corresponding position, then replaces the reduction coefficient matrix of corresponding position by the triangular matrix element after decomposingElement, whole process is original position and replaces, and does not need additionally to take memory space. And each unit internal arithmetic process ratio in moduleOther decomposition algorithm and original original position replace Algorithm are all simpler, and each unit is can degree of parallelism very high, greatly the compression of degree fortuneEvaluation time. Thereby known, the matrix decomposition module that this patent proposes has higher efficiency, and its decomposition algorithm adopting is not only transportedCalculate that precision is higher, operation use time is shorter, and computational complexity is low, can degree of parallelism high, save memory space, it is very good to haveTheory and engineering using value.

Claims

1. a module of asking matrix triangle decomposition based on improved position displacement method, is characterized in that comprising: boundary element obtainsUnit, inner element acquiring unit, upper triangular matrix resolving cell and lower triangular matrix resolving cell;

A = (\begin{matrix} a_{11} & a_{12} & ... & a_{1 i} & ... & a_{1 M} \\ a_{21} & a_{22} & ... & a_{2 i} & ... & a_{2 M} \\ . & . & . & . \\ . & . & ... & . & ... & . \\ . & . & . & . \\ a_{j 1} & a_{j 2} & ... & a_{j i} & ... & a_{j M} \\ . & . & . & . \\ . & . & ... & . & ... & . \\ . & . & . & . \\ a_{M 1} & a_{M 2} & ... & a_{M i} & ... & a_{M M} \end{matrix})

2. the module of asking matrix triangle decomposition based on improved position displacement method according to claim 1, is characterized in that:

\{\begin{matrix} n_{1 i \cdot 0} = a_{1 i} \\ n_{j 1 \cdot 0} = a_{j 1} \end{matrix} - - - (1)

n_{i i \cdot (i - 1)} = a_{i i} - Σ_{k = 1}^{i - 1} \frac{n_{i k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}} - - - (2)

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

n_{j i \cdot (i - 1)} = \{\begin{matrix} a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}, n_{i i \cdot (i - 1)} > 0 \\ - (a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}), n_{i i \cdot (i - 1)} < 0 \end{matrix} - - - (3)

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

n_{j i \cdot (i - 1)} = \{\begin{matrix} a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}, n_{j j \cdot (j - 1)} > 0 \\ - (a_{j i} - Σ_{k = 1}^{i - 1} \frac{n_{j k \cdot (k - 1)} \cdot n_{k i \cdot (k - 1)}}{n_{k k \cdot (k - 1)}}), n_{j j \cdot (j - 1)} < 0 \end{matrix} - - - (4)

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

Thereby obtain reduction coefficient matrix N be:

N = (\begin{matrix} n_{11 \cdot 0} & ... & n_{1 (i - 1) \cdot 0} & n_{1 i \cdot 0} & ... & n_{1 M \cdot 0} \\ n_{21 \cdot 0} & ... & n_{2 (i - 1) \cdot 1} & n_{2 i \cdot 0} & ... & n_{2 M \cdot 1} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ n_{j 1 \cdot 0} & ... & n_{j (i - 1) \cdot (i - 2)} & n_{j i \cdot (i - 1)} & ... & n_{j M \cdot (j - 1)} \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ n_{M 1 \cdot 0} & ... & n_{M (i - 1) \cdot (i - 2)} & n_{M i \cdot (i - 1)} & ... & n_{M M \cdot M - 1} \end{matrix})

l_ji＝0,j＝1,2,…,M-1；i＝j+1,j+2,…,M

\{\begin{matrix} l_{i i} = \sqrt{n_{i i \cdot (i - 1)}}, & n_{i i \cdot (i - 1)} > 0; & i = 1, 2, ..., M \\ l_{i i} = - \sqrt{| n_{i i \cdot (i - 1)} |}, & n_{i i \cdot (i - 1)} < 0; & i = 1, 2, ..., M \end{matrix}

\{\begin{matrix} l_{j i} = \frac{1}{\sqrt{n_{i i \cdot (i - 1)}}} \cdot n_{j i \cdot (i - 1)}, & i = 1, 2, ..., M - 1; & j = i + 1, i + 2, ..., M; n_{i i \cdot (i - 1)} > 0 \\ l_{j i} = \frac{1}{- \sqrt{| n_{i i \cdot (i - 1)} |}} \cdot n_{j i \cdot (i - 1)}, & i = 1, 2, ..., M - 1; & j = i + 1, i + 2, ..., M; n_{i i \cdot (i - 1)} < 0 \end{matrix} - - - (5)

r_ji＝0,i＝1,2,…,M-1；j＝i+1,i+2,…,M

\{\begin{matrix} r_{i i} = \sqrt{n_{i i \cdot (i - 1)}}, & n_{i i \cdot (i - 1)} > 0; & i = 1, 2, ..., M \\ r_{i i} = \sqrt{| n_{i i \cdot (i - 1)} |}, & n_{i i \cdot (i - 1)} < 0; & i = 1, 2, ..., M \end{matrix}

\{\begin{matrix} r_{j i} = \frac{1}{\sqrt{n_{j j \cdot (j - 1)}}} \cdot n_{j i \cdot (j - 1)}, & n_{j j \cdot (j - 1)} > 0; & j = 1, 2, ..., M - 1; i = j + 1, j + 2, ..., M \\ r_{j i} = \frac{1}{\sqrt{| n_{j j \cdot (j - 1)} |}} \cdot n_{j i \cdot (j - 1)}, & n_{j j \cdot (j - 1)} < 0; & j = 1, 2, ..., M - 1; i = j + 1, j + 2, ..., M \end{matrix} - - - (6) .