CN104899183A

CN104899183A - Rapid Gauss-Jordan elimination method for symbolic linear system

Info

Publication number: CN104899183A
Application number: CN201510362146.5A
Authority: CN
Inventors: 李轶; 朱广; 冯勇; 杨文强
Original assignee: Chongqing Institute of Green and Intelligent Technology of CAS
Current assignee: Chongqing Institute of Green and Intelligent Technology of CAS
Priority date: 2015-06-26
Filing date: 2015-06-26
Publication date: 2015-09-09

Abstract

The invention relates to a fast Gaussian Jordan elimination method for symbolic linear systems, which belongs to the technical field of computers. The method includes the following steps: 1) according to the engineering system problem, a mathematical physical model is established, and a linear system is constructed after Laplace transformation, and the system state space equation in the form of a symbolic matrix is listed; 2) the number of times to be eliminated is determined according to the size of the matrix, Using the fast Gaussian Jordan elimination method to directly construct the accurate result of the system state space equation simplified into a diagonal matrix; 3) Calculate the mathematical and physical relationship between the input and output of the system according to the one-to-one correspondence. This method analyzes and transforms the complex Gaussian Jordan elimination process through mathematical methods, and directly calculates the simplified state equation based on the initial state equation, eliminating the calculation of the intermediate process; in addition, the simplified matrix is a diagonal The matrix is a one-to-one correspondence, which reduces the back-substitution process of other elimination methods and greatly improves the calculation speed.

Description

A Fast Gaussian Jordan Elimination Method for Symbolic Linear Systems

技术领域technical field

本发明属于计算机技术领域，涉及一种用于符号线性系统的快速高斯约当消去方法。The invention belongs to the technical field of computers and relates to a fast Gaussian Jordan elimination method for symbolic linear systems.

背景技术Background technique

随着计算机仿真技术的迅猛发展以及虚拟设计的兴起，工业生产加工过程中对仿真技术的在适用性、准确性、实时性、可靠性等性能的要求也越来越高。With the rapid development of computer simulation technology and the rise of virtual design, the requirements for the performance of simulation technology in terms of applicability, accuracy, real-time performance and reliability are getting higher and higher in the process of industrial production and processing.

目前，业界应用的多物理场建模软件主要包括Simulink、Ansys、MapleSim、Dymola、SimulationX、MWork等，其中主流软件主要以速度较快的数值计算为主，但其在计算过程中会由于截断误差的累积造成精度的降低，以及无法保留化简结果以便下次计算，无法应用到实时性要求较高的硬件在环检测技术上。而符号计算虽然能有效的避免中间过程的阶段误差，时不变系统实时性好，能够便于研发人员研究系统输入、输出量之间的关系，但是其计算过程速度慢。At present, the multi-physics modeling software used in the industry mainly includes Simulink, Ansys, MapleSim, Dymola, SimulationX, MWork, etc. Among them, the mainstream software mainly focuses on fast numerical calculations, but it will suffer from truncation errors during the calculation process. Accumulation results in a reduction in accuracy, and the simplification results cannot be retained for the next calculation, and cannot be applied to hardware-in-the-loop detection technology with high real-time requirements. Although symbolic calculation can effectively avoid the stage error in the intermediate process, the time-invariant system has good real-time performance, and can facilitate the research and development personnel to study the relationship between the input and output of the system, but the calculation process is slow.

发明内容Contents of the invention

有鉴于此，本发明的目的在于提供一种用于符号线性系统的快速高斯约当消去方法，该方法适用于多物理场建模软件中的符号线性系统化简求解，能够极大的提高线性系统进行快速高斯约当消去符号计算的速度。In view of this, the object of the present invention is to provide a fast Gaussian Jordan elimination method for symbolic linear systems, which is applicable to the simplification and solution of symbolic linear systems in multiphysics modeling software, and can greatly improve the linearity The speed at which the system performs fast Gaussian Jordan elimination symbol calculations.

为达到上述目的，本发明提供如下技术方案：To achieve the above object, the present invention provides the following technical solutions:

一种用于符号线性系统的快速高斯约当消去方法，其特征在于：包括以下步骤：A fast Gaussian Jordan elimination method for symbolic linear systems, characterized in that: comprising the following steps:

步骤一：根据工程系统问题，建立数学物理模型，拉氏变换后构建为线性系统，列出符号矩阵形式的系统状态空间方程；Step 1: According to the engineering system problem, establish a mathematical physical model, construct a linear system after Laplace transformation, and list the system state space equation in the form of a symbolic matrix;

步骤二：根据矩阵大小确定需要进行消去的次数，利用快速高斯约当消去方法直接构造出系统状态空间方程化简为对角矩阵后的准确结果；Step 2: Determine the number of eliminations that need to be performed according to the size of the matrix, and use the fast Gaussian Jordan elimination method to directly construct the accurate result after the system state space equation is simplified to a diagonal matrix;

步骤三：根据一一对应关系，计算出该系统输入、输出量之间的数学物理关系。Step 3: Calculate the mathematical and physical relationship between the input and output quantities of the system according to the one-to-one correspondence.

进一步，步骤一中的根据工程系统问题，建立数学物理模型，拉氏变换后构建为线性系统，列出的符号矩阵形式系统状态空间方程的一般形式为：Ax＝b，其中，为状态矩阵，x＝[x₁ x₂ … x_n]^T为输出量，b＝[b₁ b₂ … b_n]^T为输入量。Further, according to the engineering system problem in step 1, a mathematical physical model is established, and a linear system is constructed after Laplace transformation, and the general form of the state space equation of the listed symbolic matrix system is: Ax=b, where, is the state matrix, x＝[x ₁ x ₂ … x _n ] ^T is the output, b＝[b ₁ b ₂ … b _n ] ^T is the input.

进一步，步骤二中的确定需要消去的次数s表述为：由步骤一所得状态空间方程的增广矩阵(A|b)＝(a_i,j)_n×m的大小n×m来确定，s＝min(n,m)，工程中通常n＝m-1，s＝n。Further, the determination of the number of times s that needs to be eliminated in step 2 is expressed as: determined by the size n×m of the augmented matrix (A|b)=(a _i,j ) _n×m of the state space equation obtained in step 1, s =min(n,m), usually n=m-1, s=n in engineering.

进一步，步骤二中的快速高斯约当(Gauss-Jordan)消去方法具体包括以下步骤：Further, the fast Gauss-Jordan (Gauss-Jordan) elimination method in step 2 specifically includes the following steps:

1)初始值设定：定义表示增广矩阵(A|b)经过k次消去后第i行第j列元素，表示增广矩阵(A|b)经过k次消去后第i行第j列元素的构造因子；给定当1≤i≤n,1≤j<m时，当1≤i≤n,j＝m时， 1) Initial value setting: definition Indicates that the augmented matrix (A|b) undergoes k times of elimination and the i-th row and j-th column element, Indicates the construction factor of the element in row i and column j of the augmented matrix (A|b) after k times of elimination; given When 1≤i≤n, 1≤j<m, When 1≤i≤n, j=m,

2)进行第k次消去后(0≤k≤s)，2) After the kth elimination (0≤k≤s),

当i>k,j>k时，元素的构造因子 $a_{i, j}^{(k)} = |\begin{matrix} a_{11}^{0} & ... & a_{1, k}^{0} & a_{1, j}^{0} \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ a_{k, 1}^{0} & ... & a_{k, k}^{0} & a_{k, j}^{0} \\ a_{i, 1}^{0} & ... & a_{i, k}^{0} & a_{i, j}^{0} \end{matrix}|;$ When i>k, j>k, the construction factor of the element $a_{i, j}^{(k)} = |\begin{matrix} a_{11}^{0} & ... & a_{1, k}^{0} & a_{1, j}^{0} \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ a_{k, 1}^{0} & ... & a_{k, k}^{0} & a_{k, j}^{0} \\ a_{i, 1}^{0} & ... & a_{i, k}^{0} & a_{i, j}^{0} \end{matrix}|;$

当i<k,j>k时，元素的构造因子 $a_{i, j}^{(k)} = |\begin{matrix} a_{11}^{0} & ... & a_{1, i - 1}^{0} & a_{1, i + 1}^{0} & ... & a_{1, k}^{0} & a_{1, j}^{0} \\ a_{21}^{0} & ... & a_{2, i - 1}^{0} & a_{2, i + 1}^{0} & ... & a_{2, k}^{0} & a_{2, j}^{0} \\ . & . & . & . & . \\ . & . & . & . & . \\ . & . & . & . & . \\ a_{k, 1}^{0} & ... & a_{k, i - 1}^{0} & a_{k, i + 1}^{0} & ... & a_{k, k}^{0} & a_{k, j}^{0} \end{matrix}|;$ When i<k,j>k, the construction factor of the element $a_{i, j}^{(k)} = |\begin{matrix} a_{11}^{0} & ... & a_{1, i - 1}^{0} & a_{1, i + 1}^{0} & ... & a_{1, k}^{0} & a_{1, j}^{0} \\ a_{twenty one}^{0} & ... & a_{2, i - 1}^{0} & a_{2, i + 1}^{0} & ... & a_{2, k}^{0} & a_{2, j}^{0} \\ . & . & . & . & . \\ . & . & . & . & . \\ . & . & . & . & . \\ a_{k, 1}^{0} & ... & a_{k, i - 1}^{0} & a_{k, i + 1}^{0} & ... & a_{k, k}^{0} & a_{k, j}^{0} \end{matrix}|;$

${a a}_{i i,, j j}^{k k} = = \{\begin{matrix} \frac{{a a}_{i i,, j j}^{((k k))}}{{a a}_{k k,, k k}^{((k k - - 11))}} & i i > > k k,, j j > > k k \\ \frac{{a a}_{i i,, j j}^{((k k - - 11))}}{{a a}_{k k,, k k}^{((k k - - 11))}} & i i = = k k,, j j > > k k \\ \frac{{a a}_{i i,, j j}^{((k k))}}{{a a}_{k k,, k k}^{((k k - - 11))}} & i i = = k k - - 11,, j j > > k k \\ \frac{{((- - 11))}^{k k - - i i + + 11} {a a}_{i i,, j j}^{((k k))}}{{a a}_{k k,, k k}^{((k k - - 11))}} & i i \leq \leq k k - - 22,, j j > > k k \end{matrix}$

经过k＝s次消去化简后的系统状态空间方程对角矩阵形式为：The diagonal matrix form of the system state space equation after k=s elimination and simplification is:

进一步，在步骤三中，根据步骤二中的化简结果，同时根据工程系统通常m＝n+1，判断出该系统输入、输出量存在一一对应关系，并计算出它们之间的数学物理关系。Further, in step 3, according to the simplification result in step 2, and according to the engineering system usually m=n+1, it is judged that there is a one-to-one correspondence between the input and output of the system, and the mathematical physics between them is calculated relation.

本发明的有益效果在于：本发明所述的方法在化简速度上远优于传统方法，避免了高斯消去法的回代过程，节约了计算时间；同时，符号计算有利于保证计算结果的精度和明确的物理关系，能够极大的提高线性系统进行快速高斯约当消去符号计算的速度。The beneficial effects of the present invention are: the method of the present invention is far superior to the traditional method in terms of simplification speed, avoids the back-substitution process of Gaussian elimination method, and saves calculation time; at the same time, symbolic calculation is beneficial to ensure the accuracy of calculation results And the clear physical relationship can greatly improve the speed of fast Gauss Jordan elimination symbol calculation for linear systems.

附图说明Description of drawings

为了使本发明的目的、技术方案和有益效果更加清楚，本发明提供如下附图进行说明：In order to make the purpose, technical scheme and beneficial effect of the present invention clearer, the present invention provides the following drawings for illustration:

图1为本发明所述方法的流程示意图；Fig. 1 is a schematic flow sheet of the method of the present invention;

图2为本发明实施案例示意图。Fig. 2 is a schematic diagram of an embodiment of the present invention.

具体实施方式Detailed ways

下面将结合附图，对本发明的优选实施例进行详细的描述。The preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

图1为本发明所述方法的流程示意图，如图所示，本方法包括以下步骤：步骤一，根据工程系统问题，建立数学物理模型，拉氏变换后构建为线性系统，列出符号矩阵形式的系统状态空间方程；步骤二，根据矩阵大小确定需要进行消去的次数，利用快速高斯约当消去方法直接构造出系统状态空间方程化简为对角矩阵后的准确结果；步骤三，根据一一对应关系，计算出该系统输入、输出量之间的数学物理关系。Fig. 1 is the schematic flow chart of the method of the present invention, as shown in the figure, this method comprises the following steps: Step 1, according to engineering system problem, establishes mathematical physics model, builds into linear system after Laplace transform, lists symbolic matrix form The system state space equation; Step 2, determine the number of times to be eliminated according to the size of the matrix, and use the fast Gaussian Jordan elimination method to directly construct the accurate result after the system state space equation is simplified into a diagonal matrix; Step 3, according to one by one Correspondence, calculate the mathematical and physical relationship between the input and output of the system.

下面将结合附图1，对本发明的优选实施例进行详细的描述：Below in conjunction with accompanying drawing 1, preferred embodiment of the present invention is described in detail:

步骤一中的工程系统的系统状态空间方程的一般形式为：The general form of the system state space equation of the engineering system in step 1 is:

其中，x＝[x₁ x₂ … x_n]^T为状态变量，y＝[y₁ y₂ … y_q]^T为输出变量，u＝[u₁ u₂ … u_p]^T为输入变量，A＝(a_i,j(t))_n×n为系统矩阵，B＝(b_i,j(t))_n×p为控制、输入或分布矩阵，C＝(c_i,j(t))_q×n为输出矩阵，D＝(d_i,j(t))_q×p为输出分布矩阵。Among them, x＝[x ₁ x ₂ ... x _n ] ^T is the state variable, y＝[y ₁ y ₂ ... y _q ] ^T is the output variable, u＝[u ₁ u ₂ ... u _p ] ^T is the input variable, A=(a _i,j (t)) _n×n is the system matrix, B=(b _i,j (t)) _n×p is the control, input or distribution matrix, C=( _ci,j (t) ) _q×n is the output matrix, D=(d _i,j (t)) _q×p is the output distribution matrix.

对于时不变系统，拉氏变换线性化后的矩阵形式的系统状态方程为：For a time-invariant system, the system state equation in matrix form after Laplace transform linearization is:

$\{\begin{matrix} ((I I s the s - - A A)) \cdot &Center Dot; X x ((s the s)) = = B B \cdot \cdot U u ((s the s)) \\ Y Y ((s the s)) = = C C \cdot &Center Dot; X x ((s the s)) + + D D. \cdot \cdot U u ((s the s)) \end{matrix}$

结合附图2，在本实施例中，状态变量 $x = {[\begin{matrix} y_{1} & {\overset{\cdot}{y}}_{1} & y_{2} & {\overset{\cdot}{y}}_{2} \end{matrix}]}^{T},$ 输出变量y＝[y₁ y₂]^T，In conjunction with accompanying drawing 2, in the present embodiment, state variable $x = {[\begin{matrix} {the y}_{1} & {\overset{\cdot}{the y}}_{1} & {the y}_{2} & {\overset{\cdot}{the y}}_{2} \end{matrix}]}^{T},$ output variable y=[y ₁ y ₂ ] ^T ,

输入变量u＝[u ₁u₂]^T，系统矩阵 $A = [\begin{matrix} 0 & 1 & 0 & 0 \\ \frac{k_{1}}{m_{1}} & \frac{η_{1}}{m_{1}} & - \frac{k_{1}}{m_{1}} & - \frac{η_{1}}{m_{1}} \\ 0 & 0 & 0 & 1 \\ 0 & 0 & \frac{k_{2}}{m_{2}} & \frac{η_{2}}{m_{2}} \end{matrix}],$ 控制、输入或分布矩阵 $B = [\begin{matrix} 0 \\ - \frac{u_{1}}{m_{1}} \\ 0 \\ - \frac{u_{2}}{m_{2}} \end{matrix}],$ 输出矩阵C＝[1 0 1 0]，输出分布矩阵D＝[0 0]。Input variable u＝[u ₁ u ₂ ] ^T , system matrix $A = [\begin{matrix} 0 & 1 & 0 & 0 \\ \frac{k_{1}}{m_{1}} & \frac{η_{1}}{m_{1}} & - \frac{k_{1}}{m_{1}} & - \frac{η_{1}}{m_{1}} \\ 0 & 0 & 0 & 1 \\ 0 & 0 & \frac{k_{2}}{m_{2}} & \frac{η_{2}}{m_{2}} \end{matrix}],$ Control, input or distribution matrix $B = [\begin{matrix} 0 \\ - \frac{u_{1}}{m_{1}} \\ 0 \\ - \frac{u_{2}}{m_{2}} \end{matrix}],$ Output matrix C=[1 0 1 0], output distribution matrix D=[0 0].

针对系统状态方程，进行拉氏变换后，其增广矩阵为For the system state equation, after Laplace transform, its augmented matrix is

$((((I I - - A A)) | | ((B B U u ((s the s)))))) = = [\begin{matrix} s the s & - - 11 & 00 & 00 & 00 \\ - - \frac{{k k}_{11}}{{m m}_{11}} & s the s - - \frac{{η η}_{11}}{{m m}_{11}} & \frac{{k k}_{11}}{{m m}_{11}} & \frac{{η η}_{11}}{{m m}_{11}} & - - \frac{{U u}_{11} ((s the s))}{{m m}_{11}} \\ 00 & 00 & s the s & - - 11 & 00 \\ 00 & 00 & - - \frac{{k k}_{22}}{{m m}_{22}} & s the s - - \frac{{η η}_{22}}{{m m}_{22}} & - - \frac{{U u}_{22} ((s the s))}{{m m}_{22}} \end{matrix}]$

所得状态空间方程的增广矩阵的大小为4×5，故需要消去的次数s＝4。The size of the augmented matrix of the obtained state space equation is 4×5, so the number of elimination times s=4 is required.

利用快速高斯约当消去方法对其进行处理：It is processed using a fast Gaussian Jordan elimination method:

1)初始值设定：定义表示增广矩阵((I-A)|(BU(s)))经过k次消去后第i行第j列元素，表示增广矩阵((I-A)|(BU(s)))经过k次消去后第i行第j列元素的构造因子；给定当1≤i≤4,1≤j<5时，当1≤i≤4,j＝5时， 1) Initial value setting: definition Indicates the augmented matrix ((IA)|(BU(s))) after k times of elimination, the i-th row and the j-th column element, Indicates the construction factor of the element in row i and column j of the augmented matrix ((IA)|(BU(s))) after k times of elimination; given When 1≤i≤4, 1≤j<5, When 1≤i≤4, j=5,

2)进行第k次消去后(0≤k≤4)，2) After the kth elimination (0≤k≤4),

步骤二中的经过4次快速高斯约当消去方法化简后的系统状态空间方程对角矩阵形式为：The diagonal matrix form of the system state space equation after 4 times of fast Gaussian Jordan elimination method in step 2 is:

${((((I I - - A A)) | | ((B B U u ((s the s))))))}^{44} = = [\begin{matrix} 11 & 00 & 00 & 00 & \frac{- - {m m}_{22} {s the s}^{22} {U u}_{11} ((s the s)) + + {η η}_{11} {sU sU}_{22} ((s the s)) + + {η η}_{22} {sU sU}_{11} ((s the s)) + + {k k}_{11} {U u}_{22} ((s the s)) + + {k k}_{22} {U u}_{11} ((s the s))}{((- - {m m}_{22} {s the s}^{22} + + {η η}_{22} s the s + + {k k}_{22})) ((- - {m m}_{11} {s the s}^{22} + + {η η}_{11} s the s + + {k k}_{11}))} \\ 00 & 11 & 00 & 00 & \frac{s the s ((- - {m m}_{22} {s the s}^{22} {U u}_{11} ((s the s)) + + {η η}_{11} {sU sU}_{22} ((s the s)) + + {η η}_{22} {sU sU}_{11} ((s the s)) + + {k k}_{11} {U u}_{22} ((s the s)) + + {k k}_{22} {U u}_{11} ((s the s))))}{((- - {m m}_{22} {s the s}^{22} + + {η η}_{22} s the s + + {k k}_{22})) ((- - {m m}_{11} {s the s}^{22} + + {η η}_{11} s the s + + {k k}_{11}))} \\ 00 & 00 & 11 & 00 & \frac{{U u}_{22} ((s the s))}{- - {m m}_{22} {s the s}^{22} + + {η η}_{22} s the s + + {k k}_{22}} \\ 00 & 00 & 00 & 11 & \frac{{U u}_{22} ((s the s)) s the s}{- - {m m}_{22} {s the s}^{22} + + {η η}_{22} s the s + + {k k}_{22}} \end{matrix}]$

根据步骤二中的化简结果由(Ιs-A)·X(s)＝B·U(s)可以判断出该系统输入、输出量存在一一对应关系，即According to the simplification result in step 2 (Ιs-A) X(s) = B U(s), it can be judged that there is a one-to-one correspondence between the input and output of the system, that is

${Y Y}_{11} ((s the s)) = = \frac{- - {m m}_{22} {s the s}^{22} {U u}_{11} ((s the s)) + + {η η}_{11} {sU sU}_{22} ((s the s)) + + {η η}_{22} {sU sU}_{11} ((s the s)) + + {k k}_{11} {U u}_{22} ((s the s)) + + {k k}_{22} {U u}_{11} ((s the s))}{((- - {m m}_{22} {s the s}^{22} + + {η η}_{22} s the s + + {k k}_{22})) ((- - {m m}_{11} {s the s}^{22} + + {η η}_{11} s the s + + {k k}_{11}))}$

${Y Y}_{22} ((s the s)) = = \frac{{U u}_{22} ((s the s))}{- - {m m}_{22} {s the s}^{22} + + {η η}_{22} s the s + + {k k}_{22}}$

将本发明方法在Maple软件上编程实现后，对本实施案例进行化简处理，传统的高斯约当消去法的程序响应总时间为0.063秒，其中CPU进行的化简时间0.016秒，而本发明方法的程序总响应时间为0.046秒，其中CPU进行的化简时间几乎为0秒。可见，本发明方法在化简速度上远优于传统方法，避免了高斯消去法的回代过程，节约了计算时间；同时，符号计算有利于保证计算结果的精度和明确的物理关系。特别地，有限元分析过程中，网格化后的单元都可以看作是一个个串联或者并联的等效的质量——阻尼——弹簧系统，与该实施案例类似，采用本发明方法对其计算精度和计算速度都有很好的保证。After the method of the present invention is programmed on the Maple software, the implementation case is simplified. The total program response time of the traditional Gaussian Jordan elimination method is 0.063 seconds, and the simplification time carried out by the CPU is 0.016 seconds, while the method of the present invention The total response time of the program is 0.046 seconds, and the reduction time of the CPU is almost 0 seconds. It can be seen that the method of the present invention is far superior to the traditional method in simplification speed, avoids the back-substitution process of the Gaussian elimination method, and saves calculation time; at the same time, the symbolic calculation is beneficial to ensure the accuracy of the calculation result and the clear physical relationship. In particular, in the finite element analysis process, the meshed units can be regarded as equivalent mass-damping-spring systems connected in series or in parallel. Similar to this implementation case, the method of the present invention is used to The calculation accuracy and calculation speed are well guaranteed.

最后说明的是，以上优选实施例仅用以说明本发明的技术方案而非限制，尽管通过上述优选实施例已经对本发明进行了详细的描述，但本领域技术人员应当理解，可以在形式上和细节上对其作出各种各样的改变，而不偏离本发明权利要求书所限定的范围。Finally, it should be noted that the above preferred embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail through the above preferred embodiments, those skilled in the art should understand that it can be described in terms of form and Various changes may be made in the details without departing from the scope of the invention defined by the claims.

Claims

1. A fast Gaussian Jordan elimination method for symbolic linear systems, characterized in that: comprising the following steps:

Step 1: According to the engineering system problem, establish a mathematical physical model, construct a linear system after Laplace transformation, and list the system state space equation in the form of a symbolic matrix;

Step 2: Determine the number of eliminations that need to be performed according to the size of the matrix, and use the fast Gaussian Jordan elimination method to directly construct the accurate result after the system state space equation is simplified to a diagonal matrix;

Step 3: Calculate the mathematical and physical relationship between the input and output quantities of the system according to the one-to-one correspondence.

2. a kind of fast Gaussian Jordan elimination method that is used for symbolic linear system according to claim 1, is characterized in that: according to engineering system problem in the step 1, set up mathematical physics model, build into linear system after Laplace transform , the general form of the state-space equation of the system listed in symbolic matrix form is: Ax=b, where, is the state matrix, x＝[x ₁ x ₂ … x _n ] ^T is the output, b＝[b ₁ b ₂ … b _n ] ^T is the input.

3. a kind of fast Gaussian Jordan elimination method that is used for symbolic linear system according to claim 2, is characterized in that: the number of times s that needs to eliminate in step 2 is expressed as: by the increase of state space equation of step 1 gained The wide matrix (A|b)=(a _i,j ) is determined by the size n× _m of n×m, s=min(n,m), usually n=m-1, s=n in engineering.

4. a kind of fast Gaussian Jordan elimination method for symbolic linear system according to claim 3, is characterized in that: the fast Gaussian Jordan elimination method in the step 2 specifically comprises the following steps:

1) Initial value setting: definition Indicates that the augmented matrix (A|b) undergoes k times of elimination and the i-th row and j-th column element, Indicates the construction factor of the element in row i and column j of the augmented matrix (A|b) after k times of elimination; given When 1≤i≤n, 1≤j<m, When 1≤i≤n, j=m,

2) After the kth elimination (0≤k≤s),

When i>k, j>k, the construction factor of the element

a_{i, j}^{(k)} = | \begin{matrix} a_{11}^{0} & ... & a_{1, k}^{0} & a_{1, j}^{0} \\ &Center Dot; & &Center Dot; & \cdot \\ \cdot & \cdot & \cdot \\ &Center Dot; & &Center Dot; & &Center Dot; \\ a_{k, 1}^{0} & ... & a_{k, k}^{0} & a_{k, j}^{0} \\ a_{i, 1}^{0} & ... & a_{i, k}^{0} & a_{i, j}^{0} \end{matrix} |;

When i<k,j>k, the construction factor of the element

a_{i, j}^{(k)} = | \begin{matrix} a_{11}^{0} & ... & a_{1, i - 1}^{0} & a_{1, i + 1}^{0} & ... & a_{1, k}^{0} & a_{1, j}^{0} \\ a_{twenty one}^{0} & ... & a_{2, i - 1}^{0} & a_{2, i + 1}^{0} & ... & a_{2, k}^{0} & a_{2, j}^{0} \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & &Center Dot; \\ &Center Dot; & &Center Dot; & &Center Dot; & \cdot & \cdot \\ a_{k, 1}^{0} & ... & a_{k, i - 1}^{0} & a_{k, i + 1}^{0} & ... & a_{k, k}^{0} & a_{k, j}^{0} \end{matrix} |;

3) Combined with step 2), the augmented matrix (A|b) is transformed into (A|b) ^k after the k-th elimination (0≤k≤s), where the elements of (A|b) ^k satisfy:

{a a}_{i i,, j j}^{k k} = = \{\begin{matrix} \frac{{a a}_{i i,, j j}^{((k k))}}{{a a}_{k k,, k k}^{((k k - - 11))}} & i i > > k k,, j j > > k k \\ \frac{{a a}_{i i,, j j}^{((k k - - 11))}}{{a a}_{k k,, k k}^{((k k - - 11))}} & i i = = k k,, j j > > k k \\ \frac{{a a}_{i i,, j j}^{((k k))}}{{a a}_{k k,, k k}^{((k k - - 11))}} & i i = = k k - - 11,, j j > > k k \\ \frac{{((- - 11))}^{k k - - i i + + 11} {a a}_{i i,, j j}^{((k k))}}{{a a}_{k k,, k k}^{((k k - - 11))}} & i i \leq \leq k k - - 22,, j j > > k k \end{matrix}

The diagonal matrix form of the system state space equation after k=s elimination and simplification is:

5. A kind of fast Gaussian Jordan elimination method for symbolic linear system according to claim 4, is characterized in that: in step 3, according to the simplification result in step 2, according to engineering system m=n usually simultaneously +1, it is judged that there is a one-to-one correspondence between the input and output of the system, and the mathematical and physical relationship between them is calculated.