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

Abstract

The invention relates to a rapid Gauss-Jordan elimination method for a symbolic linear system, and belongs to the technical field of computers. The method comprises the steps of 1) building a mathematical physical model according to the problem of an engineering system, performing Laplace transform and then arranging into a linear system, and listing a system state space equation in a symbolic matrix form; 2) determining elimination times required according to the size of the matrix, and directly producing the accurate result after simplifying a system state space equation into a diagonal matrix by the rapid Gauss-Jordan elimination method; 3) calculating the mathematical physical relationship between the input quantity and the output quantity of the system according to the one-by-one correspondence relationship. According to the method, the complex Gauss-Jordan elimination process is analyzed and converted by the mathematical method; the simplified state equation can be directly calculated according to the initial state equation; the calculation of the medium process is saved; in addition, the simplified matrix is the diagonal matrix under the one-by-one correspondence relationship; therefore, the back substitution of other elimination method is reduced, and as a result, the calculation speed is greatly increased.

Description

A kind of quick Gauss for symbol linear system about works as elimination method

Technical field

The invention belongs to field of computer technology, relate to a kind of quick Gauss for symbol linear system and about work as elimination method.

Background technology

Along with the fast development of computer simulation technique and the rise of virtual design, also more and more higher in the requirement of the performances such as applicability, accuracy, real-time, reliability to emulation technology in commercial production process.

At present, the multiple physical field modeling software of industry application mainly comprises Simulink, Ansys, MapleSim, Dymola, SimulationX, MWork etc., wherein main software is mainly based on the numerical evaluation of speed, but it can cause the reduction of precision due to the accumulation of truncation error in computation process, and abbreviation result cannot be retained so that next time calculates, cannot be applied in the higher hardware in loop detection technique of requirement of real-time.And although symbolic computation effectively can avoid the truncation errors of pilot process, time-invariant system real-time is good, and can be convenient to the relation between the input of research staff's Study system, output quantity, its computation process speed is slow.

Summary of the invention

In view of this, a kind of quick Gauss for symbol linear system is the object of the present invention is to provide about to work as elimination method, the symbol linear system abbreviation that the method is applicable in multiple physical field modeling software solves, and can greatly improve linear system and carry out quick Gauss about when the speed of cancellation symbolic computation.

For achieving the above object, the invention provides following technical scheme:

Quick Gauss for symbol linear system about works as an elimination method, it is characterized in that: comprise the following steps:

Step one: according to engineering system problem, set up mathematics physics model, be configured to linear system after Laplace transformation, lists the system state space equation of sign matrix form;

Step 2: determine the number of times needing to carry out cancellation according to matrix size, utilizes quick Gauss about when elimination method directly constructs system state space equation abbreviation for the accurate result after diagonal matrix;

Step 3: according to one-to-one relationship, calculates the mathematical physics relation between the input of this system, output quantity.

Further, in step one according to engineering system problem, set up mathematics physics model, be configured to linear system after Laplace transformation, the general type of the sign matrix formal system state space equation listed is: Ax=b, wherein, for state matrix, x=[x ₁x ₂x _n] ^tfor output quantity, b=[b ₁b ₂b _n] ^tfor input quantity.

Further, the determination in step 2 needs the number of times s of cancellation to be expressed as: by augmented matrix (A|b)=(a of step one gained state space equation _i,j) _{n × m}size n × m determine, s=min (n, m), usual n=m-1, s=n in engineering.

Further, the quick Gauss in step 2 is about when (Gauss-Jordan) elimination method specifically comprises the following steps:

1) initial value setting: definition represent augmented matrix (A|b) the i-th row jth column element after k cancellation, represent the structure requirement of augmented matrix (A|b) i-th row jth column element after k cancellation; Given as 1≤i≤n, during 1≤j<m, as 1≤i≤n, j=m,

2) carry out (0≤k≤s) after kth time cancellation,

Work as i>k, during j>k, the structure requirement of element $a_{i,j}^{\left(k\right)}=\left|\begin{array}{cccc}{a}_{11}^0& \mathrm{...}& a_{1,k}^0& a_{1,j}^0\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ a_{k,1}^0& \mathrm{...}& a_{k,k}^0& a_{k,j}^0\\ a_{i,1}^0& \mathrm{...}& a_{i,k}^0& a_{i,j}^0\end{array}\right|;$

Work as i<k, during j>k, the structure requirement of element $a_{i,j}^{\left(k\right)}=\left|\begin{array}{ccccccc}{a}_{11}^0& \mathrm{...}& a_{1,i-1}^0& a_{1,i+1}^0& \mathrm{...}& a_{1,k}^0& a_{1,j}^0\\ a_{21}^0& \mathrm{...}& a_{2,i-1}^0& a_{2,i+1}^0& \mathrm{...}& a_{2,k}^0& a_{2,j}^0\\ .& & .& .& & .& .\\ .& & .& .& & .& .\\ .& & .& .& & .& .\\ a_{k,1}^0& \mathrm{...}& a_{k,i-1}^0& a_{k,i+1}^0& \mathrm{...}& a_{k,k}^0& a_{k,j}^0\end{array}\right|;$

3) integrating step 2), after augmented matrix (A|b) carries out kth time cancellation, (0≤k≤s) changes into (A|b) ^k, wherein (A|b) ^kelement meet:

$a_{i,j}^k=\left\{\begin{array}{cc}\frac{a_{i,j}^{\left(k\right)}}{a_{k,k}^{(k-1)}}& i>k,j>k\\ \frac{a_{i,j}^{(k-1)}}{a_{k,k}^{(k-1)}}& i=k,j>k\\ \frac{a_{i,j}^{\left(k\right)}}{a_{k,k}^{(k-1)}}& i=k-1,j>k\\ \frac{{(-1)}^{k-i+1}{a}_{i,j}^{\left(k\right)}}{a_{k,k}^{(k-1)}}& i\&le;k-2,j>k\end{array}\right.$

System state space equation diagonal matrix form after k=s cancellation abbreviation is:

Further, in step 3, according to the abbreviation result in step 2, simultaneously according to the usual m=n+1 of engineering system, judge the input of this system, output quantity exists one-to-one relationship, and calculate the mathematical physics relation between them.

Beneficial effect of the present invention is: method of the present invention is much better than classic method in abbreviation speed, avoids the backward steps of Gaussian elimination, has saved computing time; Meanwhile, symbolic computation is conducive to ensureing the precision of result of calculation and clear and definite physical relation, can greatly improve linear system and carry out quick Gauss about when the speed of cancellation symbolic computation.

Accompanying drawing explanation

In order to make object of the present invention, technical scheme and beneficial effect clearly, the invention provides following accompanying drawing and being described:

Fig. 1 is the schematic flow sheet of the method for the invention;

Fig. 2 is the invention process case schematic diagram.

Embodiment

Below in conjunction with accompanying drawing, the preferred embodiments of the present invention are described in detail.

Fig. 1 is the schematic flow sheet of the method for the invention, and as shown in the figure, this method comprises the following steps: step one, according to engineering system problem, set up mathematics physics model, after Laplace transformation, be configured to linear system, list the system state space equation of sign matrix form; Step 2, determines the number of times needing to carry out cancellation according to matrix size, utilize quick Gauss about when elimination method directly constructs system state space equation abbreviation for the accurate result after diagonal matrix; Step 3, according to one-to-one relationship, calculates the mathematical physics relation between the input of this system, output quantity.

Below in conjunction with accompanying drawing 1, the preferred embodiments of the present invention are described in detail:

The general type of the system state space equation of the engineering system in step one is:

Wherein, x=[x ₁x ₂x _n] ^tfor state variable, y=[y ₁y ₂y _q] ^tfor output variable, u=[u ₁u ₂u _p] ^tfor input variable, A=(a _i,j(t)) _{n × n}for system matrix, B=(b _i,j(t)) _{n × p}for controlling, inputting or distribution matrix, C=(c _i,j(t)) _{q × n}for output matrix, D=(d _i,j(t)) _{q × p}for exporting distribution matrix.

For time-invariant system, the system state equation of the matrix form after Laplace transformation linearization is:

$\left\{\begin{array}{c}(Is-A)\&CenterDot;X\left(s\right)=B\&CenterDot;U\left(s\right)\\ Y\left(s\right)=C\&CenterDot;X\left(s\right)+D\&CenterDot;U\left(s\right)\end{array}\right.$

By reference to the accompanying drawings 2, in the present embodiment, state variable $x={\left[\begin{array}{cccc}{y}_1& {\stackrel{\&CenterDot;}{y}}_1& y_2& {\stackrel{\&CenterDot;}{y}}_2\end{array}\right]}^T,$ Output variable y=[y ₁y ₂] ^t,

Input variable u=[u ₁u ₂] ^t, system matrix $A=\left[\begin{array}{cccc}0& 1& 0& 0\\ \frac{k_1}{m_1}& \frac{{\mathrm{\&eta;}}_1}{m_1}& -\frac{k_1}{m_1}& -\frac{{\mathrm{\&eta;}}_1}{m_1}\\ 0& 0& 0& 1\\ 0& 0& \frac{k_2}{m_2}& \frac{{\mathrm{\&eta;}}_2}{m_2}\end{array}\right],$ Control, input or distribution matrix $B=\left[\begin{array}{c}0\\ -\frac{u_1}{m_1}\\ 0\\ -\frac{u_2}{m_2}\end{array}\right],$ Output matrix C=[1 01 0], exports distribution matrix D=[0 0].

For system state equation, after carrying out Laplace transformation, its augmented matrix is

$\left(\right(I-A\left)\right|\left(BU\right(s\left)\right))=\left[\begin{array}{ccccc}s& -1& 0& 0& 0\\ -\frac{k_1}{m_1}& s-\frac{{\mathrm{\&eta;}}_1}{m_1}& \frac{k_1}{m_1}& \frac{{\mathrm{\&eta;}}_1}{m_1}& -\frac{U_1\left(s\right)}{m_1}\\ 0& 0& s& -1& 0\\ 0& 0& -\frac{k_2}{m_2}& s-\frac{{\mathrm{\&eta;}}_2}{m_2}& -\frac{U_2\left(s\right)}{m_2}\end{array}\right]$

The size of the augmented matrix of gained state space equation is 4 × 5, therefore needs the number of times s=4 of cancellation.

Utilize quick Gauss about when elimination method processes it:

1) initial value setting: definition expression augmented matrix ((I-A) | (BU (s))) the i-th row jth column element after k cancellation, expression augmented matrix ((I-A) | (BU (s))) structure requirement of the i-th row jth column element after k cancellation; Given when 1≤i≤4, during 1≤j<5, when 1≤i≤4, during j=5,

2) carry out (0≤k≤4) after kth time cancellation,

Work as i>k, during j>k, the structure requirement of element $a_{i,j}^{\left(k\right)}=\left|\begin{array}{cccc}{a}_{11}^0& \mathrm{...}& a_{1,k}^0& a_{1,j}^0\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ a_{k,1}^0& \mathrm{...}& a_{k,k}^0& a_{k,j}^0\\ a_{i,1}^0& \mathrm{...}& a_{i,k}^0& a_{i,j}^0\end{array}\right|;$

Work as i<k, during j>k, the structure requirement of element $a_{i,j}^{\left(k\right)}=\left|\begin{array}{ccccccc}{a}_{11}^0& \mathrm{...}& a_{1,i-1}^0& a_{1,i+1}^0& \mathrm{...}& a_{1,k}^0& a_{1,j}^0\\ a_{21}^0& \mathrm{...}& a_{2,i-1}^0& a_{2,i+1}^0& \mathrm{...}& a_{2,k}^0& a_{2,j}^0\\ .& & .& .& & .& .\\ .& & .& .& & .& .\\ .& & .& .& & .& .\\ a_{k,1}^0& \mathrm{...}& a_{k,i-1}^0& a_{k,i+1}^0& \mathrm{...}& a_{k,k}^0& a_{k,j}^0\end{array}\right|;$

3) integrating step 2), after augmented matrix ((I-A) | (BU (s))) carries out kth time cancellation, (0≤k≤4) change ((I-A) | (BU (s))) into ^k, wherein ((I-A) | (BU (s))) ^kelement meet:

$a_{i,j}^k=\left\{\begin{array}{cc}\frac{a_{i,j}^{\left(k\right)}}{a_{k,k}^{(k-1)}}& i>k,j>k\\ \frac{a_{i,j}^{(k-1)}}{a_{k,k}^{(k-1)}}& i=k,j>k\\ \frac{a_{i,j}^{\left(k\right)}}{a_{k,k}^{(k-1)}}& i=k-1,j>k\\ \frac{{(-1)}^{k-i+1}{a}_{i,j}^{\left(k\right)}}{a_{k,k}^{(k-1)}}& i\&le;k-2,j>k\end{array}\right.$

In step 2 through 4 quick Gausses about when the system state space equation diagonal matrix form after elimination method abbreviation is:

${\left(\right(I-A\left)\right|\left(BU\right(s\left)\right))}^4=\left[\begin{array}{ccccc}1& 0& 0& 0& \frac{-m_2{s}^2{U}_1\left(s\right)+{\mathrm{\&eta;}}_1{\mathrm{sU}}_2\left(s\right)+{\mathrm{\&eta;}}_2{\mathrm{sU}}_1\left(s\right)+k_1{U}_2\left(s\right)+k_2{U}_1\left(s\right)}{(-m_2{s}^2+{\mathrm{\&eta;}}_{2}s+k_2)(-m_1{s}^2+{\mathrm{\&eta;}}_{1}s+k_1)}\\ 0& 1& 0& 0& \frac{s(-m_2{s}^2{U}_1(s)+{\mathrm{\&eta;}}_1{\mathrm{sU}}_2\left(s\right)+{\mathrm{\&eta;}}_2{\mathrm{sU}}_1\left(s\right)+k_1{U}_2\left(s\right)+k_2{U}_1\left(s\right))}{(-m_2{s}^2+{\mathrm{\&eta;}}_{2}s+k_2)(-m_1{s}^2+{\mathrm{\&eta;}}_{1}s+k_1)}\\ 0& 0& 1& 0& \frac{U_2\left(s\right)}{-m_2{s}^2+{\mathrm{\&eta;}}_{2}s+k_2}\\ 0& 0& 0& 1& \frac{U_2\left(s\right)s}{-m_2{s}^2+{\mathrm{\&eta;}}_{2}s+k_2}\end{array}\right]$

The input of this system can be judged according to the abbreviation result in step 2 by (Ι s-A) X (s)=BU (s), output quantity exists one-to-one relationship, namely

$Y_1\left(s\right)=\frac{-m_2{s}^2{U}_1\left(s\right)+{\mathrm{\&eta;}}_1{\mathrm{sU}}_2\left(s\right)+{\mathrm{\&eta;}}_2{\mathrm{sU}}_1\left(s\right)+k_1{U}_2\left(s\right)+k_2{U}_1\left(s\right)}{(-m_2{s}^2+{\mathrm{\&eta;}}_{2}s+k_2)(-m_1{s}^2+{\mathrm{\&eta;}}_{1}s+k_1)}$

$Y_2\left(s\right)=\frac{U_2\left(s\right)}{-m_2{s}^2+{\mathrm{\&eta;}}_{2}s+k_2}$

By the inventive method on Maple software after programming realization, abbreviation process is carried out to the implementation case, traditional Gauss about when elimination method program response T.T. be 0.063 second, wherein CPU 0.016 second abbreviation time of carrying out, and the program overall response time of the inventive method is 0.046 second, the abbreviation time that wherein CPU carries out is almost 0 second.Visible, the inventive method is much better than classic method in abbreviation speed, avoids the backward steps of Gaussian elimination, has saved computing time; Meanwhile, symbolic computation is conducive to ensureing the precision of result of calculation and clear and definite physical relation.Especially, in finite element analysis process, unit after gridding can regard quality---damping---spring system of the equivalence of series connection or parallel connection one by one as, similar with this case study on implementation, adopts the inventive method all to have a good guarantee to its computational accuracy and computing velocity.

What finally illustrate is, above preferred embodiment is only in order to illustrate technical scheme of the present invention and unrestricted, although by above preferred embodiment to invention has been detailed description, but those skilled in the art are to be understood that, various change can be made to it in the form and details, and not depart from claims of the present invention limited range.

Claims (5)

1. the quick Gauss for symbol linear system about works as an elimination method, it is characterized in that: comprise the following steps:

Step one: according to engineering system problem, set up mathematics physics model, be configured to linear system after Laplace transformation, lists the system state space equation of sign matrix form;

Step 2: determine the number of times needing to carry out cancellation according to matrix size, utilizes quick Gauss about when elimination method directly constructs system state space equation abbreviation for the accurate result after diagonal matrix;

Step 3: according to one-to-one relationship, calculates the mathematical physics relation between the input of this system, output quantity.

2. a kind of quick Gauss for symbol linear system according to claim 1 about works as elimination method, it is characterized in that: in step one according to engineering system problem, set up mathematics physics model, linear system is configured to after Laplace transformation, the general type of the sign matrix formal system state space equation listed is: Ax=b, wherein for state matrix, x=[x ₁x ₂x _n] ^tfor output quantity, b=[b ₁b ₂b _n] ^tfor input quantity.

3. a kind of quick Gauss for symbol linear system according to claim 2 about works as elimination method, it is characterized in that: the determination in step 2 needs the number of times s of cancellation to be expressed as: by augmented matrix (A|b)=(a of step one gained state space equation _i,j) _{n × m}size n × m determine, s=min (n, m), usual n=m-1, s=n in engineering.

4. a kind of quick Gauss for symbol linear system according to claim 3 about works as elimination method, it is characterized in that: the quick Gauss in step 2 is about when elimination method specifically comprises the following steps:

1) initial value setting: definition represent augmented matrix (A|b) the i-th row jth column element after k cancellation, represent the structure requirement of augmented matrix (A|b) i-th row jth column element after k cancellation; Given as 1≤i≤n, during 1≤j<m, as 1≤i≤n, j=m,

2) carry out (0≤k≤s) after kth time cancellation,

Work as i>k, during j>k, the structure requirement of element $a_{i,j}^{\left(k\right)}=\left|\begin{array}{cccc}{a}_{11}^0& ...& a_{1,k}^0& a_{1,j}^0\\ \&CenterDot;& & \&CenterDot;& \&CenterDot;\\ \&CenterDot;& & \&CenterDot;& \&CenterDot;\\ \&CenterDot;& & \&CenterDot;& \&CenterDot;\\ a_{k,1}^0& ...& a_{k,k}^0& a_{k,j}^0\\ a_{i,1}^0& ...& a_{i,k}^0& a_{i,j}^0\end{array}\right|;$

Work as i<k, during j>k, the structure requirement of element $a_{i,j}^{\left(k\right)}=\left|\begin{array}{ccccccc}{a}_{11}^0& ...& a_{1,i-1}^0& a_{1,i+1}^0& ...& a_{1,k}^0& a_{1,j}^0\\ a_{21}^0& ...& a_{2,i-1}^0& a_{2,i+1}^0& ...& a_{2,k}^0& a_{2,j}^0\\ \&CenterDot;& & \&CenterDot;& \&CenterDot;& & \&CenterDot;& \&CenterDot;\\ \&CenterDot;& & \&CenterDot;& \&CenterDot;& & \&CenterDot;& \&CenterDot;\\ \&CenterDot;& & \&CenterDot;& \&CenterDot;& & \&CenterDot;& \&CenterDot;\\ a_{k,1}^0& ...& a_{k,i-1}^0& a_{k,i+1}^0& ...& a_{k,k}^0& a_{k,j}^0\end{array}\right|;$

3) integrating step 2), after augmented matrix (A|b) carries out kth time cancellation, (0≤k≤s) changes into (A|b) ^k, wherein (A|b) ^kelement meet:

$a_{i,j}^k=\left\{\begin{array}{cc}\frac{a_{i,j}^{\left(k\right)}}{a_{k,k}^{(k-1)}}& i>k,j>k\\ \frac{a_{i,j}^{(k-1)}}{a_{k,k}^{(k-1)}}& i=k,j>k\\ \frac{a_{i,j}^{\left(k\right)}}{a_{k,k}^{(k-1)}}& i=k-1,j>k\\ \frac{{(-1)}^{k-i+1}{a}_{i,j}^{\left(k\right)}}{a_{k,k}^{(k-1)}}& i\&le;k-2,j>k\end{array}\right.$

System state space equation diagonal matrix form after k=s cancellation abbreviation is:

5. a kind of quick Gauss for symbol linear system according to claim 4 about works as elimination method, it is characterized in that: in step 3, according to the abbreviation result in step 2, simultaneously according to the usual m=n+1 of engineering system, judge the input of this system, output quantity exists one-to-one relationship, and calculate the mathematical physics relation between them.