How to solve a simple Matrix equation?

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Phillipp 2020-2-19

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编辑： Jon 2020-2-20

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Hello,

I have a small problem. I want to solve the multiplication of a given matrix.

For Example:

A = [-1,0,0,1;1,-1,0,0;0,1,-1,0;0,0,1,-1];
x = sym('x', [4,1]);
x(1,1)=1;
solve(A*x==[0;0;0;0])

Because I am a beginner at Matlab, I really don't know what im doing wrong. In this Example x should be equal to ones.

The goal is, to use it for a big matrix and calculate all unknown 'x'. If it is not possible to calculate them, I have to estimate the missing x, and solve it with something like a Netwon-Raphson method. But that will propably be in a later question.

Thank you for answering.

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Answer 1

Jon 2020-2-19

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https://ww2.mathworks.cn/matlabcentral/answers/506459-how-to-solve-a-simple-matrix-equation#answer_416410

编辑：Jon 2020-2-19

在 MATLAB Online 中打开

It looks like you want to find all of the vectors satisfying Ax = 0 for some singular matrix , A. If so, you can do this using

X = null(A)

This returns a basis for the null space of a matrix A.

In your case MATLAB returns X = [0.5;0.5;0.5;0.5], any multiple of this will satisfy your equation,

so for example 2*[0.5;0.5;0.5,0.5] = [1;1;1;1] is a solution and you can verify that A*[1;1;1;1]=0

To find out more about the null function type doc null on the command line in MATLAB

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Jon 2020-2-19

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So actually, now that I understand your problem, I think you can solve it quite directly using MATLAB's lsqlin function. Hopefully you have the optimization toolbox included in your MATLAB distribution.

You can type ver on the command line to see if you have the optimization toolbox.

If so you can solve your problem as shown in the attached simple example with 3 nodes and 2 pipes where I know the flow in pipe 1 and pipe 3 are 3 l/h and 8 l/h respectively.

% define tolerance for finding solution
tol = 1e-6;
% define piping network
Aeq = [1 1 -1 0;-1 -1 0 1;0 0 1 -1] % piping topology
% assign known measured values
iPipe = [1;3]; % pipe numbers where flow is measured (columns indices in Aeq)
xMeasured = [3;8] % measured flow rates
% find key dimensions
[numNodes,numPipes] = size(A);
numMeasured = length(xMeasured);
% assign constraints
beq = zeros(numNodes,1); % sum of flows into nodes must be zero
lb = zeros(numNodes,1); % lower bound, flows must be greater than zero
% assign objective function  whose norm is minimized C*x - d
C = zeros(numMeasured,numPipes);
for k = 1:numMeasured
    C(k,iPipe(k))=1; %C*x  select desired flows
end 
d = xMeasured; % so ideally C*x = d, or equivalently Cx - d = 0
% solve constrained least squares problem
[x,resnorm] = lsqlin(C,d,[],[],Aeq,beq,lb,[])
% check the size of resnorm to see if a solution was possible, it should be
% very small
if resnorm > tol
    warning('exact solution could not be found')
end

Phillipp 2020-2-20

在 MATLAB Online 中打开

I have the optimization toolbox installed.Thank you! That really helped me.

You made a tiny mistake you have to change this:

lb = zeros(numNodes,1); % lower bound, flows must be greater than zero

to this.

lb = zeros(numPipes,1); % lower bound, flows must be greater than zero

But man, you really helped me out. Thanks a lot.

Jon 2020-2-20

编辑：Jon 2020-2-20

Good catch!

Glad it seem like this approach will help you. One benefit of the least squares minimization, is that it very naturally handles the situation where you have sensor measurements of the flows but they may be noisy/slightly in error. Imagine that you in fact have all of the flows measured with noisy measurements. The measured flows will not satisfy Ax = 0 but you can find a set of flows which are as close as possible to the measured values (in the least square sense) but that satisfy Ax = 0. This is in some sense best physically consistent estimate of the overall network flows.

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How to solve a simple Matrix equation?

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