Solving System of Equations

Hi all,

I want to solve equations, when I know the number of equations, this is how I normally do

%
syms a b c d
eqn1 = 13*a -2.5*b == 8;
eqn2 = -10.5*a + 13*b - 2.5*c == 8;
eqn4 = -10.5b + 2*c + 3*d == 8;
[A,B] = equationsToMatrix([eqn1, eqn2, eqn3, eqn4], [a, b, c, d])
X = linsolve(A,B)

I want to modify this format(if possible) into the unkown equations, for example for 10 equations or 20. Since the number of equations can change, I cannot write them as shown above. If it's not possible, is there any other way to solve these unknown number of equations which are dependent each other?

%Modified??
n=10;
syms a b c d e x y z %number of variables can change as the number of equations change, so no idea what to write here
eqn1 = 13*a -2.5*b == 8;
eqn2 = -10.5*a + 13*b - 2.5*c == 8;
eqn3 = -10.5*b + 13*c -2.5*d == 8;
eqn4 = -10.5*c + 13*d - 2.5*e == 8;
...
eqnn = -10.5x + 2*y + 3*z == 8;
[A,B] = equationsToMatrix([eqn1, eqn2, eqn3], [x, y, z]) %?
X = linsolve(A,B)

Answer 1

[Fabio Freschi]

It seems you are solving a numeric system of equations. Why don't you simply put the coefficient matrix in A and the right-hand-side in an array b and use backslash?

% case 1
A = [13 -2.5 0 0;
    -10.5 13 -2.5 0
    1 1 1 1 % dummy values: equation 3 is missing in your code
    0 -10.5 2 3];
b = [8
    8
    8  % dummy value
    8];
x = A\b;

You can scale it to as many equations you like, and it is definitely faster than symbolic calulations

[Erkin Karatas]

I am trying to write a code about CFD node analysis, so the number of equations change when we change the number of nodes and the actual equations are way longer this one, and I cannot write the coefficients because they depend on other calculations, I just wrote the coefficients just to make the code simpler.

[Fabio Freschi]

It is about 15 years that I write numerical formulations for electromagnetics. You should build your coefficient matrix filling the entries using your calculations (based on FEM, FDM, ...) and then using

x = A\b;

To solve the final system. BTW: using standard discretization of PDE, you will have a sparse coefficient (stiffness) matrix. It may be of help having a look at the sparse command and the use of sparse matrices in Matlab

[Fabio Freschi]

To go further in the details, the stiffness matrix is usually built as

for i = 1:Nelements  % loop over elements
    
    Aloc = zeros(Nnodes)  % preallocation
    for j = 1:Nnodes  % loop over nodes og th i-th elem
        for k = 1:Nnodes   % loop over nodes og th i-th elem
            Aloc(i,j) = % your method
        end
    end
    ... % map local matrix to global matrix A
end

Even if there are many examples to avoid some or all these loops.

Answer 2

[Jeremy]

Typically I prefer doing it numerically by putting my equations into the form

Ax=b 

and then

x = A\b

[Erkin Karatas]

I do not know the number of equations they can be changed by the user. I am trying to write a code about CFD node analysis, so the number of nodes(number of equations) increase as the user makes the analysis finer.

[Jeremy]

Then you would modify your A matrix and b vector based on user input, but x = A\b is still going to give the correct answer as long as the problem is set up correctly

[Erkin Karatas]

I know, but how can i modiy my A matrix. How can I put the coefficients in to the matrix.