Solving System of Equations

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Erkin Karatas
Erkin Karatas 2019-11-11
评论： Fabio Freschi 2019-11-11
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)

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回答（2 个）

Fabio Freschi
Fabio Freschi 2019-11-11
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
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Fabio Freschi
Fabio Freschi 2019-11-11
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
Fabio Freschi 2019-11-11
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.

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Jeremy
Jeremy 2019-11-11
Typically I prefer doing it numerically by putting my equations into the form
Ax=b
and then
x = A\b
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Jeremy
Jeremy 2019-11-11
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
Erkin Karatas 2019-11-11
I know, but how can i modiy my A matrix. How can I put the coefficients in to the matrix.

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