MATLAB: Algebra: Linear Equations

algebrakirchhoff's first lawlinear systemsMATLABwithout symbolic toolbox

How can I solve this linear systems without using Symbolic toolbox?

a*x + b*y = c1
c*x + d*y + e*z = c2
f*y + g*z = c3
% where a,b,c,d,e,f,g and c1,c2,c3 are constants.
% I need to solve this for Kirchhoff's first law without the symbolic math toolbox.

Thank you so much.

Best Answer

If the value of these constants are known, then you can use mldivide: https://www.mathworks.com/help/matlab/ref/mldivide.html

a = 1;
b = 2;
c = 1.4;
d = 1.2;
e = 2.1;
f = 3.2;
g = 1.1;
c1 = 1;
c2 = 2;
c3 = 3;
A = [a b 0;
     c d e;
     0 f g];
 
B = [c1; c2; c3];
sol = A\B;
x = sol(1);
y = sol(2);
z = sol(3);

Related Solutions

MATLAB: Solve symbol four equations for c1 c2 c3 c4

Thanks, I solve it,

clc
clear
syms a b x y c1 c2 c3 c4 T1 T2 T3 T4  
T=c1+c2*x+c3*y+c4*x*y;
%%
r1=c1+c2*a+c3*b+c4*a*b-T1;
r2=c1-c2*a+c3*b-c4*a*b-T2;
r3=c1-c2*a-c3*b+c4*a*b-T3;
r4=c1+c2*a-c3*b-c4*a*b-T4;
%%Find C1 and C4
p13=r1+r3;
p24=r2+r4;
n1=solve(p13==0,c1);
n2=solve(p24==0,c1);
C4=solve(n1==n2,c4)
w1=solve(p13==0,c4);
w2=solve(p24==0,c4);
C1=solve(w1==w2,c1)
r1=subs(r1,c1,C1);
r1=subs(r1,c4,C4);
r2=subs(r2,c1,C1);
r2=subs(r2,c4,C4);
r3=subs(r3,c1,C1);
r3=subs(r3,c4,C4);
r4=subs(r4,c1,C1);
r4=subs(r4,c4,C4);
%%Find C3
q12=r1+r2;
q34=r3+r4;
C3=solve(q12==0,c3)
r1=subs(r1,c3,C3);
r2=subs(r2,c3,C3);
r3=subs(r3,c3,C3);
r4=subs(r4,c3,C3);
%% find C2
C2=solve(r1==0,c2)
r1=subs(r1,c2,C2);
r2=subs(r2,c2,C2);
r3=subs(r3,c2,C2);
r4=subs(r4,c2,C2);
C1
C2
C3
C4

MATLAB: How to solve this Matrix Equation

The solution is definitely not unique.

Let the two result matrices be [rab11,0;0,rab22] and [rcd11;0;0;rcd22]

Set rcd22 to any non-zero non-infinite value. Then, except for possible singularities,

rab11 = rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b11+b21*d12*d11*(c11*c22-c12*c21))*a12+a11*b11*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((d12*c21*b22*a22-b12*a21*d22*c22)*b11+d12*b12*b21*(-c21*a22+a21*c22))*a12+a11*a22*b11*c22*(-d12*b22+b12*d22))*(c11*c22-c12*c21)*(d11*d22-d12*d21))

rab22 = -rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b12+b22*d12*d11*(c11*c22-c12*c21))*a22+a21*b12*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((b11*d22*c22*a11-d12*c21*a12*b21)*b12-d12*b11*b22*(c22*a11-a12*c21))*a22-a12*a21*c22*b12*(b11*d22-b21*d12))*(c11*c22-c12*c21)*(d11*d22-d12*d21))

rcd11 = -rcd22*((d11*c11*(b11*b22-b12*b21)*a22-a21*c12*b12*(b11*d21-b21*d11))*a12+a11*a22*b11*c12*(b12*d21-d11*b22))/((c21*d12*(b11*b22-b12*b21)*a22-a21*c22*b12*(b11*d22-b21*d12))*a12+a11*a22*b11*c22*(-d12*b22+b12*d22))

You can see in this that rcd22 acts as an arbitrary scale factor for the other coefficients.

The X matrix then becomes

X(1,1) = (b22*a22*rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b11+b21*d12*d11*(c11*c22-c12*c21))*a12+a11*b11*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((d12*c21*b22*a22-b12*a21*d22*c22)*b11+d12*b12*b21*(-c21*a22+a21*c22))*a12+a11*a22*b11*c22*(-d12*b22+b12*d22))*(c11*c22-c12*c21)*(d11*d22-d12*d21))-b22*a12*rab21-b21*a22*rab12-b21*a12*rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b12+b22*d12*d11*(c11*c22-c12*c21))*a22+a21*b12*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((b11*d22*c22*a11-d12*c21*a12*b21)*b12-d12*b11*b22*(c22*a11-a12*c21))*a22-a12*a21*c22*b12*(b11*d22-b21*d12))*(c11*c22-c12*c21)*(d11*d22-d12*d21)))/((a11*a22-a12*a21)*(b11*b22-b12*b21))

X(1,2) = -(b12*a22*rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b11+b21*d12*d11*(c11*c22-c12*c21))*a12+a11*b11*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((d12*c21*b22*a22-b12*a21*d22*c22)*b11+d12*b12*b21*(-c21*a22+a21*c22))*a12+a11*a22*b11*c22*(-d12*b22+b12*d22))*(c11*c22-c12*c21)*(d11*d22-d12*d21))-b12*a12*rab21-b11*a22*rab12-b11*a12*rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b12+b22*d12*d11*(c11*c22-c12*c21))*a22+a21*b12*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((b11*d22*c22*a11-d12*c21*a12*b21)*b12-d12*b11*b22*(c22*a11-a12*c21))*a22-a12*a21*c22*b12*(b11*d22-b21*d12))*(c11*c22-c12*c21)*(d11*d22-d12*d21)))/((a11*a22-a12*a21)*(b11*b22-b12*b21))

X(2,1) = -(b22*a21*rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b11+b21*d12*d11*(c11*c22-c12*c21))*a12+a11*b11*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((d12*c21*b22*a22-b12*a21*d22*c22)*b11+d12*b12*b21*(-c21*a22+a21*c22))*a12+a11*a22*b11*c22*(-d12*b22+b12*d22))*(c11*c22-c12*c21)*(d11*d22-d12*d21))-b22*a11*rab21-b21*a21*rab12-b21*a11*rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b12+b22*d12*d11*(c11*c22-c12*c21))*a22+a21*b12*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((b11*d22*c22*a11-d12*c21*a12*b21)*b12-d12*b11*b22*(c22*a11-a12*c21))*a22-a12*a21*c22*b12*(b11*d22-b21*d12))*(c11*c22-c12*c21)*(d11*d22-d12*d21)))/((a11*a22-a12*a21)*(b11*b22-b12*b21))

X(2,2) = -(b22*a21*rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b11+b21*d12*d11*(c11*c22-c12*c21))*a12+a11*b11*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((d12*c21*b22*a22-b12*a21*d22*c22)*b11+d12*b12*b21*(-c21*a22+a21*c22))*a12+a11*a22*b11*c22*(-d12*b22+b12*d22))*(c11*c22-c12*c21)*(d11*d22-d12*d21))-b22*a11*rab21-b21*a21*rab12-b21*a11*rcd22*(((c12*c21*d12*d21-c11*c22*d11*d22)*b12+b22*d12*d11*(c11*c22-c12*c21))*a22+a21*b12*c12*c22*(d11*d22-d12*d21))*(a11*a22-a12*a21)*(b11*b22-b12*b21)/((((b11*d22*c22*a11-d12*c21*a12*b21)*b12-d12*b11*b22*(c22*a11-a12*c21))*a22-a12*a21*c22*b12*(b11*d22-b21*d12))*(c11*c22-c12*c21)*(d11*d22-d12*d21)))/((a11*a22-a12*a21)*(b11*b22-b12*b21))

This was calculated using straight-forward linear algebra by multiplying appropriate inverses on both sides to get two isolated X equations, and then solve()'ing for the corresponding components to be the same.

Best Answer

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