Solving large number of equations

I am having issues to solve like following equations. I request for help/ suggestion to solve this type of equations. Thanks!

syms X1 X2 X3 X4 X5 X6 X7 X8

p_11 = 1; p_12 = 2; p_13 = 3; p_14 = 4; p_15 = 5; p_16 = 6; p_17 = 7;
p_21 = 8; p_22 = 9; p_23 = 10; p_24 = 11; p_25 = 12; p_26 = 13; p_27 = 14;
p_31 = 15; p_32 = 16; p_33 = 17; p_34 = 18; p_35 = 19; p_36 = 20; p_37 = 21;
p_41 = 22; p_42 = 23; p_43 = 24; p_44 = 25; p_45 = 26; p_46 = 27; p_47 = 28;
p_51 = 29; p_52 = 30; p_53 = 31; p_54 = 32; p_55 = 33; p_56 = 34; p_57 = 35;
p_61 = 36; p_62 = 37; p_63 = 38; p_64 = 39; p_65 = 40; p_66 = 41; p_67 = 42;
p_71 = 43; p_72 = 44; p_73 = 45; p_74 = 46; p_75 = 47; p_76 = 48; p_77 = 49;
p_81 = 50; p_82 = 51; p_83 = 52; p_84 = 53; p_85 = 54; p_86 = 55; p_87 = 56;

C_1 = [4 3 5];
C_2 = [2 4 6];
C_3 = [5 6 8];
C_4 = [1 2 3];
C_5 = [2 3 4];
C_6 = [3 4 5];
C_7 = [2 3 5];
C_8 = [3 6 2];

for i = 1:length(C_1)
    eq1(i) = p_11*X1 + p_12*X2 + p_13*X3 + p_14*X4 + p_15*X5 + p_16*X6 + p_17*X7*sin(X8) == C_1(i);
    eq2(i) = p_21*X1 + p_22*X2 + p_23*X3 + p_24*X4 + p_25*X5 + p_26*X6 + p_27*X7*sin(X8) == C_2(i);
    eq3(i) = p_31*X1 + p_32*X2 + p_33*X3 + p_34*X4 + p_35*X5 + p_36*X6 + p_37*X7*cos(X8) == C_3(i);
    eq4(i) = p_41*X1 + p_42*X2 + p_43*X3 + p_44*X4 + p_45*X5 + p_46*X6 + p_47*X7*sin(X8) == C_4(i);
    eq5(i) = p_51*X1 + p_52*X2 + p_53*X3 + p_54*X4 + p_55*X5 + p_56*X6 + p_57*X7*sin(X8) == C_5(i);
    eq6(i) = p_61*X1 + p_62*X2 + p_63*X3 + p_64*X4 + p_65*X5 + p_66*X6 + p_67*X7*sin(X8) == C_6(i);
    eq7(i) = p_71*X1 + p_72*X2 + p_73*X3 + p_74*X4 + p_75*X5 + p_76*X6 + p_77*X7*sin(X8) == C_7(i);
    eq8(i) = p_81*X1 + p_82*X2 + p_83*X3 + p_84*X4 + p_85*X5 + p_86*X6 + p_87*X7 == C_8(i);
end

% Solve the system of equations
sol = solve([eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8], [X1, X2, X3, X4, X5, X6, X7, X8]);

% Display solutions
disp(sol)

    X1: [0x1 sym]
    X2: [0x1 sym]
    X3: [0x1 sym]
    X4: [0x1 sym]
    X5: [0x1 sym]
    X6: [0x1 sym]
    X7: [0x1 sym]
    X8: [0x1 sym]

Answer 1

[Walter Roberson]

syms X1 X2 X3 X4 X5 X6 X7 X8

p_11 = 1; p_12 = 2; p_13 = 3; p_14 = 4; p_15 = 5; p_16 = 6; p_17 = 7;
p_21 = 8; p_22 = 9; p_23 = 10; p_24 = 11; p_25 = 12; p_26 = 13; p_27 = 14;
p_31 = 15; p_32 = 16; p_33 = 17; p_34 = 18; p_35 = 19; p_36 = 20; p_37 = 21;
p_41 = 22; p_42 = 23; p_43 = 24; p_44 = 25; p_45 = 26; p_46 = 27; p_47 = 28;
p_51 = 29; p_52 = 30; p_53 = 31; p_54 = 32; p_55 = 33; p_56 = 34; p_57 = 35;
p_61 = 36; p_62 = 37; p_63 = 38; p_64 = 39; p_65 = 40; p_66 = 41; p_67 = 42;
p_71 = 43; p_72 = 44; p_73 = 45; p_74 = 46; p_75 = 47; p_76 = 48; p_77 = 49;
p_81 = 50; p_82 = 51; p_83 = 52; p_84 = 53; p_85 = 54; p_86 = 55; p_87 = 56;

C_1 = [4 3 5];
C_2 = [2 4 6];
C_3 = [5 6 8];
C_4 = [1 2 3];
C_5 = [2 3 4];
C_6 = [3 4 5];
C_7 = [2 3 5];
C_8 = [3 6 2];

for i = 1:length(C_1)
    eq1(i) = p_11*X1 + p_12*X2 + p_13*X3 + p_14*X4 + p_15*X5 + p_16*X6 + p_17*X7*sin(X8) == C_1(i);
    eq2(i) = p_21*X1 + p_22*X2 + p_23*X3 + p_24*X4 + p_25*X5 + p_26*X6 + p_27*X7*sin(X8) == C_2(i);
    eq3(i) = p_31*X1 + p_32*X2 + p_33*X3 + p_34*X4 + p_35*X5 + p_36*X6 + p_37*X7*cos(X8) == C_3(i);
    eq4(i) = p_41*X1 + p_42*X2 + p_43*X3 + p_44*X4 + p_45*X5 + p_46*X6 + p_47*X7*sin(X8) == C_4(i);
    eq5(i) = p_51*X1 + p_52*X2 + p_53*X3 + p_54*X4 + p_55*X5 + p_56*X6 + p_57*X7*sin(X8) == C_5(i);
    eq6(i) = p_61*X1 + p_62*X2 + p_63*X3 + p_64*X4 + p_65*X5 + p_66*X6 + p_67*X7*sin(X8) == C_6(i);
    eq7(i) = p_71*X1 + p_72*X2 + p_73*X3 + p_74*X4 + p_75*X5 + p_76*X6 + p_77*X7*sin(X8) == C_7(i);
    eq8(i) = p_81*X1 + p_82*X2 + p_83*X3 + p_84*X4 + p_85*X5 + p_86*X6 + p_87*X7 == C_8(i);
end

eqns = [eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8];
size(eqns)

ans = 1×2

1 24

symvar(eqns)

ans = 

You have 24 equations in 8 variables, and you are trying to solve them for 8 variables.

% Solve the system of equations
sol = solve(eqns, [X1, X2, X3, X4, X5, X6, X7, X8]);

solve() will immediately give up on trying to solve the 24 equations for 8 variables.

% Display solutions
disp(sol)

    X1: [0×1 sym]
    X2: [0×1 sym]
    X3: [0×1 sym]
    X4: [0×1 sym]
    X5: [0×1 sym]
    X6: [0×1 sym]
    X7: [0×1 sym]
    X8: [0×1 sym]

If you are trying to do a least-squared fit then solve() of more equations than variables is the wrong way to do it.

[Ismita]

Thank you so much @Walter. I found that type of result instead of any value for the variables. I am new in coding. Could you please suggest me more specifically to solve these equations? Thanks a lot!

[Torsten]

"solve" from above says your system does not have a solution.

"fsolve" as a numerical solver also cannot find a solution for your system (see below).

As you can see, the Jacobian of your system has only rank 4 instead of rank 8. Thus it will be almost impossible to solve it for arbitrary right-hand sides.

p_11 = 1; p_12 = 2; p_13 = 3; p_14 = 4; p_15 = 5; p_16 = 6; p_17 = 7;
p_21 = 8; p_22 = 9; p_23 = 10; p_24 = 11; p_25 = 12; p_26 = 13; p_27 = 14;
p_31 = 15; p_32 = 16; p_33 = 17; p_34 = 18; p_35 = 19; p_36 = 20; p_37 = 21;
p_41 = 22; p_42 = 23; p_43 = 24; p_44 = 25; p_45 = 26; p_46 = 27; p_47 = 28;
p_51 = 29; p_52 = 30; p_53 = 31; p_54 = 32; p_55 = 33; p_56 = 34; p_57 = 35;
p_61 = 36; p_62 = 37; p_63 = 38; p_64 = 39; p_65 = 40; p_66 = 41; p_67 = 42;
p_71 = 43; p_72 = 44; p_73 = 45; p_74 = 46; p_75 = 47; p_76 = 48; p_77 = 49;
p_81 = 50; p_82 = 51; p_83 = 52; p_84 = 53; p_85 = 54; p_86 = 55; p_87 = 56;
C_1 = [4 3 5];
C_2 = [2 4 6];
C_3 = [5 6 8];
C_4 = [1 2 3];
C_5 = [2 3 4];
C_6 = [3 4 5];
C_7 = [2 3 5];
C_8 = [3 6 2];
for i = 1:length(C_1)
    f1 = @(X1,X2,X3,X4,X5,X6,X7,X8) p_11*X1 + p_12*X2 + p_13*X3 + p_14*X4 + p_15*X5 + p_16*X6 + p_17*X7*sin(X8) - C_1(i);
    f2 = @(X1,X2,X3,X4,X5,X6,X7,X8) p_21*X1 + p_22*X2 + p_23*X3 + p_24*X4 + p_25*X5 + p_26*X6 + p_27*X7*sin(X8) - C_2(i);
    f3 = @(X1,X2,X3,X4,X5,X6,X7,X8) p_31*X1 + p_32*X2 + p_33*X3 + p_34*X4 + p_35*X5 + p_36*X6 + p_37*X7*cos(X8) - C_3(i);
    f4 = @(X1,X2,X3,X4,X5,X6,X7,X8) p_41*X1 + p_42*X2 + p_43*X3 + p_44*X4 + p_45*X5 + p_46*X6 + p_47*X7*sin(X8) - C_4(i);
    f5 = @(X1,X2,X3,X4,X5,X6,X7,X8) p_51*X1 + p_52*X2 + p_53*X3 + p_54*X4 + p_55*X5 + p_56*X6 + p_57*X7*sin(X8) - C_5(i);
    f6 = @(X1,X2,X3,X4,X5,X6,X7,X8) p_61*X1 + p_62*X2 + p_63*X3 + p_64*X4 + p_65*X5 + p_66*X6 + p_67*X7*sin(X8) - C_6(i);
    f7 = @(X1,X2,X3,X4,X5,X6,X7,X8) p_71*X1 + p_72*X2 + p_73*X3 + p_74*X4 + p_75*X5 + p_76*X6 + p_77*X7*sin(X8) - C_7(i);
    f8 = @(X1,X2,X3,X4,X5,X6,X7,X8) p_81*X1 + p_82*X2 + p_83*X3 + p_84*X4 + p_85*X5 + p_86*X6 + p_87*X7 - C_8(i);
    f = @(X)[f1(X(1),X(2),X(3),X(4),X(5),X(6),X(7),X(8));
             f2(X(1),X(2),X(3),X(4),X(5),X(6),X(7),X(8));
             f3(X(1),X(2),X(3),X(4),X(5),X(6),X(7),X(8));
             f4(X(1),X(2),X(3),X(4),X(5),X(6),X(7),X(8));
             f5(X(1),X(2),X(3),X(4),X(5),X(6),X(7),X(8));
             f6(X(1),X(2),X(3),X(4),X(5),X(6),X(7),X(8));
             f7(X(1),X(2),X(3),X(4),X(5),X(6),X(7),X(8));
             f8(X(1),X(2),X(3),X(4),X(5),X(6),X(7),X(8))];
    X = sym('X',[8 1]);
    J = jacobian(f(X))
    rank(J)
    X0 = [1;2;3;4;5;6;7;8];
    sol(i,:) = fsolve(f,X0)   
end

J = 

ans = 4

Solver stopped prematurely.

fsolve stopped because it exceeded the function evaluation limit,
options.MaxFunctionEvaluations = 8.000000e+02.

sol = 1×8

1.0e+06 * -0.0003 0.0204 -1.6787 0.0627 4.8308 -3.2348 0.0000 0.0000

J = 

ans = 4

Solver stopped prematurely.

fsolve stopped because it exceeded the function evaluation limit,
options.MaxFunctionEvaluations = 8.000000e+02.

sol = 2×8

1.0e+06 * -0.0003 0.0204 -1.6787 0.0627 4.8308 -3.2348 0.0000 0.0000 0.0023 -1.9292 0.0913 3.6340 0.1631 -1.9616 0.0000 0.0000

J = 

ans = 4

Solver stopped prematurely.

fsolve stopped because it exceeded the function evaluation limit,
options.MaxFunctionEvaluations = 8.000000e+02.

sol = 3×8

1.0e+06 * -0.0003 0.0204 -1.6787 0.0627 4.8308 -3.2348 0.0000 0.0000 0.0023 -1.9292 0.0913 3.6340 0.1631 -1.9616 0.0000 0.0000 0.0000 -0.1246 1.9481 -1.2147 -2.9165 2.3077 -0.0000 0.0000

[Ismita]

Thank you so much @Torsten.

Answer 2

[John D'Errico]

1. This is not even a remotely large number of equations.
2. Will you please learn to use matrices?
3. X8 appears in only one form in your equations, as X7*sin(X8). Therefore, you could transform the problem trivially, by substituting a single new unknown into the problem. That is...

Y8 = X7*sin(X8).

Replace the product X7*sin(X8) in each place it appears in your system. Your system is now linear, in the set of unknowns {X1,X2,X3,X4,X5,X67,X7,Y8}. Solve that LINEAR system. Once you have found the values of X7 and Y8, you can recover X8.

X8 = asin(Y8/X7)

Remember this means there will be infinitely many equivalent solutions, but so what? And of course, if X7 were exactly zero, then no solution exists. Finally, if it is true that abs(Y8/X7)>1, then the only solutions possible will be complex solutions. Such is life.

So just use backslash to solve the problem. AND PLEASE LEARN TO USE MATRICES.

[Ismita]

Thank you @John D'Errico

[Torsten]

@John D'Errico

X7*cos(X8) in eq3 and p_87*X7 in eq8 unfortunately hinder to convert the system to a linear one.

[Ismita]

Thanks @John D'Errico