MATLAB: Unable to find explicit solution

equationMATLABsymbol [~]

syms c l1 h1 d1 d2 d3 d4 l2 h2 k3 l3 h3 h4 k4 l4 B

q1=1/(d3^2)==((h3^2/d1^2*h1^2)+(k3^2)/(d2^2*l2^2)+l3^2/(c^2*sin(B)^2)-(2*h3*l3*cos(B))/(d1*h1*c*sin(B)));

q2=1/(d4^2)==((h4^2/d1^2*h1^2)+(k4^2)/(d2^2*l2^2)+l4^2/(c^2*sin(B)^2)-(2*h4*l4*cos(B))/(d1*h1*c*sin(B)));

                sol=solve([q1, q2], [c, B]);
                csol=sol.c
                Bsol=sol.B
the Ans is 'Unable to find explicit solution'

but when I use value

the Ans can be found

h1=1; d3=10; d4=15; h3=2; h4=1; d1=24; k3=3; k4=2; d2=18; l2=2; l3=1; l4=3;

syms c B

q1=1/(d3^2)==((h3^2/d1^2*h1^2)+(k3^2)/(d2^2*l2^2)+l3^2/(c^2*sin(B)^2)-(2*h3*l3*cos(B))/(d1*h1*c*sin(B)));

q2=1/(d4^2)==((h4^2/d1^2*h1^2)+(k4^2)/(d2^2*l2^2)+l4^2/(c^2*sin(B)^2)-(2*h4*l4*cos(B))/(d1*h1*c*sin(B)));

                sol=solve([q1, q2], [c, B]);
                csol=sol.c
                Bsol=sol.B
csol =
   (919795581000*(- (38460*21210^(1/2))/149167 - 5603167/149167)^(1/2))/669312329 + (81000*(- (38460*21210^(1/2))/149167 - 5603167/149167)^(3/2))/4487
 - (919795581000*(- (38460*21210^(1/2))/149167 - 5603167/149167)^(1/2))/669312329 - (81000*(- (38460*21210^(1/2))/149167 - 5603167/149167)^(3/2))/4487
       (919795581000*((38460*21210^(1/2))/149167 - 5603167/149167)^(1/2))/669312329 + (81000*((38460*21210^(1/2))/149167 - 5603167/149167)^(3/2))/4487
     - (919795581000*((38460*21210^(1/2))/149167 - 5603167/149167)^(1/2))/669312329 - (81000*((38460*21210^(1/2))/149167 - 5603167/149167)^(3/2))/4487

Bsol =

 -2*atan((- (38460*21210^(1/2))/149167 - 5603167/149167)^(1/2))
  2*atan((- (38460*21210^(1/2))/149167 - 5603167/149167)^(1/2))
   -2*atan(((38460*21210^(1/2))/149167 - 5603167/149167)^(1/2))
    2*atan(((38460*21210^(1/2))/149167 - 5603167/149167)^(1/2))

how can i show the Ans with symbol? thanks

Best Answer

Your query is a mathematical issue and not a matlab issue.

Let me give you an example...

Supose you want to solve analytically a 6th degree polynomial equation:

a*x^6 + b*x^5 + c*x^4 + d*x^3 + e*x^2 + f*x^1 + g =0

There is no analytical formulla for every value of coefficients a,b,c,d,e,f,g

But for some sets of these coefficients, there is indeed a compact way to find the solution:

2*x^6 -2*x^4 - 8*x^2 + 8 = 0

2(x^6 - x^4 -4*x^2 + 4) = 0

2 * (x^2 - 1) * (x^4 - 4 ) = 0

2 * (x-1) * (x+1) * (x^2 - 2) * (x^2 +2) = 0

2 * (x-1) * (x+1) * (x - sqrt(2) ) * (x + sqrt(2) ) * ( x^2 +2 ) = 0

where roots of the later equation are , 1 , -1 , sqrt(2), -sqrt(2)

Related Solutions

MATLAB: Defining variables for fsolve

Your Guess has 2 elements, suggesting you are wanting to work with two variables, but your F is @(x,y,z) suggesting you want to work with three variables.

fsolve() is going to pass one variable to the function handle. That one variable will be a vector the same length as Guess.

It appears to me that your Guess should have three elements, and it appears to me that your function handle should be:

F  = @(X) double( subs( [((g1_1.^2 + h1_1.^2).*(exp(2.*a_1)) + (g2_1.^2 + h2_1.^2).*(exp(-2.*a_1)) + A_1.*(cos(2.*b_1)) + B_1.*(sin(2.*b_1)))./((exp(2.*a_1)) + (g1_1.^2 + h1_1.^2).*(g2_1.^2 + h2_1.^2).*(exp(-2.*a_1)) + C_1.*(cos(2.*b_1)) + D_1.*(sin(2.*b_1))) - R_1; ...
       ((g1_2.^2 + h1_2.^2).*(exp(2.*a_2)) + (g2_2.^2 + h2_2.^2).*(exp(-2.*a_2)) + A_2.*(cos(2.*b_2)) + B_2.*(sin(2.*b_2)))./((exp(2.*a_2)) + (g1_2.^2 + h1_2.^2).*(g2_2.^2 + h2_2.^2).*(exp(-2.*a_2)) + C_2.*(cos(2.*b_2)) + D_2.*(sin(2.*b_2))) - R_2; ...
       ((g1_3.^2 + h1_3.^2).*(exp(2.*a_3)) + (g2_3.^2 + h2_3.^2).*(exp(-2.*a_3)) + A_3.*(cos(2.*b_3)) + B_3.*(sin(2.*b_3)))./((exp(2.*a_3)) + (g1_3.^2 + h1_3.^2).*(g2_3.^2 + h2_3.^2).*(exp(-2.*a_3)) + C_3.*(cos(2.*b_3)) + D_3.*(sin(2.*b_3))) - R_3], {x y z}, {X(1) X(2) X(3)}) );

It would not be typical to use symbolic expressions with fsolve() !

MATLAB: Fsolve inputs

I cannot test this as I do not have the toolbox with fsolve in it.

This is the structure I told you to adapt several threads ago.

function nkd = g3driver
      Guess = [1.58, 0, 25];
      nkd = fsolve(@g3, Guess);
end
function F = g3(X)
  % This program aims to back-solve for reflectance (R) using given values
  % input by hand.
    x = X(1); y = X(2); z = X(3);
    y_1  = 309;
    n0_1 = 1;
    %n1_1 = 1.580086;
    n1_1 = x;
    %k1_1 = 0;
    k1_1 = y;
    n2_1 = 5.07;
    k2_1 = 3.62;
    %d1_1 = 25;
    d1_1 = z;
    R_1  = .4335;
    y_2  = 310;
    n0_2 = 1;
    %n1_2 = 1.579925;
    n1_2 = x;
    %k1_2 = 0;
    k1_2 = y;
    n2_2 = 5.07;
    k2_2 = 3.56;
    %d1_2 = 25;
    d1_2 = z;
    R_2  = .4294;
    y_3  = 311;
    n0_3 = 1;
    %n1_3 = 1.579764;
    n1_3 = x;
    %k1_3 = 0;
    k1_3 = y;
    n2_3 = 5.08;
    k2_3 = 3.53;
    %d1_3 = 25;
    d1_3 = z;
    R_3  = .4277;
    g1_1 = (n0_1.^2 - n1_1.^2 - k1_1.^2)./((n1_1 + n2_1).^2 + k1_1.^2);
    g2_1 = (n1_1.^2 - n2_1.^2 + k1_1.^2 - k2_1.^2)./((n1_1 + n2_1).^2 + (k1_1 + k2_1).^2);
    h1_1 = (2.*n0_1.*k1_1)./((n0_1 + n1_1).^2 + k1_1.^2);
    h2_1 = (2.*(n1_1.*k2_1 - n2_1.*k1_1))./((n1_1 + n2_1).^2 + (k1_1 + k2_1).^2);
    a_1  = (2.*pi().*k1_1.*d1_1)./y_1;
    b_1  = (2.*pi().*n1_1.*d1_1)./y_1;
    A_1  = 2.*(g1_1.*g2_1 + h1_1.*h2_1);
    B_1  = 2.*(g1_1.*h2_1 - g2_1.*h1_1);
    C_1  = 2.*(g1_1.*g2_1 - h1_1.*h2_1);
    D_1  = 2.*(g1_1.*h2_1 + g2_1.*h1_1);
    g1_2 = (n0_2.^2 - n1_2.^2 - k1_2.^2)./((n1_2 + n2_2).^2 + k1_2.^2);
    g2_2 = (n1_2.^2 - n2_2.^2 + k1_2.^2 - k2_2.^2)./((n1_2 + n2_2).^2 + (k1_2 + k2_2).^2);
    h1_2 = (2.*n0_2.*k1_2)./((n0_2 + n1_2).^2 + k1_2.^2);
    h2_2 = (2.*(n1_2.*k2_2 - n2_2.*k1_2))./((n1_2 + n2_2).^2 + (k1_2 + k2_2).^2);
    a_2  = (2.*pi().*k1_2.*d1_2)./y_2;
    b_2  = (2.*pi().*n1_2.*d1_2)./y_2;
    A_2  = 2.*(g1_2.*g2_2 + h1_2.*h2_2);
    B_2  = 2.*(g1_2.*h2_2 - g2_2.*h1_2);
    C_2  = 2.*(g1_2.*g2_2 - h1_2.*h2_2);
    D_2  = 2.*(g1_2.*h2_2 + g2_2.*h1_2);
    g1_3 = (n0_3.^2 - n1_3.^2 - k1_3.^2)./((n1_3 + n2_3).^2 + k1_3.^2);
    g2_3 = (n1_3.^2 - n2_3.^2 + k1_3.^2 - k2_3.^2)./((n1_3 + n2_3).^2 + (k1_3 + k2_3).^2);
    h1_3 = (2.*n0_3.*k1_3)./((n0_3 + n1_3).^2 + k1_3.^2);
    h2_3 = (2.*(n1_3.*k2_3 - n2_3.*k1_3))./((n1_3 + n2_3).^2 + (k1_3 + k2_3).^2);
    a_3  = (2.*pi().*k1_3.*d1_3)./y_3;
    b_3  = (2.*pi().*n1_3.*d1_3)./y_3;
    A_3  = 2.*(g1_3.*g2_3 + h1_3.*h2_3);
    B_3  = 2.*(g1_3.*h2_3 - g2_3.*h1_3);
    C_3  = 2.*(g1_3.*g2_3 - h1_3.*h2_3);
    D_3  = 2.*(g1_3.*h2_3 + g2_3.*h1_3);
    F  = [((g1_1.^2 + h1_1.^2).*(exp(2.*a_1)) + (g2_1.^2 + h2_1.^2).*(exp(-2.*a_1)) + A_1.*(cos(2.*b_1)) + B_1.*(sin(2.*b_1)))./((exp(2.*a_1)) + (g1_1.^2 + h1_1.^2).*(g2_1.^2 + h2_1.^2).*(exp(-2.*a_1)) + C_1.*(cos(2.*b_1)) + D_1.*(sin(2.*b_1))) - R_1; ...
               ((g1_2.^2 + h1_2.^2).*(exp(2.*a_2)) + (g2_2.^2 + h2_2.^2).*(exp(-2.*a_2)) + A_2.*(cos(2.*b_2)) + B_2.*(sin(2.*b_2)))./((exp(2.*a_2)) + (g1_2.^2 + h1_2.^2).*(g2_2.^2 + h2_2.^2).*(exp(-2.*a_2)) + C_2.*(cos(2.*b_2)) + D_2.*(sin(2.*b_2))) - R_2; ...
               ((g1_3.^2 + h1_3.^2).*(exp(2.*a_3)) + (g2_3.^2 + h2_3.^2).*(exp(-2.*a_3)) + A_3.*(cos(2.*b_3)) + B_3.*(sin(2.*b_3)))./((exp(2.*a_3)) + (g1_3.^2 + h1_3.^2).*(g2_3.^2 + h2_3.^2).*(exp(-2.*a_3)) + C_3.*(cos(2.*b_3)) + D_3.*(sin(2.*b_3))) - R_3];
end

Best Answer

Related Solutions

MATLAB: Defining variables for fsolve

MATLAB: Fsolve inputs

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