MATLAB: Root finder solver for cubic equation

root solver

I have the following cubic equation in beta, knowing gamma, theta and M1 how can I find the roots of beta using MATLAB?

for example, knowing the following:

M1 = 8:3:20;
gamma = 1.4;
theta = 8;

Best Answer

syms M1 gamma theta_degrees tanbeta_degrees
eqn = 1 + (gamma - 1)/2 * M1.^2 * tand(theta_degrees).*tanbeta_degrees.^3 + (1-M1.^2).*tanbeta_degrees.^2 + (1+(gamma+1)/2.*M1.^2);
solbeta = solve(eqn, tanbeta_degrees);
beta_degrees = atand(solbeta)
beta_degrees = 
M1_ = 8:3:20;
gamma_ = 1.4;
theta_degrees_ = 8; %in degrees 
beta_values = subs(beta_degrees, {M1, gamma, theta_degrees}, {M1_, gamma_, theta_degrees_})
beta_values = 
beta_numeric = double(beta_values)
beta_numeric = 3×5
  -47.7576  -47.4871  -47.3704  -47.3097  -47.2741
   48.6681   48.3829   48.2600   48.1960   48.1586
   88.3627   88.3749   88.3801   88.3828   88.3844
whos beta_numeric
  Name              Size            Bytes  Class     Attributes

  beta_numeric      3x5               120  double

Here, the different columns correspond to the different M1 values, and the different rows correspond to the different roots of the cubic. Results are in degrees.

Related Solutions

MATLAB: Solving two nonlinear equations

A = 0.5:0.01:2;
alpha = 0.5;
l00 = 1;
for i = 1:numel(A)
  a = A(i);
  fun = @(l0) 1 + alpha*l0^(alpha-1) - l0 - alpha*a*(2-l0)^(alpha-1);
  L0(i) = fzero(fun,l00);
  L1(i) = 2 - L0(i);
  l00 = L0(i);
end
plot(A,L0,A,L1)
%res_L0 = 1-alpha*A.*L1.^(alpha-1)+alpha*L0.^(alpha-1)-L0;
%res_L1 = 1+alpha*A.*L1.^(alpha-1)-alpha*L0.^(alpha-1)-L1;
%plot(A,res_L0,A,res_L1)

MATLAB: Is it so hard for MATLAB to solve the following symbolic equation

This seems to work

syms rp alpha beta theta psi z
A=cos(alpha)-cos(beta);
B=sin(alpha)-sin(beta);
C=(cos(beta)-1)/tan(beta)-(cos(alpha)-1)/tan(alpha);
R=A*(cos(theta)-cos(psi))+B*cos(theta)*sin(psi);
S=A*sin(theta)*sin(psi)-B*cos(theta)+C*cos(psi);
phi=atan(-R/S);
x=-rp*(cos(theta)*cos(phi)+sin(psi)*sin(theta)*sin(phi))*cos(alpha)-rp*(-cos(theta)*sin(phi)+sin(psi)*sin(theta)*cos(phi))*sin(alpha)+(rp/tan(alpha))*(cos(psi)*sin(phi)*(cos(alpha)-1)+cos(psi)*cos(phi)*sin(alpha));
y=-cos(psi)*sin(phi)*rp;
p=[x;y;z];
T=Rot('y',theta)*Rot('x',psi)*Rot('z',phi);
a1=T*[rp;0;0];
a2=T*[rp*cos(alpha);rp*sin(alpha);0];
a3=T*[rp*cos(beta);rp*sin(beta);0];  
Con1=(p+a1)'*[0;1;0];
Con2=(p+a2)'*[-sin(alpha);cos(alpha);0];
Con3=(p+a3)'*[-sin(beta);cos(beta);0];
disp([Con1; Con2; Con3])
function output=Rot(axis,angle)
syms x y z
if axis==x
    output=[1,0,0;0,cos(angle),-sin(angle);0,sin(angle),cos(angle)];
end
if axis==y
    output=[cos(angle),0,sin(angle);0,1,0;-sin(angle),0,cos(angle)];
end
if axis==z
    output=[cos(angle),-sin(angle),0;sin(angle),cos(angle),0;0,0,1];
end
end

Best Answer

Related Solutions

MATLAB: Solving two nonlinear equations

MATLAB: Is it so hard for MATLAB to solve the following symbolic equation

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