Finding more than one root for nonlinear equation

Hello there! Hope you're all fairing well.

I used the fsolve function to solve my problem, but it only returns one answer. I want to know if there is another function that can return all the possible answer, if there are more than one. Heres what I am trying to solve:

I'm trying to solve for qb, qt, beta, and their respective derivatives.

[Torsten]

What is A ? How to get a scalar from -A^(-1)*[1;0;0] ?

[Walter Roberson]

You could try setting it all up using the Symbolic toolbox, and solve()

[Lucas Resende]

A is another Matrix! But doesnt solve only return one possível answer?

[Torsten]

I don't understand how an inverse of a matrix can be a vector of size 1x3 which is necessary to make

-A^(-1)*[1;0;0] a scalar.

[Lucas Resende]

That's true, I sent It wrong. Its going to be a 1x3 vector, and the rest of the Matrix madre of zero.

[Torsten]

But doesnt solve only return one possível answer?

"solve" tries to find all solutions to a system of equations.

When setting up the system using the Symbolic Toolbox as suggested by @Walter Roberson, you can already assume that q_t_dot and beta_dot are 0. This follows from equations (2) and (3) of your system.

[Lucas Resende]

Thank you both so much! Im going to try It right away!

Answer 1

[Torsten]

Equations (5) and (6) are two linear equations in q_t, beta and q_dot_b. Use them to express q_t and beta as linear functions of q_b_dot.

Now insert the expressions for q_t and beta in equation (4). After having done this, equations (1) and (4) will be of the form

a11*q_dot_b + a12*q_b^3 = 0

a21*q_dot_b + a22*q_b = 0

Eliminating q_dot_b gives a cubic equation for q_b that usually has 3 solutions.

Inserting backwards gives 3 solutions for all unknowns except for q_dot_t and beta_dot that were identified to be zero right at the beginning.

syms q_dot_b q_b q_t beta
syms a11 a41 a42 a43 a44 a52 a53 a54 a62 a63 a64 real
eqn1 = a11*q_b^3 + q_dot_b == 0
eqn2 = a41*q_b+a42*q_t + a43*beta + a44*q_dot_b == 0
eqn3 = a52*q_t + a53*beta + a54*q_dot_b == 0
eqn4 = a62*q_t + a63*beta + a64*q_dot_b == 0
sol = solve([eqn1,eqn2,eqn3,eqn4],[q_b,q_t,beta,q_dot_b])

sol = struct with fields:
        q_b: [3x1 sym]
        q_t: [3x1 sym]
       beta: [3x1 sym]
    q_dot_b: [3x1 sym]

char(sol.q_b)

ans = '[0; (((a41^3*(a52*a63 - a53*a62)^3)/(a11*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62)^3))^(1/2)*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62))/(a41*(a52*a63 - a53*a62)); -(((a41^3*(a52*a63 - a53*a62)^3)/(a11*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62)^3))^(1/2)*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62))/(a41*(a52*a63 - a53*a62))]'

char(sol.q_t)

ans = '[0; -((a53*a64 - a54*a63)*((a41^3*(a52*a63 - a53*a62)^3)/(a11*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62)^3))^(1/2))/(a52*a63 - a53*a62); ((a53*a64 - a54*a63)*((a41^3*(a52*a63 - a53*a62)^3)/(a11*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62)^3))^(1/2))/(a52*a63 - a53*a62)]'

char(sol.beta)

ans = '[0; ((a52*a64 - a54*a62)*((a41^3*(a52*a63 - a53*a62)^3)/(a11*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62)^3))^(1/2))/(a52*a63 - a53*a62); -((a52*a64 - a54*a62)*((a41^3*(a52*a63 - a53*a62)^3)/(a11*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62)^3))^(1/2))/(a52*a63 - a53*a62)]'

char(sol.q_dot_b)

ans = '[0; -((a41^3*(a52*a63 - a53*a62)^3)/(a11*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62)^3))^(1/2); ((a41^3*(a52*a63 - a53*a62)^3)/(a11*(a42*a53*a64 - a42*a54*a63 - a43*a52*a64 + a43*a54*a62 + a44*a52*a63 - a44*a53*a62)^3))^(1/2)]'