System of nonlinear equations
Specify a system of equations using optimization variables, and solve the system
EquationProblem object by using the
function. Add equations to the problem by creating
objects and setting them as
Equations properties of the
prob = eqnproblem; x = optimvar('x'); eqn = x^5 - x^4 + 3*x == 1/2; prob.Equations.eqn = eqn;
The problem-based approach does not support complex values in an objective function, nonlinear equalities, or nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result can be incorrect.
Equations— Problem equations
OptimizationEqualityarray | structure with
OptimizationEqualityarrays as fields
Problem equations, specified as an
OptimizationEquality array or structure with
OptimizationEquality arrays as fields.
sum(x.^2,2) == 4
Description— Problem label
''(default) | string | character vector
Problem label, specified as a string or character vector. The software does not use
Description for computation.
Description is an
arbitrary label that you can use for any reason. For example, you can share, archive, or
present a model or problem, and store descriptive information about the model or problem
"An iterative approach to the Traveling Salesman problem"
Variables— Optimization variables in object
This property is read-only.
Optimization variables in the object, specified as a structure of
To solve the nonlinear system of equations
using the problem-based approach, first define
x as a two-element optimization variable.
x = optimvar('x',2);
Create the left side of the first equation. Because this side is not a polynomial or rational function, process this expression into an optimization expression by using
ls1 = fcn2optimexpr(@(x)exp(-exp(-(x(1)+x(2)))),x);
Create the first equation.
eq1 = ls1 == x(2)*(1 + x(1)^2);
Similarly, create the left side of the second equation by using
ls2 = fcn2optimexpr(@(x)x(1)*cos(x(2))+x(2)*sin(x(1)),x);
Create the second equation.
eq2 = ls2 == 1/2;
Create an equation problem, and place the equations in the problem.
prob = eqnproblem; prob.Equations.eq1 = eq1; prob.Equations.eq2 = eq2;
Review the problem.
EquationProblem : Solve for: x eq1: arg_LHS == (x(2) .* (1 + x(1).^2)) where: anonymousFunction1 = @(x)exp(-exp(-(x(1)+x(2)))); arg_LHS = anonymousFunction1(x); eq2: arg_LHS == 0.5 where: anonymousFunction2 = @(x)x(1)*cos(x(2))+x(2)*sin(x(1)); arg_LHS = anonymousFunction2(x);
Solve the problem starting from the point
[0,0]. For the problem-based approach, specify the initial point as a structure, with the variable names as the fields of the structure. For this problem, there is only one variable,
x0.x = [0 0]; [sol,fval,exitflag] = solve(prob,x0)
Solving problem using fsolve. Equation solved. fsolve completed because the vector of function values is near zero as measured by the value of the function tolerance, and the problem appears regular as measured by the gradient.
sol = struct with fields: x: [2x1 double]
fval = struct with fields: eq1: -2.4069e-07 eq2: -3.8253e-08
exitflag = EquationSolved
View the solution point.