System of nonlinear equations


Specify a system of equations using optimization variables, and solve the system using solve.


For the full workflow, see Problem-Based Workflow for Solving Equations.


Create an EquationProblem object by using the eqnproblem function. Add equations to the problem by creating OptimizationEquality objects and setting them as Equations properties of the EquationProblem object.

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.


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Problem equations, specified as an OptimizationEquality array or structure with OptimizationEquality arrays as fields.

Example: sum(x.^2,2) == 4

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 in Description.

Example: "An iterative approach to the Traveling Salesman problem"

Data Types: char | string

This property is read-only.

Optimization variables in the object, specified as a structure of OptimizationVariable objects.

Data Types: struct

Object Functions

optimoptionsCreate optimization options
prob2structConvert optimization problem or equation problem to solver form
showDisplay information about optimization object
solveSolve optimization problem or equation problem
varindexMap problem variables to solver-based variable index
writeSave optimization object description


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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 first equation as an optimization equality expression.

eq1 = exp(-exp(-(x(1) + x(2)))) == x(2)*(1 + x(1)^2);

Similarly, create the second equation as an optimization equality expression.

eq2 = x(1)*cos(x(2)) + x(2)*sin(x(1)) == 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:

       exp(-exp(-(x(1) + x(2)))) == (x(2) .* (1 + x(1).^2))

       ((x(1) .* cos(x(2))) + (x(2) .* sin(x(1)))) == 0.5

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, x.

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: -3.8255e-08

exitflag = 

View the solution point.


Unsupported Functions Require fcn2optimexpr

If your equation functions are not composed of elementary functions, you must convert the functions to optimization expressions using fcn2optimexpr. For the present example:

ls1 = fcn2optimexpr(@(x)exp(-exp(-(x(1)+x(2)))),x);
eq1 = ls1 == x(2)*(1 + x(1)^2);
ls2 = fcn2optimexpr(@(x)x(1)*cos(x(2))+x(2)*sin(x(1)),x);
eq2 = ls2 == 1/2;

See Supported Operations on Optimization Variables and Expressions and Convert Nonlinear Function to Optimization Expression.

Introduced in R2019b