EquationProblem

System of nonlinear equations

Since R2019b

Description

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

Tip

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

Creation

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;```

Warning

The problem-based approach does not support complex values in the following: an objective function, nonlinear equalities, and nonlinear inequalities. If a function calculation has a complex value, even as an intermediate value, the final result might be incorrect.

Properties

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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`

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

Data Types: `struct`

Object Functions

 `optimoptions` Create optimization options `prob2struct` Convert optimization problem or equation problem to solver form `show` Display information about optimization object `solve` Solve optimization problem or equation problem `solvers` Determine default and valid solvers for optimization problem or equation problem `varindex` Map problem variables to solver-based variable index `write` Save optimization object description

Examples

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To solve the nonlinear system of equations

`$\begin{array}{l}\mathrm{exp}\left(-\mathrm{exp}\left(-\left({x}_{1}+{x}_{2}\right)\right)\right)={x}_{2}\left(1+{x}_{1}^{2}\right)\\ {x}_{1}\mathrm{cos}\left({x}_{2}\right)+{x}_{2}\mathrm{sin}\left({x}_{1}\right)=\frac{1}{2}\end{array}$`

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.

`show(prob)`
``` EquationProblem : Solve for: x eq1: exp((-exp((-(x(1) + x(2)))))) == (x(2) .* (1 + x(1).^2)) eq2: ((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: -2.4070e-07 eq2: -3.8255e-08 ```
```exitflag = EquationSolved ```

View the solution point.

`disp(sol.x)`
``` 0.3532 0.6061 ```

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;```

Version History

Introduced in R2019b