# eqnproblem

Create equation problem

## Syntax

``prob = eqnproblem``
``prob = eqnproblem(Name,Value)``

## Description

Use `eqnproblem` to create an equation problem.

Tip

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

example

````prob = eqnproblem` creates an equation problem with default properties.```

example

````prob = eqnproblem(Name,Value)` specifies additional options using one or more name-value pair arguments. For example, you can specify equations when constructing the problem by using the `Equations` name.```

## 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;```

When `x` is a 2-by-2 matrix, the equation

`${\mathit{x}}^{3}=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$`

is a system of polynomial equations. Here, ${x}^{3}$ means $x*x*x$ using matrix multiplication. You can easily formulate and solve this system using the problem-based approach.

First, define the variable `x` as a 2-by-2 matrix variable.

`x = optimvar('x',2,2);`

Define the equation to be solved in terms of `x`.

`eqn = x^3 == [1 2;3 4];`

Create an equation problem with this equation.

`prob = eqnproblem('Equations',eqn);`

Solve the problem starting from the point `[1 1;1 1]`.

```x0.x = ones(2); sol = 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: [2x2 double] ```

Examine the solution.

`disp(sol.x)`
``` -0.1291 0.8602 1.2903 1.1612 ```

Display the cube of the solution.

`sol.x^3`
```ans = 2×2 1.0000 2.0000 3.0000 4.0000 ```

## Input Arguments

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### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: `prob = eqnproblem('Equations',eqn)`

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`

## Output Arguments

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Equation problem, returned as an `EquationProblem` object. Typically, to complete the problem description, you specify `prob.Equations` and, for nonlinear equations, an initial point structure. Solve a complete problem by calling `solve`.

Warning

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.

## Version History

Introduced in R2019b