# eqnproblem

## Description

Use `eqnproblem`

to create an equation problem.

**Tip**

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

specifies additional options using one or more name-value pair arguments. For example, you
can specify equations when constructing the problem by using the
`prob`

= eqnproblem(`Name,Value`

)`Equations`

name.

## Examples

### Solve Nonlinear System of Equations, Problem-Based

To solve the nonlinear system of equations

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

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;

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

### Solve Nonlinear System of Polynomials, Problem-Based

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

### 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)`

`Equations`

— Problem equations

`[]`

(default) | `OptimizationEquality`

array | structure with `OptimizationEquality`

arrays as fields

Problem equations, specified as an `OptimizationEquality`

array or structure with
`OptimizationEquality`

arrays as fields.

**Example: **`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
in `Description`

.

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

**Data Types: **`char`

| `string`

## Output Arguments

`prob`

— Equation problem

`EquationProblem`

object

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 might be incorrect.

## Version History

**Introduced in R2019b**

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