Systems of Nonlinear Equations

Solve systems of nonlinear equations in serial or parallel

Find a solution to a multivariable nonlinear equation F(x) = 0. You can also solve a scalar equation or linear system of equations, or a system represented by F(x) = G(x) in the problem-based approach (equivalent to F(x) – G(x) = 0 in the solver-based approach). For nonlinear systems, solvers convert the equation-solving problem to the optimization problem of minimizing the sum of squares of the components of F, namely min(∑F_i²(x)). Linear and scalar equations have different solution algorithms; see Equation Solving Algorithms.

Before you begin to solve an optimization problem, you must choose the appropriate approach: problem-based or solver-based. For details, see First Choose Problem-Based or Solver-Based Approach.

For the problem-based approach, create problem variables, and then represent the equations in terms of these variables. For the problem-based steps to take, see Problem-Based Workflow for Solving Equations. To solve the resulting problem, use solve.

For the solver-based steps to take, including defining the objective function and choosing the appropriate solver, see Solver-Based Optimization Problem Setup.

Functions

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Solve and Analyze, Problem-Based

`eqnproblem`	Create equation problem
`evaluate`	Evaluate optimization expression or objectives and constraints in problem
`infeasibility`	Constraint violation at a point
`optimeq`	Create empty optimization equality array
`optimvar`	Create optimization variables
`prob2struct`	Convert optimization problem or equation problem to solver form
`show`	Display information about optimization object
`solve`	Solve optimization problem or equation problem
`write`	Save optimization object description

Solve Equations, Solver-Based

`fsolve`	Solve system of nonlinear equations
`fzero`	Root of nonlinear function
`lsqlin`	Solve constrained linear least-squares problems
`lsqnonlin`	Solve nonlinear least-squares (nonlinear data-fitting) problems
`checkGradients`	Check first derivative function against finite-difference approximation (Since R2023b)

Live Editor Tasks

Optimize

Optimize or solve equations in the Live Editor (Since R2020b)

Objects

`EquationProblem`	System of nonlinear equations
`OptimizationEquality`	Equalities and equality constraints
`OptimizationExpression`	Arithmetic or functional expression in terms of optimization variables
`OptimizationVariable`	Variable for optimization

Topics

Problem-Based Systems of Nonlinear Equations

Solve Nonlinear System of Equations, Problem-Based
Solve a system of nonlinear equations using the problem-based approach.
Solve Nonlinear System of Polynomials, Problem-Based
Solve a polynomial system of equations using the problem-based approach.
Follow Equation Solution as a Parameter Changes
Solve a sequence of problems using the previous solution as a start point.
Nonlinear System of Equations with Constraints, Problem-Based
Solve a system of nonlinear equations with constraints using the problem-based approach.

Solver-Based Systems of Nonlinear Equations

Solve Nonlinear System Without and Including Jacobian
Use derivatives in nonlinear equation solving.
Large System of Nonlinear Equations with Jacobian Sparsity Pattern
Solve a nonlinear system of equations with a known finite-difference sparsity pattern.
Large Sparse System of Nonlinear Equations with Jacobian
Example of solving a nonlinear system of equations that has derivatives available.
Nonlinear Systems with Constraints
Learn techniques for solving nonlinear systems of equations with constraints.

Code Generation

Code Generation in Nonlinear Equation Solving: Background
Prerequisites to generate C code for systems of nonlinear equations.
Generate Code for fsolve
Example of code generation for solving systems of nonlinear equations.
Optimization Code Generation for Real-Time Applications
Explore techniques for handling real-time requirements in generated code.

Parallel Computing

What Is Parallel Computing in Optimization Toolbox?
Use multiple processors for optimization.
Using Parallel Computing in Optimization Toolbox
Perform gradient estimation in parallel.
Improving Performance with Parallel Computing
Investigate factors for speeding optimizations.

Algorithms and Options

Equation Solving Algorithms
Solve linear systems of equations, nonlinear equations in one variable, and systems of n nonlinear equations in n variables.
Optimization Options Reference
Explore optimization options.