Nonlinear Least Squares (Curve Fitting)
Nonlinear least-squares is solving the problem min(∑||F(x_{i}) - y_{i}||^{2}), where F(x_{i}) is a nonlinear function and y_{i} is data. The problem can have bounds, linear constraints, or nonlinear constraints. These problems come from fitting curves to experimental data, estimating parameters for physical models, and others.
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 objective function and constraints in terms of these
symbolic variables. To formulate a least-squares problem, follow the
instructions in Write Objective Function for Problem-Based Least Squares. For
the problem-based steps to take, see Problem-Based Optimization Workflow. To
solve the resulting problem, use solve
.
For the solver-based steps to take, including defining the objective
function and constraints, and choosing the appropriate solver, see Solver-Based Optimization Problem Setup. To solve
the resulting problem, use lsqcurvefit
or lsqnonlin
.
Functions
Live Editor Tasks
Optimize | Optimize or solve equations in the Live Editor (Since R2020b) |
Topics
Problem-Based Nonlinear Least Squares
- Nonlinear Least-Squares, Problem-Based
Basic example of nonlinear least squares using the problem-based approach. - Nonlinear Data-Fitting Using Several Problem-Based Approaches
Solve a least-squares fitting problem using different solvers and different approaches to linear parameters. - Fit ODE Parameters Using Optimization Variables
Fit parameters of an ODE using problem-based least squares. - Compare lsqnonlin and fmincon for Constrained Nonlinear Least Squares
Compare the performance oflsqnonlin
andfmincon
on a nonlinear least-squares problem with nonlinear constraints. - Write Objective Function for Problem-Based Least Squares
Syntax rules for problem-based least squares.
Solver-Based Nonlinear Least Squares
- Nonlinear Data-Fitting
Basic example showing several ways to solve a data-fitting problem. - Banana Function Minimization
Shows how to solve for the minimum of Rosenbrock's function using different solvers, with or without gradients. - lsqnonlin with a Simulink Model
Example of fitting a simulated model. - Nonlinear Least Squares Without and Including Jacobian
Example showing the use of analytic derivatives in nonlinear least squares. - Nonlinear Curve Fitting with lsqcurvefit
Example showing how to do nonlinear data-fitting with lsqcurvefit. - Fit an Ordinary Differential Equation (ODE)
Example showing how to fit parameters of an ODE to data, or fit parameters of a curve to the solution of an ODE. - Fit a Model to Complex-Valued Data
Example showing how to solve a nonlinear least-squares problem that has complex-valued data.
Code Generation
- Code Generation in Nonlinear Least Squares: Background
Prerequisites to generate C code for nonlinear least squares. - Generate Code for lsqcurvefit or lsqnonlin
Example of code generation for nonlinear least squares. - 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
- Write Objective Function for Problem-Based Least Squares
Syntax rules for problem-based least squares. - Least-Squares (Model Fitting) Algorithms
Minimizing a sum of squares in n dimensions with only bound or linear constraints. - Optimization Options Reference
Explore optimization options.