lsqcurvefit

Solve nonlinear curve-fitting (data-fitting) problems in least-squares sense

Syntax

``x = lsqcurvefit(fun,x0,xdata,ydata)``
``x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub)``
``x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq)``
``x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,nonlcon)``
``x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options)``
``x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,nonlcon,options)``
``x = lsqcurvefit(problem)``
``````[x,resnorm] = lsqcurvefit(___)``````
``````[x,resnorm,residual,exitflag,output] = lsqcurvefit(___)``````
``````[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqcurvefit(___)``````

Description

Nonlinear least-squares solver

Find coefficients x that solve the problem

`$\underset{x}{\mathrm{min}}{‖F\left(x,xdata\right)-ydata‖}_{2}^{2}=\underset{x}{\mathrm{min}}\sum _{i}{\left(F\left(x,xdat{a}_{i}\right)-ydat{a}_{i}\right)}^{2},$`

given input data xdata, and the observed output ydata, where xdata and ydata are matrices or vectors, and F (x, xdata) is a matrix-valued or vector-valued function of the same size as ydata.

Optionally, the components of x are subject to the constraints

`$\begin{array}{c}\text{lb}\le x\\ x\le \text{ub}\\ Ax\le b\\ \text{Aeq}x=\text{beq}\\ c\left(x\right)\le 0\\ \text{ceq}\left(x\right)=0.\end{array}$`

The arguments x, lb, and ub can be vectors or matrices; see Matrix Arguments.

The `lsqcurvefit` function uses the same algorithm as `lsqnonlin`. `lsqcurvefit` simply provides a convenient interface for data-fitting problems.

Rather than compute the sum of squares, `lsqcurvefit` requires the user-defined function to compute the vector-valued function

`$F\left(x,xdata\right)=\left[\begin{array}{c}F\left(x,xdata\right)\left(1\right)\\ F\left(x,xdata\right)\left(2\right)\\ ⋮\\ F\left(x,xdata\right)\left(k\right)\end{array}\right].$`

example

````x = lsqcurvefit(fun,x0,xdata,ydata)` starts at `x0` and finds coefficients `x` to best fit the nonlinear function `fun(x,xdata)` to the data `ydata` (in the least-squares sense). `ydata` must be the same size as the vector (or matrix) `F` returned by `fun`. NotePassing Extra Parameters explains how to pass extra parameters to the vector function `fun(x)`, if necessary. ```

example

````x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub)` defines a set of lower and upper bounds on the design variables in `x`, so that the solution is always in the range `lb `≤` x `≤` ub`. You can fix the solution component `x(i)` by specifying `lb(i) = ub(i)`. NoteIf the specified input bounds for a problem are inconsistent, the output `x` is `x0` and the outputs `resnorm` and `residual` are `[]`.Components of `x0` that violate the bounds `lb ≤ x ≤ ub` are reset to the interior of the box defined by the bounds. Components that respect the bounds are not changed. ```

example

````x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq)` constrains the solution to satisfy the linear constraintsA x ≤ bAeq x = beq.```

example

````x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,nonlcon)` constrain the solution to satisfy the nonlinear constraints in the `nonlcon`(`x`) function. `nonlcon` returns two outputs, `c` and `ceq`. The solver attempts to satisfy the constraintsc ≤ 0`ceq` = 0.```

example

````x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options)` or `x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,nonlcon,options)` minimizes with the optimization options specified in `options`. Use `optimoptions` to set these options. Pass empty matrices for `lb` and `ub` and for other input arguments if the arguments do not exist.```
````x = lsqcurvefit(problem)` finds the minimum for `problem`, a structure described in `problem`.```
``````[x,resnorm] = lsqcurvefit(___)```, for any input arguments, returns the value of the squared 2-norm of the residual at `x`: `sum((fun(x,xdata)-ydata).^2)`.```

example

``````[x,resnorm,residual,exitflag,output] = lsqcurvefit(___)``` additionally returns the value of the residual `fun(x,xdata)-ydata` at the solution `x`, a value `exitflag` that describes the exit condition, and a structure `output` that contains information about the optimization process.```
``````[x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqcurvefit(___)``` additionally returns a structure `lambda` whose fields contain the Lagrange multipliers at the solution `x`, and the Jacobian of `fun` at the solution `x`.```

Examples

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Suppose that you have observation time data `xdata` and observed response data `ydata`, and you want to find parameters $x\left(1\right)$ and $x\left(2\right)$ to fit a model of the form

`$\text{ydata}=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right).$`

Input the observation times and responses.

```xdata = ... [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3]; ydata = ... [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];```

Create a simple exponential decay model.

`fun = @(x,xdata)x(1)*exp(x(2)*xdata);`

Fit the model using the starting point `x0 = [100,-1]`.

```x0 = [100,-1]; x = lsqcurvefit(fun,x0,xdata,ydata)```
```Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. ```
```x = 1×2 498.8309 -0.1013 ```

Plot the data and the fitted curve.

```times = linspace(xdata(1),xdata(end)); plot(xdata,ydata,'ko',times,fun(x,times),'b-') legend('Data','Fitted exponential') title('Data and Fitted Curve')```

Find the best exponential fit to data where the fitting parameters are constrained.

Generate data from an exponential decay model plus noise. The model is

`$y=\mathrm{exp}\left(-1.3t\right)+\epsilon ,$`

with $t$ ranging from 0 through 3, and $\epsilon$ normally distributed noise with mean 0 and standard deviation 0.05.

```rng default % for reproducibility xdata = linspace(0,3); ydata = exp(-1.3*xdata) + 0.05*randn(size(xdata));```

The problem is: given the data (`xdata`, `ydata`), find the exponential decay model $y=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right)$ that best fits the data, with the parameters bounded as follows:

`$0\le x\left(1\right)\le 3/4$`

`$-2\le x\left(2\right)\le -1.$`

```lb = [0,-2]; ub = [3/4,-1];```

Create the model.

`fun = @(x,xdata)x(1)*exp(x(2)*xdata);`

Create an initial guess.

`x0 = [1/2,-2];`

Solve the bounded fitting problem.

`x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub)`
```Local minimum found. Optimization completed because the size of the gradient is less than the value of the optimality tolerance. ```
```x = 1×2 0.7500 -1.0000 ```

Examine how well the resulting curve fits the data. Because the bounds keep the solution away from the true values, the fit is mediocre.

```plot(xdata,ydata,'ko',xdata,fun(x,xdata),'b-') legend('Data','Fitted exponential') title('Data and Fitted Curve')```

Create artificial data for a nonlinear model $y=a+b\mathrm{arctan}\left(t-{t}_{0}\right)+ct$ with parameters $a$, $b$, ${t}_{0}$, and $c$, for time $t$ from 2 to 7. Add noise to the data using `randn`.

```a = 2; % x(1) b = 4; % x(2) t0 = 5; % x(3) c = 1/2; % x(4) xdata = linspace(2,7); rng default ydata = a + b*atan(xdata - t0) + c*xdata + 1/10*randn(size(xdata));```

Plot the data.

`plot(xdata,ydata,'ro')`

Fit a nonlinear model to the data with the following constraints:

• All coefficients are between 0 and 7.

• ${x}_{1}+{x}_{2}\ge {x}_{3}+{x}_{4}$. You can write this constraint in the form `A*x <= b` using `A = [-1 -1 1 1]` and `b = 0`.

```lb = zeros(4,1); ub = 7*ones(4,1); A = [-1 -1 1 1]; b = 0;```

The `myfun` function at the end of this example creates the objective function for this model.

Solve the fitting problem starting from the point `[1 2 3 1]`.

```startpt = [1 2 3 1]; Aeq = []; beq = []; [x,res] = lsqcurvefit(@myfun,startpt,xdata,ydata,lb,ub,A,b,Aeq,beq)```
```Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. ```
```x = 1×4 2.3447 4.0972 4.9979 0.4303 ```
```res = 1.2682 ```

The returned solution is not far from the original point `[2 4 5 1/2]`. Plot the data against the curve from the solution point.

`plot(xdata,ydata,'ro',xdata,myfun(x,xdata),'b-')`

The returned solution matches the data pretty well. Is the constraint active?

`A*x(:)`
```ans = -1.0137 ```

The constraint is not active, because `A*x < 0`.

```function F = myfun(x,xdata) a = x(1); b = x(2); t0 = x(3); c = x(4); F = a + b*atan(xdata - t0) + c*xdata; end```

Create artificial data for a nonlinear model $y=a+b\mathrm{arctan}\left(t-{t}_{0}\right)+ct$ with parameters $a$, $b$, ${t}_{0}$, and $c$, for time $t$ from 2 to 7. Add noise to the data using `randn`.

```a = 2; % x(1) b = 4; % x(2) t0 = 5; % x(3) c = 1/2; % x(4) xdata = linspace(2,7); rng default ydata = a + b*atan(xdata - t0) + c*xdata + 1/10*randn(size(xdata));```

Plot the data.

`plot(xdata,ydata,'ro')`

Fit a nonlinear model to the data with the following constraints:

• All coefficients are between 0 and 7.

• ${x}_{1}^{2}+{x}_{2}^{2}\le {4}^{2}$

```lb = zeros(4,1); ub = 7*ones(4,1);```

The problem has no linear constraints.

```A = []; b = []; Aeq = []; beq = [];```

The `myfun` function at the end of this example creates the objective function for this model. The `nlcon` function at the end of this example creates the nonlinear constraint function.

Solve the fitting problem starting from the point `[1 2 3 1]`.

```startpt = [1 2 3 1]; [x,res] = lsqcurvefit(@myfun,startpt,xdata,ydata,lb,ub,A,b,Aeq,beq,@nlcon)```
```Feasible point with lower objective function value found, but optimality criteria not satisfied. See output.bestfeasible.. Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. ```
```x = 1×4 1.3806 3.7542 5.0169 0.6337 ```
```res = 1.6018 ```

The returned solution `x` is not at the original point `[2 4 5 1/2]` because the nonlinear constraint is violated at that point. Plot the data against the curve from the solution point and compute the constraint function.

`plot(xdata,ydata,'ro',xdata,myfun(x,xdata),'b-')`

`[c,ceq] = nlcon(x)`
```c = -3.1307e-06 ```
```ceq = [] ```

The nonlinear inequality constraint is active at the solution because `c = 0` at the solution.

Even though the solution point is not at the original point, the solution curve matches the data pretty well.

```function F = myfun(x,xdata) a = x(1); b = x(2); t0 = x(3); c = x(4); F = a + b*atan(xdata - t0) + c*xdata; end function [c,ceq] = nlcon(x) ceq = []; c = x(1)^2 + x(2)^2 - 4^2; end```

Compare the results of fitting with the default `'trust-region-reflective'` algorithm and the `'levenberg-marquardt'` algorithm.

Suppose that you have observation time data `xdata` and observed response data `ydata`, and you want to find parameters $x\left(1\right)$ and $x\left(2\right)$ to fit a model of the form

`$\text{ydata}=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right).$`

Input the observation times and responses.

```xdata = ... [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3]; ydata = ... [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];```

Create a simple exponential decay model.

`fun = @(x,xdata)x(1)*exp(x(2)*xdata);`

Fit the model using the starting point `x0 = [100,-1]`.

```x0 = [100,-1]; x = lsqcurvefit(fun,x0,xdata,ydata)```
```Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. ```
```x = 1×2 498.8309 -0.1013 ```

Compare the solution with that of a `'levenberg-marquardt'` fit.

```options = optimoptions('lsqcurvefit','Algorithm','levenberg-marquardt'); lb = []; ub = []; x = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options)```
```Local minimum possible. lsqcurvefit stopped because the relative size of the current step is less than the value of the step size tolerance. ```
```x = 1×2 498.8309 -0.1013 ```

The two algorithms converged to the same solution. Plot the data and the fitted exponential model.

```times = linspace(xdata(1),xdata(end)); plot(xdata,ydata,'ko',times,fun(x,times),'b-') legend('Data','Fitted exponential') title('Data and Fitted Curve')```

Compare the results of fitting with the default `'trust-region-reflective'` algorithm and the `'levenberg-marquardt'` algorithm. Examine the solution process to see which is more efficient in this case.

Suppose that you have observation time data `xdata` and observed response data `ydata`, and you want to find parameters $x\left(1\right)$ and $x\left(2\right)$ to fit a model of the form

`$\text{ydata}=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right).$`

Input the observation times and responses.

```xdata = ... [0.9 1.5 13.8 19.8 24.1 28.2 35.2 60.3 74.6 81.3]; ydata = ... [455.2 428.6 124.1 67.3 43.2 28.1 13.1 -0.4 -1.3 -1.5];```

Create a simple exponential decay model.

`fun = @(x,xdata)x(1)*exp(x(2)*xdata);`

Fit the model using the starting point `x0 = [100,-1]`.

```x0 = [100,-1]; [x,resnorm,residual,exitflag,output] = lsqcurvefit(fun,x0,xdata,ydata);```
```Local minimum possible. lsqcurvefit stopped because the final change in the sum of squares relative to its initial value is less than the value of the function tolerance. ```

Compare the solution with that of a `'levenberg-marquardt'` fit.

```options = optimoptions('lsqcurvefit','Algorithm','levenberg-marquardt'); lb = []; ub = []; [x2,resnorm2,residual2,exitflag2,output2] = lsqcurvefit(fun,x0,xdata,ydata,lb,ub,options);```
```Local minimum possible. lsqcurvefit stopped because the relative size of the current step is less than the value of the step size tolerance. ```

Are the solutions equivalent?

`norm(x-x2)`
```ans = 2.0642e-06 ```

Yes, the solutions are equivalent.

Which algorithm took fewer function evaluations to arrive at the solution?

```fprintf(['The ''trust-region-reflective'' algorithm took %d function evaluations,\n',... 'and the ''levenberg-marquardt'' algorithm took %d function evaluations.\n'],... output.funcCount,output2.funcCount)```
```The 'trust-region-reflective' algorithm took 87 function evaluations, and the 'levenberg-marquardt' algorithm took 72 function evaluations. ```

Plot the data and the fitted exponential model.

```times = linspace(xdata(1),xdata(end)); plot(xdata,ydata,'ko',times,fun(x,times),'b-') legend('Data','Fitted exponential') title('Data and Fitted Curve')```

The fit looks good. How large are the residuals?

```fprintf(['The ''trust-region-reflective'' algorithm has residual norm %f,\n',... 'and the ''levenberg-marquardt'' algorithm has residual norm %f.\n'],... resnorm,resnorm2)```
```The 'trust-region-reflective' algorithm has residual norm 9.504887, and the 'levenberg-marquardt' algorithm has residual norm 9.504887. ```

Input Arguments

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Function you want to fit, specified as a function handle or the name of a function. For the `'interior-point'` algorithm, `fun` must be a function handle. `fun` is a function that takes two inputs: a vector or matrix `x`, and a vector or matrix `xdata`. `fun` returns a vector or matrix `F`, the objective function evaluated at `x` and `xdata`.

Note

`fun` should return `fun(x,xdata)`, and not the sum-of-squares `sum((fun(x,xdata)-ydata).^2)`. `lsqcurvefit` implicitly computes the sum of squares of the components of `fun(x,xdata)-ydata`. See Examples.

The function `fun` can be specified as a function handle for a function file:

`x = lsqcurvefit(@myfun,x0,xdata,ydata)`

where `myfun` is a MATLAB® function such as

```function F = myfun(x,xdata) F = ... % Compute function values at x, xdata```

`fun` can also be a function handle for an anonymous function.

```f = @(x,xdata)x(1)*xdata.^2+x(2)*sin(xdata); x = lsqcurvefit(f,x0,xdata,ydata);```

`lsqcurvefit` passes `x` to your objective function in the shape of the `x0` argument. For example, if `x0` is a 5-by-3 array, then `lsqcurvefit` passes `x` to `fun` as a 5-by-3 array.

If the Jacobian can also be computed and the `'SpecifyObjectiveGradient'` option is `true`, set by

`options = optimoptions('lsqcurvefit','SpecifyObjectiveGradient',true)`

then the function `fun` must return a second output argument with the Jacobian value `J` (a matrix) at `x`. By checking the value of `nargout`, the function can avoid computing `J` when `fun` is called with only one output argument (in the case where the optimization algorithm only needs the value of `F` but not `J`).

```function [F,J] = myfun(x,xdata) F = ... % objective function values at x if nargout > 1 % two output arguments J = ... % Jacobian of the function evaluated at x end ```

If `fun` returns a vector (matrix) of `m` components and `x` has `n` elements, where `n` is the number of elements of `x0`, the Jacobian `J` is an `m`-by-`n` matrix where `J(i,j)` is the partial derivative of `F(i)` with respect to `x(j)`. (The Jacobian `J` is the transpose of the gradient of `F`.) For more information, see Writing Vector and Matrix Objective Functions.

Example: `@(x,xdata)x(1)*exp(-x(2)*xdata)`

Data Types: `char` | `function_handle` | `string`

Initial point, specified as a real vector or real array. Solvers use the number of elements in `x0` and the size of `x0` to determine the number and size of variables that `fun` accepts.

Example: `x0 = [1,2,3,4]`

Data Types: `double`

Input data for model, specified as a real vector or real array. The model is

`ydata = fun(x,xdata)`,

where `xdata` and `ydata` are fixed arrays, and `x` is the array of parameters that `lsqcurvefit` changes to search for a minimum sum of squares.

Example: `xdata = [1,2,3,4]`

Data Types: `double`

Response data for model, specified as a real vector or real array. The model is

`ydata = fun(x,xdata)`,

where `xdata` and `ydata` are fixed arrays, and `x` is the array of parameters that `lsqcurvefit` changes to search for a minimum sum of squares.

The `ydata` array must be the same size and shape as the array `fun(x0,xdata)`.

Example: `ydata = [1,2,3,4]`

Data Types: `double`

Lower bounds, specified as a real vector or real array. If the number of elements in `x0` is equal to the number of elements in `lb`, then `lb` specifies that

`x(i) >= lb(i)` for all `i`.

If `numel(lb) < numel(x0)`, then `lb` specifies that

`x(i) >= lb(i)` for ```1 <= i <= numel(lb)```.

If `lb` has fewer elements than `x0`, solvers issue a warning.

Example: To specify that all x components are positive, use ```lb = zeros(size(x0))```.

Data Types: `double`

Upper bounds, specified as a real vector or real array. If the number of elements in `x0` is equal to the number of elements in `ub`, then `ub` specifies that

`x(i) <= ub(i)` for all `i`.

If `numel(ub) < numel(x0)`, then `ub` specifies that

`x(i) <= ub(i)` for ```1 <= i <= numel(ub)```.

If `ub` has fewer elements than `x0`, solvers issue a warning.

Example: To specify that all x components are less than 1, use ```ub = ones(size(x0))```.

Data Types: `double`

Linear inequality constraints, specified as a real matrix. `A` is an `M`-by-`N` matrix, where `M` is the number of inequalities, and `N` is the number of variables (number of elements in `x0`). For large problems, pass `A` as a sparse matrix.

`A` encodes the `M` linear inequalities

`A*x <= b`,

where `x` is the column vector of `N` variables `x(:)`, and `b` is a column vector with `M` elements.

For example, consider these inequalities:

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30,

Specify the inequalities by entering the following constraints.

```A = [1,2;3,4;5,6]; b = [10;20;30];```

Example: To specify that the x components sum to 1 or less, use ```A = ones(1,N)``` and `b = 1`.

Data Types: `double`

Linear inequality constraints, specified as a real vector. `b` is an `M`-element vector related to the `A` matrix. If you pass `b` as a row vector, solvers internally convert `b` to the column vector `b(:)`. For large problems, pass `b` as a sparse vector.

`b` encodes the `M` linear inequalities

`A*x <= b`,

where `x` is the column vector of `N` variables `x(:)`, and `A` is a matrix of size `M`-by-`N`.

For example, consider these inequalities:

x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30.

Specify the inequalities by entering the following constraints.

```A = [1,2;3,4;5,6]; b = [10;20;30];```

Example: To specify that the x components sum to 1 or less, use ```A = ones(1,N)``` and `b = 1`.

Data Types: `double`

Linear equality constraints, specified as a real matrix. `Aeq` is an `Me`-by-`N` matrix, where `Me` is the number of equalities, and `N` is the number of variables (number of elements in `x0`). For large problems, pass `Aeq` as a sparse matrix.

`Aeq` encodes the `Me` linear equalities

`Aeq*x = beq`,

where `x` is the column vector of `N` variables `x(:)`, and `beq` is a column vector with `Me` elements.

For example, consider these inequalities:

x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20,

Specify the inequalities by entering the following constraints.

```Aeq = [1,2,3;2,4,1]; beq = [10;20];```

Example: To specify that the x components sum to 1, use `Aeq = ones(1,N)` and `beq = 1`.

Data Types: `double`

Linear equality constraints, specified as a real vector. `beq` is an `Me`-element vector related to the `Aeq` matrix. If you pass `beq` as a row vector, solvers internally convert `beq` to the column vector `beq(:)`. For large problems, pass `beq` as a sparse vector.

`beq` encodes the `Me` linear equalities

`Aeq*x = beq`,

where `x` is the column vector of `N` variables `x(:)`, and `Aeq` is a matrix of size `Me`-by-`N`.

For example, consider these equalities:

x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20.

Specify the equalities by entering the following constraints.

```Aeq = [1,2,3;2,4,1]; beq = [10;20];```

Example: To specify that the x components sum to 1, use `Aeq = ones(1,N)` and `beq = 1`.

Data Types: `double`

Nonlinear constraints, specified as a function handle. `nonlcon` is a function that accepts a vector or array `x` and returns two arrays, `c(x)` and `ceq(x)`.

• `c(x)` is the array of nonlinear inequality constraints at `x`. `lsqcurvefit` attempts to satisfy

 `c(x) <= 0` for all entries of `c`. (1)
• `ceq(x)` is the array of nonlinear equality constraints at `x`. `lsqcurvefit` attempts to satisfy

 `ceq(x) = 0` for all entries of `ceq`. (2)

For example,

`x = lsqcurvefit(@myfun,x0,xdata,ydata,lb,ub,A,b,Aeq,beq,@mycon,options)`

where `mycon` is a MATLAB function such as

```function [c,ceq] = mycon(x) c = ... % Compute nonlinear inequalities at x. ceq = ... % Compute nonlinear equalities at x.```
If the gradients of the constraints can also be computed and the `SpecifyConstraintGradient` option is `true`, as set by
`options = optimoptions('lsqcurvefit','SpecifyConstraintGradient',true)`
then `nonlcon` must also return, in the third and fourth output arguments, `GC`, the gradient of `c(x)`, and `GCeq`, the gradient of `ceq(x)`. `GC` and `GCeq` can be sparse or dense. If `GC` or `GCeq` is large, with relatively few nonzero entries, save running time and memory in the `'interior-point'` algorithm by representing them as sparse matrices. For more information, see Nonlinear Constraints.

Data Types: `function_handle`

Optimization options, specified as the output of `optimoptions` or a structure such as `optimset` returns.

Some options apply to all algorithms, and others are relevant for particular algorithms. See Optimization Options Reference for detailed information.

Some options are absent from the `optimoptions` display. These options appear in italics in the following table. For details, see View Optimization Options.

 All Algorithms `Algorithm` Choose between `'trust-region-reflective'` (default), `'levenberg-marquardt'`, and `'interior-point'`.The `Algorithm` option specifies a preference for which algorithm to use. It is only a preference, because certain conditions must be met to use each algorithm. For the trust-region-reflective algorithm, the number of elements of `F` returned by `fun` must be at least as many as the length of `x`.The `'interior-point'` algorithm is the only algorithm that can solve problems with linear or nonlinear constraints. If you include these constraints in your problem and do not specify an algorithm, the solver automatically switches to the `'interior-point'` algorithm. The `'interior-point'` algorithm calls a modified version of the `fmincon` `'interior-point'` algorithm.For more information on choosing the algorithm, see Choosing the Algorithm. `CheckGradients` Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. Choices are `false` (default) or `true`.For `optimset`, the name is `DerivativeCheck` and the values are `'on'` or `'off'`. See Current and Legacy Option Names.The `CheckGradients` option will be removed in a future release. To check derivatives, use the `checkGradients` function. Diagnostics Display diagnostic information about the function to be minimized or solved. Choices are `'off'` (default) or `'on'`. DiffMaxChange Maximum change in variables for finite-difference gradients (a positive scalar). The default is `Inf`. DiffMinChange Minimum change in variables for finite-difference gradients (a positive scalar). The default is `0`. `Display` Level of display (see Iterative Display): `'off'` or `'none'` displays no output.`'iter'` displays output at each iteration, and gives the default exit message.`'iter-detailed'` displays output at each iteration, and gives the technical exit message.`'final'` (default) displays just the final output, and gives the default exit message.`'final-detailed'` displays just the final output, and gives the technical exit message. `FiniteDifferenceStepSize` Scalar or vector step size factor for finite differences. When you set `FiniteDifferenceStepSize` to a vector `v`, the forward finite differences `delta` are`delta = v.*sign′(x).*max(abs(x),TypicalX);`where `sign′(x) = sign(x)` except `sign′(0) = 1`. Central finite differences are`delta = v.*max(abs(x),TypicalX);`A scalar `FiniteDifferenceStepSize` expands to a vector. The default is `sqrt(eps)` for forward finite differences, and `eps^(1/3)` for central finite differences. For `optimset`, the name is `FinDiffRelStep`. See Current and Legacy Option Names. `FiniteDifferenceType` Finite differences, used to estimate gradients, are either `'forward'` (default), or `'central'` (centered). `'central'` takes twice as many function evaluations, but should be more accurate.The algorithm is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to avoid evaluating at a point outside bounds. For `optimset`, the name is `FinDiffType`. See Current and Legacy Option Names. `FunctionTolerance` Termination tolerance on the function value, a nonnegative scalar. The default is `1e-6`. See Tolerances and Stopping Criteria.For `optimset`, the name is `TolFun`. See Current and Legacy Option Names. FunValCheck Check whether function values are valid. `'on'` displays an error when the function returns a value that is `complex`, `Inf`, or `NaN`. The default `'off'` displays no error. `MaxFunctionEvaluations` Maximum number of function evaluations allowed, a nonnegative integer. The default is `100*numberOfVariables` for the `'trust-region-reflective'` algorithm, `200*numberOfVariables` for the `'levenberg-marquardt'` algorithm, and `3000` for the `'interior-point'` algorithm. See Tolerances and Stopping Criteria and Iterations and Function Counts.For `optimset`, the name is `MaxFunEvals`. See Current and Legacy Option Names. `MaxIterations` Maximum number of iterations allowed, a nonnegative integer. The default is `400` for the `'trust-region-reflective'` and `'levenberg-marquardt'` algorithms, and `1000` for the `'interior-point'` algorithm. See Tolerances and Stopping Criteria and Iterations and Function Counts.For `optimset`, the name is `MaxIter`. See Current and Legacy Option Names. `OptimalityTolerance` Termination tolerance on the first-order optimality (a nonnegative scalar). The default is `1e-6`. See First-Order Optimality Measure.Internally, the `'levenberg-marquardt'` algorithm uses an optimality tolerance (stopping criterion) of `1e-4` times `FunctionTolerance` and does not use `OptimalityTolerance`. For `optimset`, the name is `TolFun`. See Current and Legacy Option Names. `OutputFcn` Specify one or more user-defined functions that an optimization function calls at each iteration. Pass a function handle or a cell array of function handles. The default is none (`[]`). See Output Function and Plot Function Syntax. `PlotFcn` Plots various measures of progress while the algorithm executes; select from predefined plots or write your own. Pass a name, a function handle, or a cell array of names or function handles. For custom plot functions, pass function handles. The default is none (`[]`): `'optimplotx'` plots the current point.`'optimplotfunccount'` plots the function count.`'optimplotfval'` plots the function value.`'optimplotresnorm'` plots the norm of the residuals.`'optimplotstepsize'` plots the step size.`'optimplotfirstorderopt'` plots the first-order optimality measure. Custom plot functions use the same syntax as output functions. See Output Functions for Optimization Toolbox and Output Function and Plot Function Syntax.For `optimset`, the name is `PlotFcns`. See Current and Legacy Option Names. `SpecifyObjectiveGradient` If `false` (default), the solver approximates the Jacobian using finite differences. If `true`, the solver uses a user-defined Jacobian (defined in `fun`), or Jacobian information (when using `JacobMult`), for the objective function. For `optimset`, the name is `Jacobian`, and the values are `'on'` or `'off'`. See Current and Legacy Option Names. `StepTolerance` Termination tolerance on `x`, a nonnegative scalar. The default is `1e-6` for the `'trust-region-reflective'` and `'levenberg-marquardt'` algorithms, and `1e-10` for the `'interior-point'` algorithm. See Tolerances and Stopping Criteria.For `optimset`, the name is `TolX`. See Current and Legacy Option Names. `TypicalX` Typical `x` values. The number of elements in `TypicalX` is equal to the number of elements in `x0`, the starting point. The default value is `ones(numberofvariables,1)`. The solver uses `TypicalX` for scaling finite differences for gradient estimation. `UseParallel` When `true`, the solver estimates gradients in parallel. Disable by setting to the default, `false`. See Parallel Computing. Trust-Region-Reflective Algorithm `JacobianMultiplyFcn` Jacobian multiply function, specified as a function handle. For large-scale structured problems, this function computes the Jacobian matrix product `J*Y`, `J'*Y`, or `J'*(J*Y)` without actually forming `J`. For `lsqnonlin` the function is of the form`W = jmfun(Jinfo,Y,flag)`where `Jinfo` contains the data that helps to compute ```J*Y ``` (or `J'*Y`, or `J'*(J*Y)`). For `lsqcurvefit` the function is of the form`W = jmfun(Jinfo,Y,flag,xdata)`where `xdata` is the data passed in the `xdata` argument.The data `Jinfo` is the second argument returned by the objective function `fun`:```[F,Jinfo] = fun(x) % or [F,Jinfo] = fun(x,xdata)````lsqcurvefit` passes the data `Jinfo`, `Y`, `flag`, and, for `lsqcurvefit`, `xdata`, and your function `jmfun` computes a result as specified next. `Y` is a matrix whose size depends on the value of `flag`. Let `m` specify the number of components of the objective function `fun`, and let `n` specify the number of problem variables in `x`. The Jacobian is of size `m`-by-`n` as described in `fun`. The `jmfun` function returns one of these results:If `flag == 0` then `W = J'*(J*Y)` and `Y` has size `n`-by-`2`. If `flag > 0` then `W = J*Y` and `Y` has size `n`-by-`1`.If `flag < 0` then `W = J'*Y` and `Y` has size `m`-by-`1`.In each case, `J` is not formed explicitly. The solver uses `Jinfo` to compute the multiplications. See Passing Extra Parameters for information on how to supply values for any additional parameters `jmfun` needs.Note`'SpecifyObjectiveGradient'` must be set to `true` for the solver to pass `Jinfo` from `fun` to `jmfun`.See Minimization with Dense Structured Hessian, Linear Equalities and Jacobian Multiply Function with Linear Least Squares for similar examples.For `optimset`, the name is `JacobMult`. See Current and Legacy Option Names. JacobPattern Sparsity pattern of the Jacobian for finite differencing. Set `JacobPattern(i,j) = 1` when `fun(i)` depends on `x(j)`. Otherwise, set ```JacobPattern(i,j) = 0```. In other words, `JacobPattern(i,j) = 1` when you can have ∂`fun(i)`/∂`x(j)` ≠ 0.Use `JacobPattern` when it is inconvenient to compute the Jacobian matrix `J` in `fun`, though you can determine (say, by inspection) when `fun(i)` depends on `x(j)`. The solver can approximate `J` via sparse finite differences when you give `JacobPattern`.If the structure is unknown, do not set `JacobPattern`. The default behavior is as if `JacobPattern` is a dense matrix of ones. Then the solver computes a full finite-difference approximation in each iteration. This can be expensive for large problems, so it is usually better to determine the sparsity structure. MaxPCGIter Maximum number of PCG (preconditioned conjugate gradient) iterations, a positive scalar. The default is `max(1,numberOfVariables/2)`. For more information, see Large Scale Nonlinear Least Squares. PrecondBandWidth Upper bandwidth of preconditioner for PCG, a nonnegative integer. The default `PrecondBandWidth` is `Inf`, which means a direct factorization (Cholesky) is used rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution. Set `PrecondBandWidth` to `0` for diagonal preconditioning (upper bandwidth of 0). For some problems, an intermediate bandwidth reduces the number of PCG iterations. `SubproblemAlgorithm` Determines how the iteration step is calculated. The default, `'factorization'`, takes a slower but more accurate step than `'cg'`. See Trust-Region-Reflective Least Squares. TolPCG Termination tolerance on the PCG iteration, a positive scalar. The default is `0.1`. Levenberg-Marquardt Algorithm InitDamping Initial value of the Levenberg-Marquardt parameter, a positive scalar. Default is `1e-2`. For details, see Levenberg-Marquardt Method. ScaleProblem `'jacobian'` can sometimes improve the convergence of a poorly scaled problem; the default is `'none'`. Interior-Point Algorithm `BarrierParamUpdate ` Specifies how `fmincon` updates the barrier parameter (see fmincon Interior Point Algorithm). The options are:`'monotone'` (default)`'predictor-corrector'`This option can affect the speed and convergence of the solver, but the effect is not easy to predict. `ConstraintTolerance` Tolerance on the constraint violation, a nonnegative scalar. The default is `1e-6`. See Tolerances and Stopping Criteria.For `optimset`, the name is `TolCon`. See Current and Legacy Option Names. InitBarrierParam Initial barrier value, a positive scalar. Sometimes it might help to try a value above the default `0.1`, especially if the objective or constraint functions are large. `SpecifyConstraintGradient` Gradient for nonlinear constraint functions defined by the user. When set to the default, `false`, `lsqcurvefit` estimates gradients of the nonlinear constraints by finite differences. When set to `true`, `lsqcurvefit` expects the constraint function to have four outputs, as described in `nonlcon`.For `optimset`, the name is `GradConstr` and the values are `'on'` or `'off'`. See Current and Legacy Option Names. `SubproblemAlgorithm` Determines how the iteration step is calculated. The default, `'factorization'`, is usually faster than `'cg'` (conjugate gradient), though `'cg'` might be faster for large problems with dense Hessians. See fmincon Interior Point Algorithm.For `optimset`, the values are `'cg'` and `'ldl-factorization'`. See Current and Legacy Option Names.

Example: ```options = optimoptions('lsqcurvefit','FiniteDifferenceType','central')```

Problem structure, specified as a structure with the following fields:

Field NameEntry

`objective`

Objective function

`x0`

Initial point for `x`

`xdata`

Input data for objective function

`ydata`

Output data to be matched by objective function

`Aineq`

Matrix for linear inequality constraints

`bineq`

Vector for linear inequality constraints

`Aeq`

Matrix for linear equality constraints

`beq`

Vector for linear equality constraints
`lb`Vector of lower bounds
`ub`Vector of upper bounds

`nonlcon`

Nonlinear constraint function

`solver`

`'lsqcurvefit'`

`options`

Options created with `optimoptions`

You must supply at least the `objective`, `x0`, `solver`, `xdata`, `ydata`, and `options` fields in the `problem` structure.

Data Types: `struct`

Output Arguments

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Solution, returned as a real vector or real array. The size of `x` is the same as the size of `x0`. Typically, `x` is a local solution to the problem when `exitflag` is positive. For information on the quality of the solution, see When the Solver Succeeds.

Squared norm of the residual, returned as a nonnegative real. `resnorm` is the squared 2-norm of the residual at `x`: `sum((fun(x,xdata)-ydata).^2)`.

Value of objective function at solution, returned as an array. In general, ```residual = fun(x,xdata)-ydata```.

Reason the solver stopped, returned as an integer.

 `1` Function converged to a solution `x`. `2` Change in `x` is less than the specified tolerance, or Jacobian at `x` is undefined. `3` Change in the residual is less than the specified tolerance. `4` Relative magnitude of search direction is smaller than the step tolerance. `0` Number of iterations exceeds `options.MaxIterations` or number of function evaluations exceeded `options.MaxFunctionEvaluations`. `-1` A plot function or output function stopped the solver. `-2` No feasible point found. The bounds `lb` and `ub` are inconsistent, or the solver stopped at an infeasible point.

Information about the optimization process, returned as a structure with fields:

 `firstorderopt` Measure of first-order optimality `iterations` Number of iterations taken `funcCount` The number of function evaluations `cgiterations` Total number of PCG iterations (`'trust-region-reflective'` and `'interior-point'` algorithms) `stepsize` Final displacement in `x` `constrviolation` Maximum of constraint functions (`'interior-point'` algorithm) `bestfeasible` Best (lowest objective function) feasible point encountered (`'interior-point'` algorithm). A structure with these fields:`x``fval``firstorderopt``constrviolation`If no feasible point is found, the `bestfeasible` field is empty. For this purpose, a point is feasible when the maximum of the constraint functions does not exceed `options.ConstraintTolerance`.The `bestfeasible` point can differ from the returned solution point `x` for a variety of reasons. For an example, see Obtain Best Feasible Point. `algorithm` Optimization algorithm used `message` Exit message

Lagrange multipliers at the solution, returned as a structure with fields:

 `lower` Lower bounds corresponding to `lb` `upper` Upper bounds corresponding to `ub` `ineqlin` Linear inequalities corresponding to `A` and `b` `eqlin` Linear equalities corresponding to `Aeq` and `beq` `ineqnonlin` Nonlinear inequalities corresponding to the `c` in `nonlcon` `eqnonlin` Nonlinear equalities corresponding to the `ceq` in `nonlcon`

Jacobian at the solution, returned as a real matrix. `jacobian(i,j)` is the partial derivative of `fun(i)` with respect to `x(j)` at the solution `x`.

For problems with active constraints at the solution, `jacobian` is not useful for estimating confidence intervals.

Limitations

• The trust-region-reflective algorithm does not solve underdetermined systems; it requires that the number of equations, i.e., the row dimension of F, be at least as great as the number of variables. In the underdetermined case, `lsqcurvefit` uses the Levenberg-Marquardt algorithm.

• `lsqcurvefit` can solve complex-valued problems directly. Note that constraints do not make sense for complex values, because complex numbers are not well-ordered; asking whether one complex value is greater or less than another complex value is nonsensical. For a complex problem with bound constraints, split the variables into real and imaginary parts. Do not use the `'interior-point'` algorithm with complex data. See Fit a Model to Complex-Valued Data.

• The preconditioner computation used in the preconditioned conjugate gradient part of the trust-region-reflective method forms JTJ (where J is the Jacobian matrix) before computing the preconditioner. Therefore, a row of J with many nonzeros, which results in a nearly dense product JTJ, can lead to a costly solution process for large problems.

• If components of x have no upper (or lower) bounds, `lsqcurvefit` prefers that the corresponding components of `ub` (or `lb`) be set to `inf` (or `-inf` for lower bounds) as opposed to an arbitrary but very large positive (or negative for lower bounds) number.

You can use the trust-region reflective algorithm in `lsqnonlin`, `lsqcurvefit`, and `fsolve` with small- to medium-scale problems without computing the Jacobian in `fun` or providing the Jacobian sparsity pattern. (This also applies to using `fmincon` or `fminunc` without computing the Hessian or supplying the Hessian sparsity pattern.) How small is small- to medium-scale? No absolute answer is available, as it depends on the amount of virtual memory in your computer system configuration.

Suppose your problem has `m` equations and `n` unknowns. If the command `J = sparse(ones(m,n))` causes an `Out of memory` error on your machine, then this is certainly too large a problem. If it does not result in an error, the problem might still be too large. You can find out only by running it and seeing if MATLAB runs within the amount of virtual memory available on your system.

Algorithms

The Levenberg-Marquardt and trust-region-reflective methods are based on the nonlinear least-squares algorithms also used in `fsolve`.

• The default trust-region-reflective algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [1] and [2]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See Trust-Region-Reflective Least Squares.

• The Levenberg-Marquardt method is described in references [4], [5], and [6]. See Levenberg-Marquardt Method.

The `'interior-point'` algorithm uses the `fmincon` `'interior-point'` algorithm with some modifications. For details, see Modified fmincon Algorithm for Constrained Least Squares.

Alternative Functionality

App

The Optimize Live Editor task provides a visual interface for `lsqcurvefit`.

References

[1] Coleman, T.F. and Y. Li. “An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds.” SIAM Journal on Optimization, Vol. 6, 1996, pp. 418–445.

[2] Coleman, T.F. and Y. Li. “On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds.” Mathematical Programming, Vol. 67, Number 2, 1994, pp. 189–224.

[3] Dennis, J. E. Jr. “Nonlinear Least-Squares.” State of the Art in Numerical Analysis, ed. D. Jacobs, Academic Press, pp. 269–312.

[4] Levenberg, K. “A Method for the Solution of Certain Problems in Least-Squares.” Quarterly Applied Mathematics 2, 1944, pp. 164–168.

[5] Marquardt, D. “An Algorithm for Least-squares Estimation of Nonlinear Parameters.” SIAM Journal Applied Mathematics, Vol. 11, 1963, pp. 431–441.

[6] Moré, J. J. “The Levenberg-Marquardt Algorithm: Implementation and Theory.” Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics 630, Springer Verlag, 1977, pp. 105–116.

[7] Moré, J. J., B. S. Garbow, and K. E. Hillstrom. User Guide for MINPACK 1. Argonne National Laboratory, Rept. ANL–80–74, 1980.

[8] Powell, M. J. D. “A Fortran Subroutine for Solving Systems of Nonlinear Algebraic Equations.” Numerical Methods for Nonlinear Algebraic Equations, P. Rabinowitz, ed., Ch.7, 1970.

Version History

Introduced before R2006a

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