lsqnonlin

📝Syntax

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

📥Input Arguments

Parameter	Description
fun	function returning a residual array.
x0	initial point.
lb, ub	bounds, optionally empty.
A, b, Aeq, beq	linear inequality and equality constraints, optionally empty.
nonlcon	nonlinear constraint function returning [c, ceq], optionally empty.
options	solver options.

📤Output Arguments

Parameter	Description
x	estimated solution, with the shape of x0.
resnorm	squared residual norm sum(fun(x).^2).
residual	residual at x, with the shape returned by fun.
exitflag	reason the solver stopped: 1 (gradient below tolerance), 2 (step below StepTolerance), 3 (residual change below FunctionTolerance), 4 (search direction below StepTolerance), 0 (iteration or evaluation limit), -1 (stopped by output function), -2 (inconsistent bounds).
output	structure with firstorderopt, iterations, funcCount, cgiterations, algorithm, stepsize, message, bestfeasible and constrviolation fields.
lambda	Lagrange multipliers structure with lower, upper, eqlin, ineqlin, eqnonlin and ineqnonlin fields.
jacobian	final finite-difference or user-provided Jacobian.

📄Description

lsqnonlin solves nonlinear least-squares problems min sum(fun(x).^2), optionally subject to bounds and constraints.

The Algorithm option selects the engine: 'trust-region-reflective' (default), 'levenberg-marquardt' (also accepts bounds) or 'interior-point'. Linear or nonlinear constraints automatically use the interior-point algorithm.

The default MaxFunctionEvaluations is 100*numberOfVariables, MaxIterations is 400 and FunctionTolerance and StepTolerance are 1e-6. The Display option supports 'off', 'none', 'final', 'final-detailed', 'notify', 'notify-detailed', 'iter' and 'iter-detailed'.

If Jacobian is 'on' or SpecifyObjectiveGradient is true, fun must also return the Jacobian of the residuals.

💡Examples

fun = @(x) [x(1) - 2; x(2) + 1];
[x, resnorm] = lsqnonlin(fun, [0; 0])

🔗See Also

fsolve lsqnonneg

Used Functions

optimoptions

📚Bibliography

K. Levenberg, "A method for the solution of certain non-linear problems in least squares", Quarterly of Applied Mathematics, 1944. D. W. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters", SIAM Journal on Applied Mathematics, 1963.