Derived Types

Defines a Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimizer for minimization of smooth functions of multiple variables. The method uses a quasi-Newton inverse Hessian approximation and the update $$B_{k+1} = B_k + \frac{y_k y_k^T}{y_k^T s_k} - \frac{B_k s_k s_k^T B_k}{s_k^T B_k s_k}$$ with $s_k = x_{k+1}-x_k$ and $y_k = \nabla f(x_{k+1})-\nabla f(x_k)$.

Defines a derivative-free solver for a scalar equation $f(x)=0$ based on Brent's method. The method combines bisection with secant and inverse quadratic interpolation to maintain a bracket and converge to a root.

A least-squares solver that enforces bounds on the solution values, typically expressed as $x_i^L \le x_i \le x_i^U$. The solver applies the bounds during the iteration so that each trial step remains feasible.

Defines a constrained least-squares solver using Powell's trust region method. The method seeks a correction $p$ that reduces the residual vector $r(x)$ while remaining inside a trust region of size $\Delta$ and the bounds $x^L \le x \le x^U$. The trial step is formed from a Gauss-Newton step $$J p_{gn} = -r$$ a steepest-descent step $$p_{sd} = -\alpha g, \qquad g = J^T r, \qquad \alpha = \frac{|g\|^2}{|Jg\|^2}$$ and a dogleg combination $$p = p_{sd} + t \left( p_{gn} - p_{sd} \right)$$ with $t$ chosen so that $|p\| = \Delta$. If the trust-region step is rejected, a backtracking line search is used to improve the step acceptance.

A base class for optimization of an equation of multiple variables.

A base class for various solvers of nonlinear systems of equations.

A base class for various solvers of equations of one variable.

Defines a type capable of encapsulating an equation of one variable of the form: f(x) = 0.

Defines a type capable of encapsulating an equation of N variables.

Defines a set of parameters that describe the behavior of the iteration process.

Defines a Levenberg-Marquardt solver for unconstrained least-squares problems. The update is obtained from the damped normal equations $$(J^T J + \lambda I) \Delta x = -J^T r$$ where $r(x)$ is the residual vector and $x_{k+1}=x_k+\Delta x$.

Defines a type capable of performing an inexact, backtracking line search to find a point as far along the specified direction vector that is usable for unconstrained minimization problems.

A base class for optimizers that improve a trial step with a line search. The search direction $p_k$ is combined with a step length $\alpha_k$ through $$x_{k+1} = x_k + \alpha_k p_k$$ and the accepted step satisfies a sufficient decrease condition.

A base class for nonlinear solvers that improve convergence by combining a search direction with a line search. At each iteration the step is taken as $$x_{k+1} = x_k + \alpha_k p_k$$ where the step length $\alpha_k$ is chosen to satisfy a sufficient decrease condition such as the Armijo rule $$f(x_k + \alpha_k p_k) \le f(x_k) + c_1 \alpha_k \nabla f(x_k)^T p_k$$ .

Defines a direct-search optimizer based on Nelder and Mead's simplex method. The algorithm maintains a simplex of $n+1$ vertices and updates it by reflection, expansion, contraction, and shrinking steps to reduce the objective value $f(x)$.

Defines a safeguarded Newton solver for a scalar equation $f(x)=0$. The iteration uses the Newton update $$x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$$ and retains a bracketing interval so that the step remains bounded by the root bracket.

Defines a Newton solver for systems of nonlinear equations. At each iteration the correction is obtained from the linearized system $$J(x_k) \Delta x_k = -F(x_k), \qquad x_{k+1} = x_k + \Delta x_k$$ where $J(x_k)$ is the Jacobian matrix of $F$.

Defines a polynomial, and associated routines for performing polynomial operations.

Defines a quasi-Newton solver based upon Broyden's method. The algorithm maintains an approximate Jacobian $B_k$ and updates the iterate from the nonlinear system $F(x)=0$ using $$B_k s_k = -F(x_k), \qquad x_{k+1} = x_k + s_k$$ with the rank-one Jacobian correction $$B_{k+1} = B_k + \frac{(y_k - B_k s_k)s_k^T}{s_k^T s_k}$$ where $y_k = F(x_{k+1}) - F(x_k)$.

Defines a pair of numeric values.

Defines a type capable of encapsulating a system of nonlinear equations of the form: F(X) = 0. This type is used to establish the system of equations to solve, and provides a means for computing the Jacobian matrix for the system of equations, and any other ancillary operations that may be needed by the solver.