constrained_least_squares_solver Derived Type

type, public, extends(constrained_equation_solver) :: constrained_least_squares_solver

Defines a constrained least-squares solver using Powell's trust region method. In the event the trust-region approach is slow to converge a backtracking type line search will be utilized. The solver also utilizes a Coleman-Li scaling approach that works to improve stability when the solution is near a constraint.

The trust region approach assumes a radius of , which can be initially defined by the user but is automatically altered by the solver, and is implemented as follows.

Gauss-Newton Step (Solved via QR decomposition):

The gradient vector for the steepest descent is calculated by utilizing the product of the Jacobian and the residual as follows.

The steepest descent step is then as follows. where .

Finally, the dogleg is computed as follows. where is found such that .


Contents

Type-Bound Procedures


Type-Bound Procedures

procedure, public :: apply_limits => ces_apply_limits

  • private subroutine ces_apply_limits(this, x)

    Applies the limits to the solution vector.

    Arguments

    Type IntentOptional Attributes Name
    class(constrained_equation_solver), intent(in) :: this

    The constrained_equation_solver object.
    real(kind=real64), intent(inout), dimension(:) :: x

    On input, the solution vector. On output, the clamped solution vector.

procedure, public :: get_fcn_tolerance => es_get_fcn_tol

  • private pure function es_get_fcn_tol(this) result(x)

    Gets the convergence on function value tolerance.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(in) :: this

    The equation_solver object.

    Return Value real(kind=real64)

    The tolerance value.

procedure, public :: get_gradient_tolerance => es_get_grad_tol

  • private pure function es_get_grad_tol(this) result(x)

    Gets the convergence on slope of the gradient vector tolerance.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(in) :: this

    The equation_solver object.

    Return Value real(kind=real64)

    The tolerance value.

procedure, public :: get_lower_limits => ces_get_lower_bounds

  • private pure function ces_get_lower_bounds(this) result(rst)

    Gets the array of lower bounds constraints.

    Arguments

    Type IntentOptional Attributes Name
    class(constrained_equation_solver), intent(in) :: this

    The constrained_equation_solver object.

    Return Value real(kind=real64), allocatable, dimension(:)

    The limit array.

procedure, public :: get_max_fcn_evals => es_get_max_eval

  • private pure function es_get_max_eval(this) result(n)

    Gets the maximum number of function evaluations allowed during a single solve.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(in) :: this

    The equation_solver object.

    Return Value integer(kind=int32)

    The maximum number of function evaluations.

procedure, public :: get_print_status => es_get_print_status

  • private pure function es_get_print_status(this) result(x)

    Gets a logical value determining if iteration status should be printed.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(in) :: this

    The equation_solver object.

    Return Value logical

    True if the iteration status should be printed; else, false.

procedure, public :: get_step_scaling_factor => cls_get_factor

procedure, public :: get_trust_region_radius => cls_get_radius

procedure, public :: get_upper_limits => ces_get_upper_bounds

  • private pure function ces_get_upper_bounds(this) result(rst)

    Gets the array of upper bounds constraints.

    Arguments

    Type IntentOptional Attributes Name
    class(constrained_equation_solver), intent(in) :: this

    The constrained_equation_solver object.

    Return Value real(kind=real64), allocatable, dimension(:)

    The limit array.

procedure, public :: get_var_tolerance => es_get_var_tol

  • private pure function es_get_var_tol(this) result(x)

    Gets the convergence on change in variable tolerance.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(in) :: this

    The equation_solver object.

    Return Value real(kind=real64)

    The tolerance value.

procedure, public :: set_fcn_tolerance => es_set_fcn_tol

  • private subroutine es_set_fcn_tol(this, x)

    Sets the convergence on function value tolerance.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(inout) :: this

    The equation_solver object.
    real(kind=real64), intent(in) :: x

    The tolerance value.

procedure, public :: set_gradient_tolerance => es_set_grad_tol

  • private subroutine es_set_grad_tol(this, x)

    Sets the convergence on slope of the gradient vector tolerance.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(inout) :: this

    The equation_solver object.
    real(kind=real64), intent(in) :: x

    The tolerance value.

procedure, public :: set_lower_limits => ces_set_lower_bounds

  • private subroutine ces_set_lower_bounds(this, x)

    Sets the array of lower bounds constraints.

    Arguments

    Type IntentOptional Attributes Name
    class(constrained_equation_solver), intent(inout) :: this

    The constrained_equation_solver object.
    real(kind=real64), intent(in), dimension(:) :: x

    The limit array.

procedure, public :: set_max_fcn_evals => es_set_max_eval

  • private subroutine es_set_max_eval(this, n)

    Sets the maximum number of function evaluations allowed during a single solve.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(inout) :: this

    The equation_solver object.
    integer(kind=int32), intent(in) :: n

    The maximum number of function evaluations.

procedure, public :: set_print_status => es_set_print_status

  • private subroutine es_set_print_status(this, x)

    Sets a logical value determining if iteration status should be printed.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(inout) :: this

    The equation_solver object.
    logical, intent(in) :: x

    True if the iteration status should be printed; else, false.

procedure, public :: set_step_scaling_factor => cls_set_factor

procedure, public :: set_trust_region_radius => cls_set_radius

procedure, public :: set_upper_limits => ces_set_upper_bounds

  • private subroutine ces_set_upper_bounds(this, x)

    Sets the array of upper bounds constraints.

    Arguments

    Type IntentOptional Attributes Name
    class(constrained_equation_solver), intent(inout) :: this

    The constrained_equation_solver object.
    real(kind=real64), intent(in), dimension(:) :: x

    The limit array.

procedure, public :: set_var_tolerance => es_set_var_tol

  • private subroutine es_set_var_tol(this, x)

    Sets the convergence on change in variable tolerance.

    Arguments

    Type IntentOptional Attributes Name
    class(equation_solver), intent(inout) :: this

    The equation_solver object.
    real(kind=real64), intent(in) :: x

    The tolerance value.

procedure, public :: solve => cls_solve

  • private subroutine cls_solve(this, fcn, x, fvec, ib, args, err)

    Applies the constrained least-squares solver to solve the nonlinear least-squares problem.

    Arguments

    Type IntentOptional Attributes Name
    class(constrained_least_squares_solver), intent(inout) :: this

    The constrained_least_squares_solver object.
    class(vecfcn_helper), intent(in) :: fcn

    The vecfcn_helper object containing the equations to solve.
    real(kind=real64), intent(inout), dimension(:) :: x

    On input, an N-element array containing an initial estimate to the solution. On output, the updated solution estimate. N is the number of variables.
    real(kind=real64), intent(out), dimension(:) :: fvec

    An M-element array that, on output, will contain the values of each equation as evaluated at the variable values given in x.
    type(iteration_behavior), optional :: ib

    An optional output, that if provided, allows the caller to obtain iteration performance statistics.
    class(*), intent(inout), optional :: args

    An optional argument to allow the user to communicate with the routine.
    class(errors), intent(inout), optional, target :: err

    An error handling object.