% -*- texinfo -*-
% @deftypefn {Function File} {@var{x} =} nlnewmark(@var{m}, @var{c}, @var{q}, @var{f}, @var{dt}, @var{x0} = 0, @var{x'0} = 0, @var{alpha} = 1/2, @var{beta} = 1/4, @var{flags} = "")
% Computes the solution of non-linear second-order differential equations of the form
%
% @example
% @var{m} @var{x}'' + @var{c} @var{x}' + @var{q}(@var{x}, @var{x}', @var{x}'') = @var{f}
% @end example
%
% where @var{x}' denotes the first time derivative of @var{x} and @var{q} is
% a non-linear function.
%
% If the function is called without the assigning a return value
% then @var{x} is plotted versus time.
%
% @strong{Inputs}
%
% @table @var
% @item m
% The mass of the body.
% @item c
% Viscous damping of the system.
% @item q
% The name of a function
% that returns the value of the resisting force for a given
% displacement. The form of Q must be:
%
% @example
% @var{F} = Q( @var{x})
% @end example
%
% where @var{F} is the restoring force for the state vector @var{x}
% = [@var{u}, @var{u'}, @var{u''}]; displacement
% (@var{u}), velocity (@var{u'}), and acceleration.
% @item f
% The forcing function as a time sampled or impulse vector
% (see @strong{Special Cases}).
% @item dt
% The time step -- assumed to be constant
% @item x0
% Initial displacement, default is zero
% @item x'0
% Initial velocity, default is zero
% @item alpha
% Alpha Coefficient -- Controls 'artificial damping' of the system.
% Unless you have a really good reason, this should be 1/2 which is
% the default.
% @item beta
% Beta Coefficient -- This coefficient is used to estimate the form of the
% system acceleration between time steps. Values between 1/4 and 1/6 are
% common. The default is 1/4 which is unconditionally stable.
% @item flags
% A string value which defines special cases. The cases are defined by
% unique characters as explained in @strong{Special Cases} below.
% @end table
%
% @strong{Outputs}
%
% @table @var
% @item x
% Matrix of size (3, @code{length(@var{f})}) with time series of displacement
% (@var{x}(1,:)), velocity (@var{x}(2,:)), and acceleration (@var{x}(3,:))
% @end table
%
% @strong{Special Cases}
%
% The @var{flags} variable is used to define special cases of analysis as
% follows.
%
% 'i' - Impulse forcing function. The forcing function, @var{f} is a
% vector of impulses instead of a sampled time history.
% 'n' - The stiffness is non-linear. In this case, @var{k} is a string
% which contains the name of a function defining the non-linear
% stiffness.
%
% @end deftypefn