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    Metadata details:
    filename: solveivp.m
    path of file: ./repos/chebfun/@chebop
    
    the code of the file until where you have to start completion: function varargout = solveivp(N, rhs, pref, varargin)
%SOLVEIVP    Solve an IVP by reforming it to a first order system.
%
%   U = SOLVEIVP(N, RHS), where N is a CHEBOP and RHS is a CHEBMATRIX, CHEBFUN
%   or a vector of doubles attempts to solve the IVP
%
%       N(U) = RHS + boundary conditions specified by N
%
%   Observe that U = SOLVEIVP(N, RHS), where N specifies an initial/final-value
%   problem (IVP/FVP), has the same effect as U = N\RHS, but this method allows
%   greater flexibility than CHEBOP backslash, as described below. Problems are
%   determined to be an IVP/FVP as follows:
%       * N.LBC is non-empty, N.RBC and N.BC are empty => IVP.
%       * N.RBC is non-empty, N.LBC and N.BC are empty => FVP.
%   Otherwise, problems are considered to be boundary-value problems, and
%   U=N\RHS will in general have the same effect as U = SOLVEBVP(N, RHS).
%
%   If successful, the solution returned, U, is a CHEBFUN if N specifies a
%   scalar problem, and a CHEBMATRIX if N specifies a coupled systems of
%   ordinary differential equations. See note below on how to call the method
%   with multiple outputs. This method solves both linear and nonlinear problems
%   be automatically converting them to a coupled first-order system, which can
%   then be solved using MATLAB's built in solvers.
%
%   U = SOLVEIVP(N, RHS, PREF) is the same as above, using the preferences
%   specified by the CHEBOPPREF variable PREF.
%
%   [U, INFO] = SOLVEIVP(N, RHS, PREF) is the same as above, but also returns
%   the MATLAB struct INFO, which contains useful information about the solution
%   process. The fields of INFO are as follows (more may be added in future
%   versions):
%       SOLVER: The MATLAB solver used when solving the problem.
%
%   [U, V, ...] = SOLVEBVP(N, ...), where N specifies a coupled system of ODEs,
%   returns CHEBFUNs U, V, ... for individual solution components, rather than a
%   CHEBMATRIX.
%
%
%   Note 1: CHEBOP allows the RHS of coupled system of ODEs to be a scalar,
%   e.g., one can both call
%       N = chebop(@(x, u, v) [diff(u) - v.^2 ; u - diff(v)], [0 3]);
%       N.lbc = @(u, v) [u - 1 ; v + 1];
%       uv = solveivp(N, 0);
%   and
%       uv = solveivp(N, [0; 0]);
%
%
%   Note 2: The solver tries to construct global CHEBFUNs to represent the
%   solutions of ODEs if possible (however, breakpoints in the domain and
%   coefficients do get respected). Turn on the global CHEBFUN splitting option
%   if you wish to obtain solutions with further breakpoints, e.g.
%       % Solve van der Pol equation without and with splitting
%       N = chebop(@(t,u) diff(u,2)-5.*(1-u.^2).*diff(u)+u, [0 20]);
%       N.lbc = @(u) [u-2; diff(u)];
%       uNoSplit = N\0
%       % Turn on splitting with max length of each piece equal to 300
%       chebfunpref.setDefaults('splitting', true)
%       chebfunpref.setDefaults({'splitPrefs','splitLength'}, 300)
%       uSplit = N\0
%       % Turn splitting back off
%       chebfunpref.setDefaults('splitting', false)
%       
%
% See also: CHEBOP, CHEBOP/MLDIVIDE, CHEBOPPREF, CHEBOP/SOLVEBVP,
% CHEBFUN/ODE113, CHEBFUN/ODE15S, CHEBFUN/ODE45, CHEBFUN/ODESOL, TREEVAR. 

% Copyright 2017 by The University of Oxford and The Chebfun Developers.
% See http://www.chebfun.org/ for Chebfun information.

% Developer note:
%   U = SOLVEIVP(N, RHS, PREF, DISPLAYINFO) allows passing in a function handle
%   to a displaying method that is called after solution has been computed.
%   This allows separating the displaying process for regular CHEBOP use and

    
function varargout = solveivp(N, rhs, pref, varargin)
%SOLVEIVP    Solve an IVP by reforming it to a first order system.
%
%   U = SOLVEIVP(N, RHS), where N is a CHEBOP and RHS is a CHEBMATRIX, CHEBFUN
%   or a vector of doubles attempts to solve the IVP
%
%       N(U) = RHS + boundary conditions specified by N
%
%   Observe that U = SOLVEIVP(N, RHS), where N specifies an initial/final-value
%   problem (IVP/FVP), has the same effect as U = N\RHS, but this method allows
%   greater flexibility than CHEBOP backslash, as described below. Problems are
%   determined to be an IVP/FVP as follows:
%       * N.LBC is non-empty, N.RBC and N.BC are empty => IVP.
%       * N.RBC is non-empty, N.LBC and N.BC are empty => FVP.
%   Otherwise, problems are considered to be boundary-value problems, and
%   U=N\RHS will in general have the same effect as U = SOLVEBVP(N, RHS).
%
%   If successful, the solution returned, U, is a CHEBFUN if N specifies a
%   scalar problem, and a CHEBMATRIX if N specifies a coupled systems of
%   ordinary differential equations. See note below on how to call the method
%   with multiple outputs. This method solves both linear and nonlinear problems
%   be automatically converting them to a coupled first-order system, which can
%   then be solved using MATLAB's built in solvers.
%
%   U = SOLVEIVP(N, RHS, PREF) is the same as above, using the preferences
%   specified by the CHEBOPPREF variable PREF.
%
%   [U, INFO] = SOLVEIVP(N, RHS, PREF) is the same as above, but also returns
%   the MATLAB struct INFO, which contains useful information about the solution
%   process. The fields of INFO are as follows (more may be added in future
%   versions):
%       SOLVER: The MATLAB solver used when solving the problem.
%
%   [U, V, ...] = SOLVEBVP(N, ...), where N specifies a coupled system of ODEs,
%   returns CHEBFUNs U, V, ... for individual solution components, rather than a
%   CHEBMATRIX.
%
%
%   Note 1: CHEBOP allows the RHS of coupled system of ODEs to be a scalar,
%   e.g., one can both call
%       N = chebop(@(x, u, v) [diff(u) - v.^2 ; u - diff(v)], [0 3]);
%       N.lbc = @(u, v) [u - 1 ; v + 1];
%       uv = solveivp(N, 0);
%   and
%       uv = solveivp(N, [0; 0]);
%
%
%   Note 2: The solver tries to construct global CHEBFUNs to represent the
%   solutions of ODEs if possible (however, breakpoints in the domain and
%   coefficients do get respected). Turn on the global CHEBFUN splitting option
%   if you wish to obtain solutions with further breakpoints, e.g.
%       % Solve van der Pol equation without and with splitting
%       N = chebop(@(t,u) diff(u,2)-5.*(1-u.^2).*diff(u)+u, [0 20]);
%       N.lbc = @(u) [u-2; diff(u)];
%       uNoSplit = N\0
%       % Turn on splitting with max length of each piece equal to 300
%       chebfunpref.setDefaults('splitting', true)
%       chebfunpref.setDefaults({'splitPrefs','splitLength'}, 300)
%       uSplit = N\0
%       % Turn splitting back off
%       chebfunpref.setDefaults('splitting', false)
%       
%
% See also: CHEBOP, CHEBOP/MLDIVIDE, CHEBOPPREF, CHEBOP/SOLVEBVP,
% CHEBFUN/ODE113, CHEBFUN/ODE15S, CHEBFUN/ODE45, CHEBFUN/ODESOL, TREEVAR. 

% Copyright 2017 by The University of Oxford and The Chebfun Developers.
% See http://www.chebfun.org/ for Chebfun information.

% Developer note:
%   U = SOLVEIVP(N, RHS, PREF, DISPLAYINFO) allows passing in a function handle
%   to a displaying method that is called after solution has been computed.
%   This allows separating the displaying process for regular CHEBOP use and
%   CHEBGUI. See chebop/displayIVPinfo() and chebgui/displayIVPinfo() for more
%   details.

%% SETUP
% Check inputs
if ( nargin < 2 )
    % Default right-hand side.
    rhs = 0;
elseif (isa(rhs, 'chebfun') || isa(rhs, 'chebmatrix') )
    % Ensure that the RHS lives on the same domain as the CHEBOP N.
    assert( (rhs.domain(1) == N.domain(1)) && ...
            (rhs.domain(end