matlabsuanfa.rar_dynamic programming_动态规划_动态规划 MATLAB_整数规划

function [tout,yout,varargout] = ode45(odefile,tspan,y0,options,varargin) %[tout,yout] = ode45('ypfun', tspan, y0, options) %这里字符串ypfun是用以表示f(t, y)的M文件名, %tspan=[t0, tfinal]表示自变量初值t0和终值tf % y0表示初值向量y0，可选参数options为用odeset设置精度等参数。 % 输出列向量tout表示节点 (t0 , t1 , … , tn)' % 输出矩阵yout 表示数值解，每一列对应y的一个分量 % 若无输出参数，则作出图形。 %例 解微分方程 % y' = y-2t/y, y(0)=1, 0<t<1 % 先写M函数quadeg5fun.m % function f=quadeg5fun(t,y) % f=y-2*t./y; % f=f(:); %保证f为一个列向量 % 再用 % [t,y]=ode45('quadeg5fun',[0,1],1) % plot(t,y); % %ODE45 Solve non-stiff differential equations, medium order method. % [T,Y] = ODE45('F',TSPAN,Y0) with TSPAN = [T0 TFINAL] integrates the % system of differential equations y' = F(t,y) from time T0 to TFINAL with % initial conditions Y0. 'F' is a string containing the name of an ODE % file. Function F(T,Y) must return a column vector. Each row in % solution array Y corresponds to a time returned in column vector T. To % obtain solutions at specific times T0, T1, ..., TFINAL (all increasing % or all decreasing), use TSPAN = [T0 T1 ... TFINAL]. % % [T,Y] = ODE45('F',TSPAN,Y0,OPTIONS) solves as above with default % integration parameters replaced by values in OPTIONS, an argument % created with the ODESET function. See ODESET for details. Commonly % used options are scalar relative error tolerance 'RelTol' (1e-3 by % default) and vector of absolute error tolerances 'AbsTol' (all % components 1e-6 by default). % % [T,Y] = ODE45('F',TSPAN,Y0,OPTIONS,P1,P2,...) passes the additional % parameters P1,P2,... to the ODE file as F(T,Y,FLAG,P1,P2,...) (see % ODEFILE). Use OPTIONS = [] as a place holder if no options are set. % % It is possible to specify TSPAN, Y0 and OPTIONS in the ODE file (see % ODEFILE). If TSPAN or Y0 is empty, then ODE45 calls the ODE file % [TSPAN,Y0,OPTIONS] = F([],[],'init') to obtain any values not supplied % in the ODE45 argument list. Empty arguments at the end of the call list % may be omitted, e.g. ODE45('F'). % % As an example, the commands % % options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4 1e-5]); % ode45('rigidode',[0 12],[0 1 1],options); % % solve the system y' = rigidode(t,y) with relative error tolerance 1e-4 % and absolute tolerances of 1e-4 for the first two components and 1e-5 % for the third. When called with no output arguments, as in this % example, ODE45 calls the default output function ODEPLOT to plot the % solution as it is computed. % % [T,Y,TE,YE,IE] = ODE45('F',TSPAN,Y0,OPTIONS) with the Events property in % OPTIONS set to 'on', solves as above while also locating zero crossings % of an event function defined in the ODE file. The ODE file must be % coded so that F(T,Y,'events') returns appropriate information. See % ODEFILE for details. Output TE is a column vector of times at which % events occur, rows of YE are the corresponding solutions, and indices in % vector IE specify which event occurred. % % See also ODEFILE and % other ODE solvers: ODE23, ODE113, ODE15S, ODE23S, ODE23T, ODE23TB % options handling: ODESET, ODEGET % output functions: ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINT % odefile examples: ORBITODE, ORBT2ODE, RIGIDODE, VDPODE % ODE45 is an implementation of the explicit Runge-Kutta (4,5) pair of % Dormand and Prince called variously RK5(4)7FM, DOPRI5, DP(4,5) and DP54. % It uses a "free" interpolant of order 4 communicated privately by % Dormand and Prince. Local extrapolation is done. % Details are to be found in The MATLAB ODE Suite, L. F. Shampine and % M. W. Reichelt, SIAM Journal on Scientific Computing, 18-1, 1997. % Mark W. Reichelt and Lawrence F. Shampine, 6-14-94 % Copyright (c) 1984-98 by The MathWorks, Inc. % $Revision: 5.53 $ $Date: 1997/11/21 23:31:04 $ true = 1; false = ~true; nsteps = 0; % stats nfailed = 0; % stats nfevals = 0; % stats npds = 0; % stats ndecomps = 0; % stats nsolves = 0; % stats if nargin == 0 error('Not enough input arguments. See ODE45.'); elseif ~isstr(odefile) & ~isa(odefile, 'inline') error('First argument must be a single-quoted string. See ODE45.'); end if nargin == 1 tspan = []; y0 = []; options = []; elseif nargin == 2 y0 = []; options = []; elseif nargin == 3 options = []; elseif ~isempty(options) & ~isa(options,'struct') if (length(tspan) == 1) & (length(y0) == 1) & (min(size(options)) == 1) tspan = [tspan; y0]; y0 = options; options = []; varargin = {}; msg = sprintf('Use ode45(''%s'',tspan,y0,...) instead.',odefile); warning(['Obsolete syntax. ' msg]); else error('Correct syntax is ode45(''odefile'',tspan,y0,options).'); end end % Get default tspan and y0 from odefile if none are specified. if isempty(tspan) | isempty(y0) if (nargout(odefile) < 3) & (nargout(odefile) ~= -1) msg = sprintf('Use ode45(''%s'',tspan,y0,...) instead.',odefile); error(['No default parameters in ' upper(odefile) '. ' msg]); end [def_tspan,def_y0,def_options] = feval(odefile,[],[],'init',varargin{:}); if isempty(tspan) tspan = def_tspan; end if isempty(y0) y0 = def_y0; end if isempty(options) options = def_options; else options = odeset(def_options,options); end end % Test that tspan is internally consistent. tspan = tspan(:); ntspan = length(tspan); if ntspan == 1 t0 = 0; next = 1; else t0 = tspan(1); next = 2; end tfinal = tspan(ntspan); if t0 == tfinal error('The last entry in tspan must be different from the first entry.'); end tdir = sign(tfinal - t0); if any(tdir * (tspan(2:ntspan) - tspan(1:ntspan-1)) <= 0) error('The entries in tspan must strictly increase or decrease.'); end t = t0; y = y0(:); neq = length(y); % Get options, and set defaults. rtol = odeget(options,'RelTol',1e-3); if (length(rtol) ~= 1) | (rtol <= 0) error('RelTol must be a positive scalar.'); end if rtol < 100 * eps rtol = 100 * eps; warning(['RelTol has been increased to ' num2str(rtol) '.']); end atol = odeget(options,'AbsTol',1e-6); if any(atol <= 0) error('AbsTol must be positive.'); end normcontrol = strcmp(odeget(options,'NormControl','off'),'on'); if normcontrol if length(atol) ~= 1 error('Solving with NormControl ''on'' requires a scalar AbsTol.'); end normy = norm(y); else if (length(atol) ~= 1) & (length(atol) ~= neq) error(sprintf(['Solving %s requires a scalar AbsTol, ' ... 'or a vector AbsTol of length %d'],upper(odefile),neq)); end atol = atol(:); end threshold = atol / rtol; % By default, hmax is 1/10 of the interval. hmax = min(abs(tfinal-t), abs(odeget(options,'MaxStep',0.1*(tfinal-t)))); if hmax <= 0 error('Option ''MaxStep'' must be greater than zero.'); end htry = abs(odeget(options,'InitialStep')); if htry <= 0 error('Option ''InitialStep'' must be greater than zero.'); end haveeventfun = strcmp(odeget(options,'Events','off'),'on'); if haveeventfun valt = feval(odefile,t,y,'events',varargin{:}); teout = []; yeout = []; ieout = []; end if nargout > 0 outfun = odeget(options,'OutputFcn'); else outfun = odeget(options,'OutputFcn','odeplot'); end if isempty(outfun) haveoutfun = false; else haveoutfun = true; outputs = odeget(options,'OutputSel',1:neq); end refine = odeget(options,'Refine',4); % Necessary for smooth plots. printstats = strcmp(odeget(options,'Stats','off'),'on'); if strcmp(odege