1. 简述
函数语法
x = lsqnonlin(fun,x0)
函数用于：
解决非线性最小二乘(非线性数据拟合)问题
解决非线性最小二乘曲线拟合问题的形式
变量x的约束上下限为ub和lb，
x = lsqnonlin(fun,x0)从x0点开始，找到fun中描述的函数的最小平方和。函数fun应该返回一个向量(或数组)，而不是值的平方和。(该算法隐式地计算了fun(x)元素的平方和。)
2. 代码
主程序：
%% 用lsqnonlin求解最小二乘问题
clear all
x0 = [0.3 0.4]; % 初值点
x,resnorm\] = lsqnonlin(@f1211,x0) % 调用最优化函数求 x 和 平方和残差 子程序： function \[xCurrent,Resnorm,FVAL,EXITFLAG,OUTPUT,LAMBDA,JACOB\] = lsqnonlin(FUN,xCurrent,LB,UB,options,varargin) %LSQNONLIN solves non-linear least squares problems. % LSQNONLIN attempts to solve problems of the form: % min sum {FUN(X).\^2} where X and the values returned by FUN can be % X vectors or matrices. % % LSQNONLIN implements two different algorithms: trust region reflective % and Levenberg-Marquardt. Choose one via the option Algorithm: for % instance, to choose Levenberg-Marquardt, set % OPTIONS = optimoptions('lsqnonlin', 'Algorithm','levenberg-marquardt'), % and then pass OPTIONS to LSQNONLIN. % % X = LSQNONLIN(FUN,X0) starts at the matrix X0 and finds a minimum X to % the sum of squares of the functions in FUN. FUN accepts input X % and returns a vector (or matrix) of function values F evaluated % at X. NOTE: FUN should return FUN(X) and not the sum-of-squares % sum(FUN(X).\^2)). (FUN(X) is summed and squared implicitly in the % algorithm.) % % X = LSQNONLIN(FUN,X0,LB,UB) defines a set of lower and upper bounds on % the design variables, X, so that the solution is in the range LB \<= X % \<= UB. Use empty matrices for LB and UB if no bounds exist. Set LB(i) % = -Inf if X(i) is unbounded below; set UB(i) = Inf if X(i) is % unbounded above. % % X = LSQNONLIN(FUN,X0,LB,UB,OPTIONS) minimizes with the default % optimization parameters replaced by values in OPTIONS, an argument % created with the OPTIMOPTIONS function. See OPTIMOPTIONS for details. % Use the SpecifyObjectiveGradient option to specify that FUN also % returns a second output argument J that is the Jacobian matrix at the % point X. If FUN returns a vector F of m components when X has length n, % then J is an m-by-n matrix where J(i,j) is the partial derivative of % F(i) with respect to x(j). (Note that the Jacobian J is the transpose % of the gradient of F.) % % X = LSQNONLIN(PROBLEM) solves the non-linear least squares problem % defined in PROBLEM. PROBLEM is a structure with the function FUN in % PROBLEM.objective, the start point in PROBLEM.x0, the lower bounds in % PROBLEM.lb, the upper bounds in PROBLEM.ub, the options structure in % PROBLEM.options, and solver name 'lsqnonlin' in PROBLEM.solver. Use % this syntax to solve at the command line a problem exported from % OPTIMTOOL. % % \[X,RESNORM\] = LSQNONLIN(FUN,X0,...) returns % the value of the squared 2-norm of the residual at X: sum(FUN(X).\^2). % % \[X,RESNORM,RESIDUAL\] = LSQNONLIN(FUN,X0,...) returns the value of the % residual at the solution X: RESIDUAL = FUN(X). % % \[X,RESNORM,RESIDUAL,EXITFLAG\] = LSQNONLIN(FUN,X0,...) returns an % EXITFLAG that describes the exit condition. Possible values of EXITFLAG % and the corresponding exit conditions are listed below. See the % documentation for a complete description. % % 1 LSQNONLIN converged to a solution. % 2 Change in X too small. % 3 Change in RESNORM too small. % 4 Computed search direction too small. % 0 Too many function evaluations or iterations. % -1 Stopped by output/plot function. % -2 Bounds are inconsistent. % % \[X,RESNORM,RESIDUAL,EXITFLAG,OUTPUT\] = LSQNONLIN(FUN,X0,...) returns a % structure OUTPUT with the number of iterations taken in % OUTPUT.iterations, the number of function evaluations in % OUTPUT.funcCount, the algorithm used in OUTPUT.algorithm, the number % of CG iterations (if used) in OUTPUT.cgiterations, the first-order % optimality (if used) in OUTPUT.firstorderopt, and the exit message in % OUTPUT.message. % % \[X,RESNORM,RESIDUAL,EXITFLAG,OUTPUT,LAMBDA\] = LSQNONLIN(FUN,X0,...) % returns the set of Lagrangian multipliers, LAMBDA, at the solution: % LAMBDA.lower for LB and LAMBDA.upper for UB. % % \[X,RESNORM,RESIDUAL,EXITFLAG,OUTPUT,LAMBDA,JACOBIAN\] = LSQNONLIN(FUN, % X0,...) returns the Jacobian of FUN at X. % % Examples % FUN can be specified using @: % x = lsqnonlin(@myfun,\[2 3 4\]) % % where myfun is a MATLAB function such as: % % function F = myfun(x) % F = sin(x); % % FUN can also be an anonymous function: % % x = lsqnonlin(@(x) sin(3\*x),\[1 4\]) % % If FUN is parameterized, you can use anonymous functions to capture the % problem-dependent parameters. Suppose you want to solve the non-linear % least squares problem given in the function myfun, which is % parameterized by its second argument c. Here myfun is a MATLAB file % function such as % % function F = myfun(x,c) % F = \[ 2\*x(1) - exp(c\*x(1)) % -x(1) - exp(c\*x(2)) % x(1) - x(2) \]; % % To solve the least squares problem for a specific value of c, first % assign the value to c. Then create a one-argument anonymous function % that captures that value of c and calls myfun with two arguments. % Finally, pass this anonymous function to LSQNONLIN: % % c = -1; % define parameter first % x = lsqnonlin(@(x) myfun(x,c),\[1;1\]) % % See also OPTIMOPTIONS, LSQCURVEFIT, FSOLVE, @, INLINE. % Copyright 1990-2018 The MathWorks, Inc. % ------------Initialization---------------- defaultopt = struct(... 'Algorithm','trust-region-reflective',... 'DerivativeCheck','off',... 'Diagnostics','off',... 'DiffMaxChange',Inf,... 'DiffMinChange',0,... 'Display','final',... 'FinDiffRelStep', \[\], ... 'FinDiffType','forward',... 'FunValCheck','off',... 'InitDamping', 0.01, ... 'Jacobian','off',... 'JacobMult',\[\],... 'JacobPattern','sparse(ones(Jrows,Jcols))',... 'MaxFunEvals',\[\],... 'MaxIter',400,... 'MaxPCGIter','max(1,floor(numberOfVariables/2))',... 'OutputFcn',\[\],... 'PlotFcns',\[\],... 'PrecondBandWidth',Inf,... 'ScaleProblem','none',... 'TolFun', 1e-6,... 'TolFunValue', 1e-6, ... 'TolPCG',0.1,... 'TolX',1e-6,... 'TypicalX','ones(numberOfVariables,1)',... 'UseParallel',false ); % If just 'defaults' passed in, return the default options in X if nargin==1 \&\& nargout \<= 1 \&\& strcmpi(FUN,'defaults') xCurrent = defaultopt; return end if nargin \< 5 options = \[\]; if nargin \< 4 UB = \[\]; if nargin \< 3 LB = \[\]; end end end problemInput = false; if nargin == 1 if isa(FUN,'struct') problemInput = true; \[FUN,xCurrent,LB,UB,options\] = separateOptimStruct(FUN); else % Single input and non-structure. error(message('optim:lsqnonlin:InputArg')); end end % No options passed. Set options directly to defaultopt after allDefaultOpts = isempty(options); % Prepare the options for the solver options = prepareOptionsForSolver(options, 'lsqnonlin'); % Set options to default if no options were passed. if allDefaultOpts % Options are all default options = defaultopt; end if nargin \< 2 \&\& \~problemInput error(message('optim:lsqnonlin:NotEnoughInputs')) end % Check for non-double inputs msg = isoptimargdbl('LSQNONLIN', {'X0','LB','UB'}, ... xCurrent,LB, UB); if \~isempty(msg) error('optim:lsqnonlin:NonDoubleInput',msg); end caller = 'lsqnonlin'; \[funfcn,mtxmpy,flags,sizes,\~,xstart,lb,ub,EXITFLAG,Resnorm,FVAL,LAMBDA, ... JACOB,OUTPUT,earlyTermination\] = lsqnsetup(FUN,xCurrent,LB,UB,options,defaultopt, ... allDefaultOpts,caller,nargout,length(varargin)); if earlyTermination return % premature return because of problem detected in lsqnsetup() end xCurrent(:) = xstart; % reshape back to user shape before evaluation % Catch any error in user objective during initial evaluation only switch funfcn{1} case 'fun' try initVals.F = feval(funfcn{3},xCurrent,varargin{:}); catch userFcn_ME optim_ME = MException('optim:lsqnonlin:InvalidFUN', ... getString(message('optim:lsqnonlin:InvalidFUN'))); userFcn_ME = addCause(userFcn_ME,optim_ME); rethrow(userFcn_ME) end initVals.J = \[\]; case 'fungrad' try \[initVals.F,initVals.J\] = feval(funfcn{3},xCurrent,varargin{:}); catch userFcn_ME optim_ME = MException('optim:lsqnonlin:InvalidFUN', ... getString(message('optim:lsqnonlin:InvalidFUN'))); userFcn_ME = addCause(userFcn_ME,optim_ME); rethrow(userFcn_ME) end case 'fun_then_grad' try initVals.F = feval(funfcn{3},xCurrent,varargin{:}); catch userFcn_ME optim_ME = MException('optim:lsqnonlin:InvalidFUN', ... getString(message('optim:lsqnonlin:InvalidFUN'))); userFcn_ME = addCause(userFcn_ME,optim_ME); rethrow(userFcn_ME) end try initVals.J = feval(funfcn{4},xCurrent,varargin{:}); catch userFcn_ME optim_ME = MException('optim:lsqnonlin:InvalidFUN', ... getString(message('optim:lsqnonlin:InvalidJacobFun'))); userFcn_ME = addCause(userFcn_ME,optim_ME); rethrow(userFcn_ME) end otherwise error(message('optim:lsqnonlin:UndefCallType')) end % Check for non-double data typed values returned by user functions if \~isempty( isoptimargdbl('LSQNONLIN', {'F','J'}, initVals.F, initVals.J) ) error('optim:lsqnonlin:NonDoubleFunVal',getString(message('optimlib:commonMsgs:NonDoubleFunVal','LSQNONLIN'))); end % Flag to determine whether to look up the exit msg. flags.makeExitMsg = logical(flags.verbosity) \|\| nargout \> 4; \[xCurrent,Resnorm,FVAL,EXITFLAG,OUTPUT,LAMBDA,JACOB\] = ... lsqncommon(funfcn,xCurrent,lb,ub,options,defaultopt,allDefaultOpts,caller,... initVals,sizes,flags,mtxmpy,varargin{:}); **3.** **运行结果** 