26 用lsqnonlin求解最小二乘问题（matlab程序）

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