非线性方程组最小二乘法高斯牛顿最小二乘法matlab代码
时间: 2023-05-14 15:03:03 浏览: 191
非线性方程组最小二乘法是解决非线性方程组问题的一种方法,它利用最小二乘法的思想来求解问题,可以有效地解决很多实际问题。而高斯牛顿最小二乘法是其中的一种算法,也是比较常用的一种。
在matlab中,可以利用以下代码来实现非线性方程组最小二乘法的计算:
function [x, resnorm, residual, exitflag, output, lambda, jacobian] = lsqnonlin(fun,x0,LB,UB,options,varargin)
其中,fun是需要求解的非线性方程组,x0是变量的初始值,LB和UB是变量的上下界,options是优化选项,varargin是额外的参数。该函数将求解结果返回给x、resnorm、residual、exitflag、output、lambda和jacobian这七个变量。
而高斯牛顿最小二乘法的matlab代码实现如下:
function [x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqnonlin(FUN,x0,lb,ub,options,varargin) % FUN - function handle
% x0 - starting point
% lb - lower bound
% ub - upper bound
% options - optimization options
% varargin - additional arguments for function handle
% x - solution vector
% resnorm - residual norm squared
% residual - residual vector
% exitflag - optimization exit flag
% output - optimization output
% lambda - Lagrange multipliers
% jacobian - Jacobian matrix
% Initialize variables
x = x0;
resnorm = Inf;
exitflag = -1;
lambda = [];
jacobian = [];
% Run optimization until successful or maximum number of iterations is reached
for iter = 1:options.MaxIter
[F,J] = feval(FUN,x,varargin{:});
residual = F;
resnorm = norm(residual,2)^2;
% Check for successful optimization
if resnorm <= options.TolFun
exitflag = 1;
output.iterations = iter;
break;
end
% Compute next point using Gauss-Newton update
p = -(J'*J)\(J'*residual);
x = x + p;
% Project onto feasible region
if ~isempty(lb)
x(x < lb) = lb(x < lb);
end
if ~isempty(ub)
x(x > ub) = ub(x > ub);
end
end
% Return Lagrange multipliers and Jacobian matrix
if nargout > 5
lambda = (-J'*J)\(J'*residual);
end
if nargout > 6
jacobian = J;
end
% Create output structure
if exitflag ~= 1
output.iterations = iter;
end
output.funcCount = iter;
output.algorithm = 'Gauss-Newton';
output.message = sprintf('Optimization terminated.');
% Display warning if maximum number of iterations is reached
if iter == options.MaxIter && exitflag ~= 1
warning('lsqnonlin:MaxIterReached','Maximum number of iterations reached without convergence.');
end
该代码使用了feval函数来求解非线性方程组,使用了高斯牛顿法求解最小化问题。其中,options是优化选项,可以设置包括最大迭代次数、函数值容许误差等多个参数。函数返回求解的解向量x,残差的平方和resnorm,残差向量residual,退出标志exitflag,优化输出output,拉格朗日乘数lambda和雅各比矩阵jacobian。