1.m.txt

该程序包含矩阵最大特征值上、下界估计、线性方程组的求解、非线性方程组的一个最新迭代方法、数值积分、微分方程数值解！

%本程序是利用改进后的单纯形算法！


function [x,fval,exitflag,output] = N_M(funfcn,x,options,varargin)
defaultopt = struct('Display','notify','MaxIter','200*numberOfVariables',...
   'MaxFunEvals','200*numberOfVariables','TolX',1e-6,'TolFun',1e-6);
if nargin==1 & nargout <= 1 & isequal(funfcn,'defaults')
   x = defaultopt;
   return
end

if nargin < 2, 
   error('FMINSEARCH requires at least two input arguments'); 
end

if nargin<3, options = []; end
n = prod(size(x));
numberOfVariables = n;

printtype = optimget(options,'Display',defaultopt,'fast');
tolx = optimget(options,'TolX',defaultopt,'fast');
tolf = optimget(options,'TolFun',defaultopt,'fast');
maxfun = optimget(options,'MaxFunEvals',defaultopt,'fast');
maxiter = optimget(options,'MaxIter',defaultopt,'fast');

% In case the defaults were gathered from calling: optimset('fminsearch'):
if ischar(maxfun)
   if isequal(lower(maxfun),'200*numberofvariables')
      maxfun = 200*numberOfVariables;
   else
      error('Option ''MaxFunEvals'' must be an integer value if not the default.')
   end
end
if ischar(maxiter)
   if isequal(lower(maxiter),'200*numberofvariables')
      maxiter = 200*numberOfVariables;
   else
      error('Option ''MaxIter'' must be an integer value if not the default.')
   end
end

switch printtype
case 'notify'
   prnt = 1;
case {'none','off'}
   prnt = 0;
case 'iter'
   prnt = 3;
case 'final'
   prnt = 2;
case 'simplex'
   prnt = 4;
otherwise
   prnt = 1;
end

header = ' Iteration   Func-count     min f(x)         Procedure';

% Convert to inline function as needed.
funfcn = fcnchk(funfcn,length(varargin));

n = prod(size(x));
% Initialize parameters
rho = 1; chi = 2.65; psi = 0.5; sigma = 0.5;
onesn = ones(1,n);
two2np1 = 2:n+1;
one2n = 1:n;

% Set up a simplex near the initial guess.
xin = x(:); % Force xin to be a column vector
v = zeros(n,n+1); fv = zeros(1,n+1);
v(:,1) = xin;    % Place input guess in the simplex! (credit L.Pfeffer at Stanford)
x(:) = xin;    % Change x to the form expected by funfcn 
fv(:,1) = feval(funfcn,x,varargin{:}); 

% Following improvement suggested by L.Pfeffer at Stanford
usual_delta = 0.05;             % 5 percent deltas for non-zero terms
zero_term_delta = 0.00025;      % Even smaller delta for zero elements of x
for j = 1:n
   y = xin;
   if y(j) ~= 0
      y(j) = (1 + usual_delta)*y(j);
   else 
      y(j) = zero_term_delta;
   end  
   v(:,j+1) = y;
   x(:) = y; f = feval(funfcn,x,varargin{:});
   fv(1,j+1) = f;
end     

% sort so v(1,:) has the lowest function value 
[fv,j] = sort(fv);
v = v(:,j);

how = 'initial';
itercount = 1;
func_evals = n+1;
if prnt == 3
   disp(' ')
   disp(header)
   disp([sprintf(' %5.0f        %5.0f     %12.6g         ', itercount, func_evals, fv(1)), how]) 
elseif prnt == 4
   clc
   formatsave = get(0,{'format','formatspacing'});
   format compact
   format short e
   disp(' ')
   disp(how)
   v
   fv
   func_evals
end
exitflag = 1;

while func_evals < maxfun & itercount < maxiter
   if max(max(abs(v(:,two2np1)-v(:,onesn)))) <= tolx & ...
         max(abs(fv(1)-fv(two2np1))) <= tolf
      break
   end
   how = '';
   % 计算反射点
   xbar = sum(v(:,one2n), 2)/n;
   xr = (1 + rho)*xbar - rho*v(:,end);
   x(:) = xr; fxr = feval(funfcn,x,varargin{:});
   func_evals = func_evals+1;
   
   if fxr < fv(:,1)
      % 计算扩展点
      xe = (1 + rho*chi)*xbar - rho*chi*v(:,end);
      x(:) = xe; fxe = feval(funfcn,x,varargin{:});
      func_evals = func_evals+1;
      if fxe < fxr
         v(:,end) = xe;
         fv(:,end) = fxe;
         how = 'expand';
      else
         v(:,end) = xr; 
         fv(:,end) = fxr;
         how = 'reflect';
      end
   else % fv(:,1) <= fxr
      if fxr < fv(:,n)
         v(:,end) = xr; 
         fv(:,end) = fxr;
         how = 'reflect';
      else % fxr >= fv(:,n) 
         % Perform contraction
         if fxr < fv(:,end)
            % Perform an outside contraction
            xc = (1 + psi*rho)*xbar - psi*rho*v(:,end);
            x(:) = xc; fxc = feval(funfcn,x,varargin{:});
            func_evals = func_evals+1;
            
            if fxc <= fxr
               v(:,end) = xc; 
               fv(:,end) = fxc;
               how = 'contract outside';
            else
               % perform a shrink
               how = 'shrink'; 
            end
         else
            % Perform an inside contraction
            xcc = (1-psi)*xbar + psi*v(:,end);
            x(:) = xcc; fxcc = feval(funfcn,x,varargin{:});
            func_evals = func_evals+1;
            
            if fxcc < fv(:,end)
               v(:,end) = xcc;
               fv(:,end) = fxcc;
               how = 'contract inside';
            else
               % perform a shrink
               how = 'shrink';
            end
         end
         if strcmp(how,'shrink')
            for j=two2np1
               v(:,j)=v(:,1)+sigma*(v(:,j) - v(:,1));
               x(:) = v(:,j); fv(:,j) = feval(funfcn,x,varargin{:});
            end
            func_evals = func_evals + n;
         end
      end
   end
   [fv,j] = sort(fv);
   v = v(:,j);
   itercount = itercount + 1;
   if prnt == 3
   disp([sprintf(' %5.0f        %5.0f     %12.6g         ', itercount, func_evals, fv(1)), how]) 
   elseif prnt == 4
      disp(' ')
      disp(how)
      v
      fv
      func_evals
   end  
end   % while
x(:) = v(:,1);
if prnt == 4,
   set(0,{'format','formatspacing'},formatsave);
end
output.iterations = itercount;
output.funcCount = func_evals;
output.algorithm = 'Nelder-Mead simplex direct search';
fval = min(fv); 
if func_evals >= maxfun 
   if prnt > 0
      disp(' ')
      disp('Exiting: Maximum number of function evaluations has been exceeded')
      disp('         - increase MaxFunEvals option.')
      msg = sprintf('         Current function value: %f \n', fval);
      disp(msg)
   end
   exitflag = 0;
elseif itercount >= maxiter 
   if prnt > 0
      disp(' ')
      disp('Exiting: Maximum number of iterations has been exceeded')
      disp('         - increase MaxIter option.')
      msg = sprintf('         Current function value: %f \n', fval);
      disp(msg)
   end
   exitflag = 0; 
else
   if prnt > 1
      convmsg1 = sprintf([ ...
         '\nOptimization terminated successfully:\n',...
         ' the current x satisfies the termination criteria using OPTIONS.TolX of %e \n',...
         ' and F(X) satisfies the convergence criteria using OPTIONS.TolFun of %e \n'
          ],tolx, tolf);
      disp(convmsg1)
   end
   exitflag = 1;
end