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matlab算法整理_matlab写递推式

作者：菜鸟追梦旅行 | 2024-03-01 21:02:35

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matlab写递推式

目录

递推关系式的作图程序

顶点覆盖近似算法

方程求根

哈密尔顿回路

画等温线

递推关系式的作图程序

  1. x(1)=1;x(2)=10;
  2. hold on
  3. for t=3:10
  4.     x(t)=0.8*x(t-1)+1.5*x(t-2);    %递推关系式，注意MATLAB里向量的下标不能从零开始。
  5. end
  6. plot(2:10,x(2:10))

顶点覆盖近似算法

  1. %只是这种近似算法结果很差
  2.  
  3. %首先输入关联矩阵F及节点个数n
  4. F=[0 1 0 0 0 0 0;
  5.     1 0 1 0 0 0 0;
  6.     0 1 0 1 1 1 0;
  7.     0 0 1 0 0 1 0;
  8.     0 0 1 0 0 1 0;
  9.     0 0 1 1 1 0 1;
  10.     0 0 0 0 0 1 0];
  11. n=7;
  12. C=[];
  13. l=0;
  14. for i=1:n
  15.     for j=1:n
  16.         if F(i,j)~=0
  17.             if l==0
  18.                 C=[i j];l=2;
  19.             else 
  20.                 p=0;q=0;
  21.                 for a=1:l
  22.                     if C(a)==i
  23.                         p=1;
  24.                     end
  25.                     if C(a)==j
  26.                         q=1;
  27.                     end
  28.                 end
  29.                 if p==0
  30.                     l=l+1;C(l)=i;
  31.                 end 
  32.                 if q==0
  33.                     l=l+1;C(l)=j;
  34.                 end 
  35.                 F(i,:)=zeros(1,n);
  36.                 F(:,j)=zeros(n,1);
  37.             end
  38.         end
  39.     end
  40. end
  41. disp(C);

方程求根

  1. % 方程求根
  2. % inv - 逆矩阵
  3. % roots - 多项式的根
  4. % fzero - 一元函数零点
  5. % fsolve - 非线性方程组
  6. % solve - 符号方程解
  7. % *newton - 牛顿迭代法解非线性方程

INV.m

  1. %求逆矩阵
  2. %用法 B=inv(A) 其中A为数值或符号方阵,B返回A的逆
  3. %例如
  4. %    inv([1 2;3 4])                    %数值
  5. %    syms a b c d;inv([[a,b;c,d])      %符号
  6. %
  7. %INV    Matrix inverse.
  8. %   INV(X) is the inverse of the square matrix X.
  9. %   A warning message is printed if X is badly scaled or
  10. %   nearly singular.
  11. %
  12. %   See also SLASH, PINV, COND, CONDEST, NNLS, LSCOV.
  13.  
  14. %   Copyright (c) 1984-98 by The MathWorks, Inc.
  15. %   $Revision: 5.5 $  $Date: 1997/11/21 23:38:31 $
  16. %   Built-in function.

ROOTS.m

  1. function r = roots(c)
  2. %roots(p)可求得多项式p的所有复根.
  3. %例   x^3+2x^2-3=0
  4. %   求解
  5. %      roots([1 2 0 -3])
  6. %
  7. %ROOTS  Find polynomial roots.
  8. %   ROOTS(C) computes the roots of the polynomial whose coefficients
  9. %   are the elements of the vector C. If C has N+1 components,
  10. %   the polynomial is C(1)*X^N + ... + C(N)*X + C(N+1).
  11. %
  12. %   See also POLY, RESIDUE, FZERO.
  13.  
  14. %   J.N. Little 3-17-86
  15. %   Copyright (c) 1984-98 by The MathWorks, Inc.
  16. %   $Revision: 5.6 $  $Date: 1997/11/21 23:41:05 $
  17.  
  18. % ROOTS finds the eigenvalues of the associated companion matrix.
  19.  
  20. if size(c,1)>1 & size(c,2)>1
  21.     error('Must be a vector.')
  22. end
  23. c = c(:).';
  24. n = size(c,2);
  25. r = zeros(0,1);
  26.  
  27. inz = find(c);
  28. if isempty(inz),
  29.     % All elements are zero
  30.     return
  31. end
  32.  
  33. % Strip leading zeros and throw away.  
  34. % Strip trailing zeros, but remember them as roots at zero.
  35. nnz = length(inz);
  36. c = c(inz(1):inz(nnz));
  37. r = zeros(n-inz(nnz),1);
  38.  
  39. % Polynomial roots via a companion matrix
  40. n = length(c);
  41. if n > 1
  42.     a = diag(ones(1,n-2),-1);
  43.     a(1,:) = -c(2:n) ./ c(1);
  44.     r = [r;eig(a)];
  45. end

FSOLVE.m

  1. function [x,FVAL,EXITFLAG,OUTPUT,JACOB] = fsolve(FUN,x,options,varargin)
  2. %优化工具箱函数,可求多元非线性方程组的实根. 用法与fzero类似。
  3. %例 先写一个M函数rooteg4fun.m
  4. %                  function  y=rooteg4fun(x)
  5. %                  y(1)=4*x(1)-x(2)+exp(x(1))/10-1;
  6. %                  y(2)=-x(1)+4*x(2)+x(1).^2/8;
  7. %   使用
  8. %      [x,f,h]=fsolve('rooteg4fun',[0,0]) %初值x(1)=0,x(2)=0 
  9. %   x返回解向量，f返回误差向量，h>0表明算法收敛
  10. %   注意：方程变量必须拼成一个向量变量，即用x(1),x(2),...
  11. %
  12. %FSOLVE Solves nonlinear equations by a least squares method.
  13. %
  14. %   FSOLVE solves equations of the form:
  15. %             
  16. %   F(X)=0    where F and X may be vectors or matrices.   
  17. %
  18. %   X=FSOLVE(FUN,X0) starts at the matrix X0 and tries to solve the 
  19. %   equations described in FUN. FUN is usually an M-file which returns 
  20. %   an evaluation of the equations for a particular value of X: F=FUN(X).
  21. %
  22. %   X=FSOLVE(FUN,X0,OPTIONS) minimizes with the default optimization
  23. %   parameters replaced by values in the structure OPTIONS, an argument
  24. %   created with the OPTIMSET function.  See OPTIMSET for details.  Used
  25. %   options are Display, TolX, TolFun, DerivativeCheck, Diagnostics, Jacobian,
  26. %   JacobPattern, LineSearchType, LevenbergMarquardt, MaxFunEvals, MaxIter, 
  27. %   DiffMinChange and DiffMaxChange, LargeScale, MaxPCGIter, PrecondBandWidth, 
  28. %   TolPCG, TypicalX. Use the Jacobian option to specify that FUN may be called 
  29. %   with two output arguments where the second, J, is the Jacobian matrix: 
  30. %   [F,J] = feval(FUN,X). If FUN returns a vector (matrix) of m components when 
  31. %   X has length n, then J is an m-by-n matrix where J(i,j) is the partial 
  32. %   derivative of F(i) with respect to x(j). (Note that the Jacobian J is the 
  33. %   transpose of the gradient of F.)
  34. %
  35. %   X=FSOLVE(FUN,X0,OPTIONS,P1,P2,...) passes the problem-dependent 
  36. %   parameters P1,P2,... directly to the function FUN: FUN(X,P1,P2,...).  
  37. %   Pass an empty matrix for OPTIONS to use the default values. 
  38. %
  39. %   [X,FVAL]=FSOLVE(FUN,X0,...) returns the value of the objective function
  40. %    at X. 
  41. %
  42. %   [X,FVAL,EXITFLAG]=FSOLVE(FUN,X0,...) returns a string EXITFLAG that 
  43. %   describes the exit condition of FSOLVE.  
  44. %   If EXITFLAG is:
  45. %      > 0 then FSOLVE converged to a solution X.
  46. %      0   then the maximum number of function evaluations was reached.
  47. %      < 0 then FSOLVE did not converge to a solution.
  48. %
  49. %   [X,FVAL,EXITFLAG,OUTPUT]=FSOLVE(FUN,X0,...) returns a structure OUTPUT
  50. %   with the number of iterations taken in OUTPUT.iterations, the number of
  51. %   function evaluations in OUTPUT.funcCount, the algorithm used in OUTPUT.algorithm,
  52. %   the number of CG iterations (if used) in OUTPUT.cgiterations, and the first-order 
  53. %   optimality (if used) in OUTPUT.firstorderopt.
  54. %
  55. %   [X,FVAL,EXITFLAG,OUTPUT,JACOB]=FSOLVE(FUN,X0,...) returns the 
  56. %   Jacobian of FUN at X.  
  57.  
  58. %   Copyright (c) 1990-98 by The MathWorks, Inc.
  59. %   $Revision: 1.26 $  $Date: 1998/10/22 19:28:31 $
  60. %   Andy Grace 7-9-90.
  61.  
  62. %   Grandfathered FSOLVE call for Optimization Toolbox versions prior to 2.0:
  63. %   [X,OPTIONS]=FSOLVE(FUN,X0,OPTIONS,GRADFUN,P1,P2,...)
  64. %
  65. % ------------Initialization----------------
  66.  
  67. defaultopt = optimset('display','final','LargeScale','on', ...
  68.    'TolX',1e-6,'TolFun',1e-6,'DerivativeCheck','off',...
  69.    'Jacobian','off','MaxFunEvals','100*numberOfVariables',...
  70.    'Diagnostics','off',...
  71.    'DiffMaxChange',1e-1,'DiffMinChange',1e-8,...
  72.    'PrecondBandWidth',0,'TypicalX','ones(numberOfVariables,1)','MaxPCGIter','max(1,floor(numberOfVariables/2))', ...
  73.    'TolPCG',0.1,'MaxIter',400,'JacobPattern',[], ...
  74.    'LineSearchType','quadcubic','LevenbergMarq','off'); 
  75. % If just 'defaults' passed in, return the default options in X
  76. if nargin==1 & nargout <= 1 & isequal(FUN,'defaults')
  77.    x = defaultopt;
  78.    return
  79. end
  80.  
  81. if nargin < 2, error('FSOLVE requires two input arguments');end
  82. if nargin < 3, options=[]; end
  83.  
  84. % These are added so that we can have the same code as in lsqnonlin which
  85. %  actually has upper and lower bounds.
  86. LB = []; UB = [];
  87.  
  88. %[x,FVAL,EXITFLAG,OUTPUT,JACOB] = fsolve(FUNin,x,options,varargin)
  89. % Note: don't send varargin in as a comma separated list!!
  90. numargin = nargin; numargout = nargout;
  91. [calltype, GRADFUN, varargin] = parse_call(FUN,options,numargin,numargout,varargin);
  92.  
  93. if isequal(calltype,'new')  % fsolve version 2.*
  94.    
  95.    xstart=x(:);
  96.    numberOfVariables=length(xstart);
  97.    
  98.    large = 'large-scale';
  99.    medium = 'medium-scale';
  100.    
  101.    l = []; u = [];
  102.    
  103.    options = optimset(defaultopt,options);
  104.    switch optimget(options,'display')
  105.    case {'off','none'}
  106.       verbosity = 0;
  107.    case 'iter'
  108.       verbosity = 2;
  109.    case 'final'
  110.       verbosity = 1;
  111.    case 'testing'
  112.       verbosity = Inf;
  113.    otherwise
  114.       verbosity = 1;
  115.    end
  116.    diagnostics = isequal(optimget(options,'diagnostics','off'),'on');
  117.    
  118.    gradflag =  strcmp(optimget(options,'Jacobian'),'on');
  119.    line_search = strcmp(optimget(options,'largescale','off'),'off'); % 0 means trust-region, 1 means line-search
  120.    
  121.    % Convert to inline function as needed
  122.    if ~isempty(FUN)  % will detect empty string, empty matrix, empty cell array
  123.       [funfcn, msg] = fprefcnchk(FUN,'fsolve',length(varargin),gradflag);
  124.    else
  125.       errmsg = sprintf('%s\n%s', ...
  126.          'FUN must be a function name, valid string expression, or inline object;', ...
  127.          ' or, FUN may be a cell array that contains these type of objects.');
  128.       error(errmsg)
  129.    end
  130.    
  131.    x(:) = xstart;
  132.    switch funfcn{1}
  133.    case 'fun'
  134.       fuser = feval(funfcn{3},x,varargin{:});
  135.       f = fuser(:);
  136.       nfun=length(f);
  137.       JAC = zeros(nfun,numberOfVariables);
  138.    case 'fungrad'
  139.       [fuser,JAC] = feval(funfcn{3},x,varargin{:});
  140.       f = fuser(:);
  141.       nfun=length(f);
  142.    case 'fun_then_grad'
  143.       fuser = feval(funfcn{3},x,varargin{:}); 
  144.       f = fuser(:);
  145.       JAC = feval(funfcn{4},x,varargin{:});
  146.       nfun=length(f);
  147.       
  148.    otherwise
  149.       error('Undefined calltype in FSOLVE');
  150.    end
  151.    
  152.    % check size of JAC
  153.    [Jrows, Jcols]=size(JAC);
  154.    if Jrows~=nfun | Jcols ~=numberOfVariables
  155.       errstr = sprintf('%s\n%s%d%s%d\n',...
  156.          'User-defined Jacobian is not the correct size:',...
  157.          '    the Jacobian matrix should be ',nfun,'-by-',numberOfVariables);
  158.       error(errstr);
  159.    end
  160.    
  161.    YDATA = []; caller = 'fsolve';
  162.    
  163.    % trustregion and enough equations (as many as variables) 
  164.    if ~line_search & nfun >= numberOfVariables 
  165.       OUTPUT.algorithm = large;
  166.       
  167.       % trust region and not enough equations -- switch to line_search
  168.    elseif ~line_search & nfun < numberOfVariables 
  169.       warnstr = sprintf('%s\n%s\n', ...
  170.          'Large-scale method requires at least as many equations as variables; ',...
  171.          '   switching to line-search method instead.');
  172.       warning(warnstr);
  173.       OUTPUT.algorithm = medium;
  174.       
  175.       % line search and no bounds  
  176.    elseif line_search & isempty(l) & isempty(u)
  177.       OUTPUT.algorithm = medium;
  178.       
  179.       % line search and  bounds  and enough equations, switch to trust region 
  180.    elseif line_search & (~isempty(LB) | ~isempty(UB))  & nfun >= numberOfVariables
  181.       warnstr = sprintf('%s\n%s\n', ...
  182.          'Line-search method does not handle bound constraints; ',...
  183.          '   switching to trust-region method instead.');
  184.       warning(warnstr);
  185.       OUTPUT.algorithm = large;
  186.       
  187.       % can't handle this one:   
  188.    elseif line_search & (~isempty(LB) | ~isempty(UB))  & nfun < numberOfVariables
  189.       errstr = sprintf('%s\n%s\n%s\n', ...
  190.          'Line-search method does not handle bound constraints ',...
  191.          '   and trust-region method requires at least as many equations as variables; ',...
  192.          '   aborting.');
  193.       error(errstr);
  194.       
  195.    end
  196.    
  197.    if diagnostics > 0
  198.       % Do diagnostics on information so far
  199.       constflag = 0; gradconstflag = 0; non_eq=0;non_ineq=0;lin_eq=0;lin_ineq=0;
  200.       confcn{1}=[];c=[];ceq=[];cGRAD=[];ceqGRAD=[];
  201.       hessflag = 0; HESS=[];
  202.       msg = diagnose('fsolve',OUTPUT,gradflag,hessflag,constflag,gradconstflag,...
  203.          line_search,options,xstart,non_eq,...
  204.          non_ineq,lin_eq,lin_ineq,LB,UB,funfcn,confcn,f,JAC,HESS,c,ceq,cGRAD,ceqGRAD);
  205.       
  206.    end
  207.    
  208.    % Execute algorithm
  209.    if isequal(OUTPUT.algorithm, large)
  210.       if ~gradflag
  211.          Jstr = optimget(options,'JacobPattern',[]);
  212.          if isempty(Jstr)  
  213.             % Put this code separate as it might generate OUT OF MEMORY error
  214.             Jstr = sparse(ones(Jrows,Jcols));
  215.          end
  216.       else
  217.          Jstr = [];
  218.       end
  219.       l = []; u = []; computeLambda = 0;
  220.       [x,FVAL,LAMBDA,JACOB,EXITFLAG,OUTPUT]=...
  221.          snls(funfcn,x,l,u,verbosity,options,f,JAC,YDATA,caller,Jstr,computeLambda,varargin{:});
  222.    else 
  223.       [x,FVAL,JACOB,EXITFLAG,OUTPUT] = ...
  224.          nlsq(funfcn,x,verbosity,options,f,JAC,YDATA,caller,varargin{:});
  225.       
  226.    end
  227.    
  228.    Resnorm = FVAL'*FVAL;  % assumes FVAL still a vector
  229.    if Resnorm > 10*optimget(options,'TolFun',1e-4) & verbosity>0
  230.       if verbosity > 0
  231.          disp('Optimizer is stuck at a minimum that is not a root')
  232.          disp('Try again with a new starting guess')
  233.       end
  234.       EXITFLAG = -1;
  235.    end
  236.    
  237.    % Reset FVAL to shape of the user-function output, fuser
  238.    FVAL = reshape(FVAL,size(fuser));
  239.    
  240.    % end FSOLVE 2.*
  241. else % version 1.5 FSOLVE
  242.    
  243.    if length(options)<5; 
  244.       options(5)=0; 
  245.    end
  246.    % Switch methods making Gauss Newton the default method.
  247.    if options(5)==0; options(5)=1; else options(5)=0; end
  248.    
  249.    % Convert to inline function as needed.
  250.    if ~isempty(FUN)
  251.       [funfcn, msg] = fcnchk(FUN,length(varargin));
  252.       if ~isempty(msg)
  253.          error(msg);
  254.       end
  255.    else
  256.       error('FUN must be a function name or valid expression.')
  257.    end
  258.    
  259.    if ~isempty(GRADFUN)
  260.       [gradfcn, msg] = fcnchk(GRADFUN,length(varargin));
  261.       if ~isempty(msg)
  262.          error(msg);
  263.       end
  264.    else
  265.       gradfcn = [];
  266.    end
  267.    
  268.    [x,options] = nlsqold(funfcn,x,options,gradfcn,varargin{:});
  269.    
  270.    if options(8)>10*options(3) & options(1)>0
  271.       disp('Optimizer is stuck at a minimum that is not a root')
  272.       disp('Try again with a new starting guess')
  273.    end
  274.    
  275.    % Set the second output argument FVAL to be options as in old calling syntax
  276.    FVAL = options;
  277.    
  278.    % end fsolve version 1.5.*
  279.    
  280. end
  281.  
  282. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  283. function [calltype, GRADFUN, otherargs] = parse_call(FUN,options,numargin,numargout,otherargs)
  284. % PARSE_CALL Determine which calling syntax is being used: the FSOLVE prior to 2.0, or
  285. %    in version 2.0 or later of the Toolbox.
  286. %    old call: [X,OPTIONS]=FSOLVE(FUN,X0,OPTIONS,GRADFUN,varargin)
  287. %    new call: [X,FVAL,EXITFLAG,OUTPUT,JACOB]=FSOLVE(FUN,X0,OPTIONS,varargin)
  288.  
  289. if numargout > 2               % [X,FVAL,EXITFLAG,...]=FSOLVE (...)
  290.    calltype = 'new';
  291.    GRADFUN = []; 
  292. elseif isa(FUN,'cell')         % FUN == {...}
  293.    calltype = 'new';
  294.    GRADFUN = [];
  295. elseif ~isempty(options) & isa(options,'double')   % OPTIONS == scalar or and array
  296.    calltype = 'old';
  297.    if length(otherargs) > 0
  298.       GRADFUN = otherargs{1};
  299.       otherargs = otherargs(2:end);
  300.    else
  301.       GRADFUN = [];
  302.    end
  303. elseif isa(options,'struct')   % OPTIONS has fields
  304.    calltype = 'new';
  305.    GRADFUN = [];
  306. else                           % Ambiguous
  307.    warnstr = sprintf('%s\n%s\n%s\n',...
  308.       'Cannot determine from calling sequence whether to use new (2.0 or later) FSOLVE ', ...
  309.       'function or grandfathered FSOLVE function.  Assuming new syntax; if call was grandfathered', ...
  310.       'FSOLVE syntax, this may give unexpected results.');
  311.    warning(warnstr)
  312.    calltype = 'new';
  313.    GRADFUN = [];
  314. end
  315.  
  316. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  317. function [allfcns,msg] = fprefcnchk(funstr,caller,lenVarIn,gradflag)
  318. %PREFCNCHK Pre- and post-process function expression for FUNCHK.
  319. %   [ALLFCNS,MSG] = PREFUNCHK(FUNSTR,CALLER,lenVarIn,GRADFLAG) takes
  320. %   the (nonempty) expression FUNSTR from CALLER with LenVarIn extra arguments,
  321. %   parses it according to what CALLER is, then returns a string or inline
  322. %   object in ALLFCNS.  If an error occurs, this message is put in MSG.
  323. %
  324. %   ALLFCNS is a cell array: 
  325. %    ALLFCNS{1} contains a flag 
  326. %    that says if the objective and gradients are together in one function 
  327. %    (calltype=='fungrad') or in two functions (calltype='fun_then_grad')
  328. %    or there is no gradient (calltype=='fun'), etc.
  329. %    ALLFCNS{2} contains the string CALLER.
  330. %    ALLFCNS{3}  contains the objective function
  331. %    ALLFCNS{4}  contains the gradient function (transpose of Jacobian).
  332. %  
  333. %    NOTE: we assume FUNSTR is nonempty.
  334. % Initialize
  335. msg='';
  336. allfcns = {};
  337. funfcn = [];
  338. gradfcn = [];
  339.  
  340. if gradflag
  341.    calltype = 'fungrad';
  342. else
  343.    calltype = 'fun';
  344. end
  345.  
  346. % {fun}
  347. if isa(funstr, 'cell') & length(funstr)==1
  348.    % take the cellarray apart: we know it is nonempty
  349.    if gradflag
  350.       calltype = 'fungrad';0
  351.    end
  352.    [funfcn, msg] = fcnchk(funstr{1},lenVarIn);
  353.    if ~isempty(msg)
  354.       error(msg);
  355.    end
  356.    
  357.    % {fun,[]}      
  358. elseif isa(funstr, 'cell') & length(funstr)==2 & isempty(funstr{2})
  359.    if gradflag
  360.       calltype = 'fungrad';
  361.    end
  362.    [funfcn, msg] = fcnchk(funstr{1},lenVarIn);
  363.    if ~isempty(msg)
  364.       error(msg);
  365.    end  
  366.    
  367.    % {fun, grad}   
  368. elseif isa(funstr, 'cell') & length(funstr)==2 % and ~isempty(funstr{2})
  369.    
  370.    [funfcn, msg] = fcnchk(funstr{1},lenVarIn);
  371.    if ~isempty(msg)
  372.       error(msg);
  373.    end  
  374.    [gradfcn, msg] = fcnchk(funstr{2},lenVarIn);
  375.    if ~isempty(msg)
  376.       error(msg);
  377.    end
  378.    calltype = 'fun_then_grad';
  379.    if ~gradflag
  380.       warnstr = ...
  381.          sprintf('%s\n%s\n%s\n','Jacobian function provided but OPTIONS.Jacobian=''off'';', ...
  382.          '  ignoring Jacobian function and using finite-differencing.', ...
  383.          '  Rerun with OPTIONS.Jacobian=''on'' to use Jacobian function.');
  384.       warning(warnstr);
  385.       calltype = 'fun';
  386.    end   
  387.    
  388. elseif ~isa(funstr, 'cell')  %Not a cell; is a string expression, function name string or inline object
  389.    [funfcn, msg] = fcnchk(funstr,lenVarIn);
  390.    if ~isempty(msg)
  391.       error(msg);
  392.    end   
  393.    if gradflag % gradient and function in one function/M-file
  394.       gradfcn = funfcn; % Do this so graderr will print the correct name
  395.    end  
  396. else
  397.    errmsg = sprintf('%s\n%s', ...
  398.       'FUN must be a function name, valid string expression, or inline object;', ...
  399.       ' or, FUN may be a cell array that contains these type of objects.');
  400.    error(errmsg)
  401. end
  402.  
  403. allfcns{1} = calltype;
  404. allfcns{2} = caller;
  405. allfcns{3} = funfcn;
  406. allfcns{4} = gradfcn;
  407. allfcns{5}=[];
  408.

SOLVE.m

  1. function varargout = solve(varargin)
  2. %求各种类型方程组的解析解, 
  3. %当找不到解析解时,将自动寻求原点附近的一个近似解, 并可达任意精度.
  4. %用法
  5. %   solve('方程1','方程2',...,'方程N','变量1','变量2',...,'变量M')
  6. %SOLVE  Symbolic solution of algebraic equations.
  7. %   SOLVE('eqn1','eqn2',...,'eqnN')
  8. %   SOLVE('eqn1','eqn2',...,'eqnN','var1,var2,...,varN')
  9. %   SOLVE('eqn1','eqn2',...,'eqnN','var1','var2',...'varN')
  10. %
  11. %   The eqns are symbolic expressions or strings specifying equations.  The
  12. %   vars are symbolic variables or strings specifying the unknown variables.
  13. %   SOLVE seeks zeros of the expressions or solutions of the equations.
  14. %   If not specified, the unknowns in the system are determined by FINDSYM.
  15. %   If no analytical solution is found and the number of equations equals
  16. %   the number of dependent variables, a numeric solution is attempted.
  17. %
  18. %   Three different types of output are possible.  For one equation and one
  19. %   output, the resulting solution is returned, with multiple solutions to
  20. %   a nonlinear equation in a symbolic vector.  For several equations and
  21. %   an equal number of outputs, the results are sorted in lexicographic
  22. %   order and assigned to the outputs.  For several equations and a single
  23. %   output, a structure containing the solutions is returned.
  24. %
  25. %   Examples:
  26. %
  27. %      solve('p*sin(x) = r') chooses 'x' as the unknown and returns
  28. %
  29. %        ans =
  30. %        asin(r/p)
  31. %
  32. %      [x,y] = solve('x^2 + x*y + y = 3','x^2 - 4*x + 3 = 0') returns
  33. %
  34. %        x =
  35. %        [ 1]
  36. %        [ 3]
  37. %
  38. %        y =
  39. %        [    1]
  40. %        [ -3/2]
  41. %
  42. %      S = solve('x^2*y^2 - 2*x - 1 = 0','x^2 - y^2 - 1 = 0') returns
  43. %      the solutions in a structure.
  44. %
  45. %        S =
  46. %          x: [8x1 sym]
  47. %          y: [8x1 sym]
  48. %
  49. %      [u,v] = solve('a*u^2 + v^2 = 0','u - v = 1') regards 'a' as a
  50. %      parameter and solves the two equations for u and v.
  51. %
  52. %      S = solve('a*u^2 + v^2','u - v = 1','a,u') regards 'v' as a
  53. %      parameter, solves the two equations, and returns S.a and S.u.
  54. %
  55. %      [a,u,v] = solve('a*u^2 + v^2','u - v = 1','a^2 - 5*a + 6') solves
  56. %      the three equations for a, u and v.
  57. %
  58. %      [x,y] = solve('sin(x+y)-exp(x)*y = 0','x^2-y = 2') cannot find
  59. %      an analytic solution, so returns a numeric solution.
  60. %
  61. %   See also DSOLVE.
  62.  
  63. %   Copyright (c) 1993-98 by The MathWorks, Inc.
  64. %   $Revision: 1.13 $  $Date: 1997/11/29 01:06:35 $
  65.  
  66. % Set the Explicit solution (for equations of order 4 or less)
  67. % option on in the Maple Workspace.
  68.  
  69. maple('_EnvExplicit := true;');
  70.  
  71. % Collect input arguments together in either equation or variable lists.
  72.  
  73. eqns = [];
  74. vars = [];
  75. for k = 1:nargin
  76.    v = varargin{k};
  77.    if all(isletter(v) | ('0'<=v & v<='9') | v == '_' | v == ',')
  78.       vars = [vars ',' v];
  79.    else
  80.       [t,stat] = maple(v);
  81.       if stat
  82.          error(['''' v ''' is not a valid expression or equation.'])
  83.       end
  84.       eqns = [eqns ',' v];   
  85.    end
  86. end
  87.  
  88. if isempty(eqns)
  89.    warning('List of equations is empty.')
  90.    varargout = cell(1,nargout);
  91.    varargout{1} = sym([]);
  92.    return
  93. else
  94.    eqns(1) = '{'; eqns(end+1) = '}';
  95. end
  96. neqns = sum(commas(eqns)) + 1;
  97.  
  98. % Determine default variables and sort variable list.
  99.  
  100. if isempty(vars)
  101.    vars = ['[' findsym(sym(eqns),neqns) ']'];
  102. else
  103.    vars(1) = '['; vars(end+1) = ']';
  104. end
  105. v = vars;
  106. [vars,stat] = maple('sort',v);
  107. if stat
  108.    error(['''' v ''' is not a valid variable list.'])
  109. end
  110. vars(1) = '{'; vars(end) = '}';
  111. nvars = sum(commas(vars)) + 1;
  112.  
  113. if neqns ~= nvars
  114.    warning([int2str(neqns) ' equations in ' int2str(nvars) ' variables.'])
  115. end
  116.  
  117. % Seek analytic solution.
  118.  
  119. [R,stat] = maple('solve',eqns,vars);
  120.  
  121. % If no analytic solution, seek numerical solution.
  122.  
  123. if (isempty(R) & (nvars == neqns))
  124.    [R,stat] = maple('fsolve',eqns,vars);
  125. end
  126.  
  127. if stat
  128.    error(R)
  129. end
  130.  
  131. % If still no solution, give up.
  132.  
  133. if isempty(R) | findstr(R,'fsolve')
  134.    warning('Explicit solution could not be found.');
  135.    varargout = cell(1,nargout);
  136.    varargout{1} = sym([]);
  137.    return
  138. end
  139.  
  140. % Expand any RootOf.
  141.  
  142. while ~isempty(findstr(R,'RootOf'))
  143.    k = min(findstr(R,'RootOf'));
  144.    p = findstr(R,'{'); p = max(p(p<k));
  145.    q = findstr(R,'}'); q = min(q(q>k));
  146.    s = R(p:q);
  147.    t = s(min(findstr(s,'RootOf'))+6:end);
  148.    e = min(find(cumsum((t=='(')-(t==')'))==0));
  149.    if isempty(find(commas(t(2:e-1))))
  150.       % RootOf with one argument, possibly an imbedded RootOf.
  151.       [s,stat] = maple('allvalues',s,'dependent');
  152.       if stat
  153.          error(s)
  154.       end
  155.    else
  156.       % RootOf with a good estimate as the second argument.
  157.       s = maple('evalf',s);
  158.    end
  159.    R = [R(1:p-1) s R(q+1:end)];
  160. end
  161.  
  162. % Parse the result.
  163.  
  164. if nvars == 1 & nargout <= 1
  165.  
  166.    % One variable and at most one output.
  167.    % Return a single scalar or vector sym.
  168.  
  169.    S = sym([]);
  170.    c = find(commas(R) | R == '}');
  171.    for p = find(R == '=')
  172.       q = min(c(c>p));
  173.       t = trim(R(p+1:q-1));  % The solution (xk)
  174.       S = [S; sym(t)];
  175.    end
  176.    varargout{1} = S;
  177.  
  178. else
  179.  
  180.    % Several variables.
  181.    % Create a skeleton structure.
  182.  
  183.    c = [1 find(commas(vars)) length(vars)];
  184.    S = [];
  185.    for j = 1:nvars
  186.       v = trim(vars(c(j)+1:c(j+1)-1));
  187.       S = setfield(S,v,[]);
  188.    end
  189.  
  190.    % Complete the structure.
  191.  
  192.    c = [1 find(commas(R) | R == '{' | R == '}') length(R)];
  193.    for p = find(R == '=')
  194.       q = max(c(c<p));
  195.       v = trim(R(q+1:p-1));  % The variable (x)
  196.       q = min(c(c>p));
  197.       t = trim(R(p+1:q-1));  % The solution (xk)
  198.       S = setfield(S,v,[getfield(S,v); sym(t)]);
  199.    end
  200.    
  201.    if nargout <= 1
  202.  
  203.       % At most one output, return the structure.
  204.       varargout{1} = S;
  205.  
  206.    elseif nargout == nvars
  207.  
  208.       % Same number of outputs as variables.
  209.       % Match results in lexicographic order to outputs.
  210.       v = fieldnames(S);
  211.       for j = 1:nvars
  212.          varargout{j} = getfield(S,v{j});
  213.       end
  214.  
  215.    else
  216.       error([int2str(nvars) ' variables does not match ' ...
  217.              int2str(nargout) ' outputs.'])
  218.    end
  219. end
  220.  
  221. %-------------------------
  222.  
  223. function s = trim(s);
  224. %TRIM  TRIM(s) deletes any leading or trailing blanks.
  225. while s(1) == ' ', s(1) = []; end
  226. while s(end) == ' ', s(end) = []; end
  227.  
  228. %-------------------------
  229.  
  230. function c = commas(s)
  231. %COMMAS  COMMAS(s) is true for commas not inside parentheses.
  232. p = cumsum((s == '(') - (s == ')'));
  233. c = (s == ',') & (p == 0);

NEWTON.m

  1. function x=newton(f_name,x0,tol)
  2. % Purpose: Solve a nonlinear equation by Newton iteration
  3. % Synopsis:     x=newton('f_name',x0)      
  4. %               x=newton('f_name',x0,tol)  
  5. %
  6. %            x: return a root of f(x)=0 near x0
  7. %            f_name:name of the function f(x) that defines the equation
  8. %            x0: initial guess
  9. %            tol: tolerence(Default:1e-6)
  10.  
  11. % L.J.Hu 1-8-1998
  12.  
  13. h=0.0001;M=500;
  14. if nargin<3, tol=1e-6;end
  15. x=x0;xb=x-999;
  16. n=0;
  17. while abs(x-xb)>tol 
  18.   xb=x;
  19.   if n>M break;end
  20.   y=feval(f_name, x);
  21.   fprintf('  n=%3.0f,  x=%12.5e,  y=%12.5e, \n',n,x,y)
  22.   y_driv=(feval(f_name,x+h)-y)/h;
  23.   x=xb-y/y_driv;
  24.   n=n+1;
  25. end
  26. fprintf('  n=%3.0f,  x=%12.5e,  y=%12.5e, ',n,x,y)
  27. if n>500,
  28.  fprintf('\n');disp('Warning: iterations exceeds the limitation, probable devergent');
  29. end

哈密尔顿回路

TSP模拟退火

 accept.m

  1. function y=accept(t,df)
  2. y=(df<0)|(((df/t)<88)&(exp(-df/t)>rand(1,1)));

annealing.m

  1. %function R=annealing(N,L,s,t,dt,C,R)
  2. %N为问题规模，即节点个数；L可取较大值，如500、1000；
  3. %s取1、2等；t为初始温度，参考范围为0.5--2；
  4. %dt为衰减因子，一般不小于0.9;
  5. %C为边权矩阵，应是一个强连通图的边权矩阵
  6. %R为初始路径，结果路径也存放在R中
  7. %L、s、t、dt应通过多次试验来确定，以获得优化的结果
  8. %参考《非数值并行算法--模拟退火算法》科学出版社
  9.  
  10. function R=annealing(N,L,s,t,dt,C,R)
  11. s0=0;
  12. while 1
  13.     a=0;
  14.     for k=1:L
  15.         [r,df]=calculate(R,C,N);
  16.         if accept(t,df)
  17.             R=r;a=1;
  18.             disp(cost_sum(R,C,N));
  19.         end
  20.     end
  21.     t=t*dt
  22.     if a==0
  23.         s0=s0+1;
  24.     else 
  25.         s0=0;
  26.     end
  27.     if s0==s
  28.         break;
  29.     end
  30. end

calculate.m

  1. function [r,df]=calculate(R,C,N)
  2. judge=rand(1,1);
  3. if judge<0.5
  4.     r=exchange2(R);
  5.     df=cost_sum(r,C,N)-cost_sum(R,C,N);
  6. else 
  7.     r=exchange3(R);
  8.     df=cost_sum(r,C,N)-cost_sum(R,C,N);
  9. end
  10.

cost_sum.m

  1. function y=cost_sum(x,C,N)
  2. y=0;
  3. for i=1:(N-1)
  4.     y=y+C(x(i),x(i+1));
  5. end
  6. y=y+C(x(N),x(1));
  7.

exchange2.m

  1. function r=exchange2(R)
  2. N=length(R);
  3. I=1+fix(unifrnd(0,N));
  4. J=1+fix(unifrnd(0,N-1));
  5. if I==J
  6.     J=J+1;
  7. end
  8. if J<I
  9.     I=I+J;
  10.     J=I-J;
  11.     I=I-J;
  12. end
  13. r=R;
  14. if J-I~=1&J-I~=N-1
  15.     for p=1:(J-I)
  16.         r(I+p)=R(J-p+1);
  17.     end
  18. end
  19. if J-I==1
  20.     r(I)=R(J);
  21.     r(J)=R(I);
  22. end
  23. if J-I==N-1
  24.     for p=1:(N-2)
  25.         r(p+2)=R(N+1-p)
  26.     end
  27. end

exchange3.m

  1. function r=exchange3(R)
  2. N=length(R);
  3. K=fix(unifrnd(0,N));
  4. J=fix(unifrnd(0,N-1));
  5. I=fix(unifrnd(0,N-2));
  6. if I==J
  7.     J=J+1;
  8. end
  9. if I==K
  10.     K=K+2;
  11. end
  12. if J==K
  13.     K=K+1;
  14. end
  15.  
  16. if I>J
  17.     I=I+J;
  18.     J=I-J;
  19.     I=I-J;
  20. end
  21. if I>K
  22.     I=I+K;
  23.     K=I-K;
  24.     I=I-K;
  25. end
  26. if J>K
  27.     J=J+K;
  28.     K=J-K;
  29.     J=J-K;
  30. end
  31.  
  32.  
  33. r=R;
  34.     if J-I~=1&K-J~=1&K-I~=N-1
  35.         for q=1:(J-I)
  36.             r(I+q)=R(J+1-q);
  37.         end
  38.         for q=1:(K-J)
  39.             r(J+q)=R(K+1-q);
  40.         end
  41.     end
  42.     if J-I==1&K-J==1
  43.         r(K)=R(J);r(J)=R(K);
  44.     end
  45.     if J-I==1&K-J~=1&K-I~=N-1
  46.         for q=1:(K-J)
  47.             r(I+q)=R(I+1+q);
  48.         end
  49.         r(K)=R(J);
  50.     end
  51.     if K-J==1&J-I~=1&K~=N
  52.         for q=1:(J-I)
  53.             r(I+1+q)=R(I+q);
  54.         end
  55.         r(I+1)=R(K);
  56.     end
  57.     if I==1&J==2&K==N
  58.         for q=1:(N-2)
  59.             r(1+q)=R(2+q);
  60.         end
  61.         r(N)=R(2);
  62.     end
  63.     if I==1&J==(N-1)&K==N
  64.         for q=1:(N-2)
  65.             r(q)=R(1+q);
  66.         end
  67.         r(N-1)=R(1);
  68.     end
  69.     if J-I~=1&K-I==N-1
  70.         for q=1:(J-1)
  71.             r(q)=R(1+q);
  72.         end
  73.         r(J)=R(1);
  74.     end
  75.     if J==(N-1)&K==N&J-I~=1
  76.         r(J+1)=R(N);
  77.         for q=1:(N-J-1)
  78.             r(J+1+q)=R(J+q);
  79.         end
  80.     end
  81.

 

 三边交换简单算法

 bianquan.m

  1. N=13;
  2. for i=1:N
  3.     for j=1:N
  4.         C(i,j)=inf;
  5.     end
  6. end
  7. for i=1:N
  8.     C(i,i)=0;
  9. end
  10. C(1,2)=6.0;C(1,13)=12.9;
  11. C(2,3)=5.9;C(2,4)=10.3;
  12. C(3,4)=12.2;C(3,5)=17.6;
  13. C(4,13)=8.8;C(4,7)=7.4;
  14. C(4,5)=11.5;
  15. C(5,2)=17.6;C(5,6)=8.2;
  16. C(6,9)=14.9;C(6,7)=20.3;
  17. C(7,9)=19.0;C(7,8)=7.3;
  18. C(8,9)=8.1;C(8,13)=9.2;
  19. C(9,10)=10.3;
  20. C(10,11)=7.7;
  21. C(11,12)=7.2;
  22. C(12,13)=7.9;
  23. for i=1:N
  24.     for j=1:N
  25.         if  C(i,j) < inf
  26.             C(j,i)=C(i,j);
  27.         end
  28.     end
  29. end
  30. for i=1:N
  31.     C(i,i)=0;
  32. end
  33.  
  34. R=[4 7 6 5 3 2 1 13 12 11 10 9 8];

cost_sum.m

  1. function y=cost_sum(x,C,N)
  2. y=0;
  3. for i=1:(N-1)
  4.     y=y+C(x(i),x(i+1));
  5. end
  6. y=y+C(x(N),x(1));
  7.

jiaohuan3.m

  1. R=1:N;
  2.  
  3. n=0;
  4. for I=1:(N-2)
  5.     for J=(I+1):(N-1)
  6.         for K=(J+1):N
  7.             n=n+1;
  8.             Z(n,:)=[I J K];
  9.         end
  10.     end
  11. end
  12.  
  13. for m=1:(N*(N-1)*(N-2)/6)
  14.     I=Z(m,1);J=Z(m,2);K=Z(m,3);
  15.     r=R;
  16.     if J-I~=1&K-J~=1&K-I~=N-1
  17.         for q=1:(J-I)
  18.             r(I+q)=R(J+1-q);
  19.         end
  20.         for q=1:(K-J)
  21.             r(J+q)=R(K+1-q);
  22.         end
  23.     end
  24.     if J-I==1&K-J==1
  25.         r(K)=R(J);r(J)=R(K);
  26.     end
  27.     if J-I==1&K-J~=1&K-I~=N-1
  28.         for q=1:(K-J)
  29.             r(I+q)=R(I+1+q);
  30.         end
  31.         r(K)=R(J);
  32.     end
  33.     if K-J==1&J-I~=1&K~=N
  34.         for q=1:(J-I)
  35.             r(I+1+q)=R(I+q);
  36.         end
  37.         r(I+1)=R(K);
  38.     end
  39.     if I==1&J==2&K==N
  40.         for q=1:(N-2)
  41.             r(1+q)=R(2+q);
  42.         end
  43.         r(N)=R(2);
  44.     end
  45.     if I==1&J==(N-1)&K==N
  46.         for q=1:(N-2)
  47.             r(q)=R(1+q);
  48.         end
  49.         r(N-1)=R(1);
  50.     end
  51.     if J-I~=1&K-I==N-1
  52.         for q=1:(J-1)
  53.             r(q)=R(1+q);
  54.         end
  55.         r(J)=R(1);
  56.     end
  57.     if J==(N-1)&K==N&J-I~=1
  58.         r(J+1)=R(N);
  59.         for q=1:(N-J-1)
  60.             r(J+1+q)=R(J+q);
  61.         end
  62.     end
  63.     if cost_sum(r,C,N)<cost_sum(R,C,N)
  64.         R=r
  65.     end
  66. end
  67. fprintf('总长为%f\n',cost_sum(R,C,N))

        提供一种求解最优哈密尔顿的算法---三边交换调整法，要求在运行jiaohuan3（三交换法）之前，给定邻接矩阵C和节点个数N，结果路径存放于R中。
    
        bianquan.m文件给出了一个参数实例，可在命令窗口中输入bianquan,得到邻接矩阵C和节点个数N以及一个任意给出的路径R,，回车后再输入jiaohuan3,得到了最优解。
  
        由于没有经过大量的实验，又是近似算法，对于网络比较复杂的情况，可以尝试多运行几次jiaohuan3,看是否能到进一步的优化结果。

画等温线

dengwen.m

  1. clear;close all;
  2. fphn=fopen('hunan.txt','r');
  3. hnb=fgetl(fphn);
  4. hnmap=fscanf(fphn,'%f %f',[2,59]); % It has 59 rows now.湖南省界经纬度
  5. fclose(fphn);
  6. hnmap=hnmap';
  7. xa=hnmap(:,[1]);
  8. ya=hnmap(:,[2]);
  9.  
  10. fp=fopen('LATLON57.txt','r');
  11. LL57=fscanf(fp,'%d %f %f',[3,97]); % It has 97 rows now.湖南省97县名称号码，经纬度
  12. fclose(fp);
  13. LL57=LL57';
  14. x=LL57(:,[3])/10;
  15. y=LL57(:,[2])/10;
  16.  
  17.  
  18. fpy=fopen('etw00100.txt','r');
  19. ymd57=fscanf(fpy,'%d',[3,1]);%实在不懂这句是什么意思，并且后面也没有用到这个变量
  20. yu97=fscanf(fpy,'%d %f %f',[3,97]); % It has 97 rows now.湖南省97县温度
  21. fclose(fpy);
  22. yu97=yu97';
  23. z=yu97(:,[2]);%湖南省97县温度
  24.  
  25. hold on;
  26. plot(xa,ya,'.','markersize',5,'color','red');%湖南省界
  27.  
  28. plot(x,y,'.','markersize',6);%湖南省97县位置
  29. [xi,yi]=meshgrid(linspace(min(x),max(x),25),linspace(min(y),max(y),25));
  30. zi=griddata(x,y,z,xi,yi,'cubic');%湖南省97县温度立方插值
  31. hold on;[c,h]=contour(xi,yi,zi,'b-');%温度等值线，注意是平面的等值线
  32. clabel(c,h);hold off;
  1. %三个txt数据文件的意义及其数据格式举例
  2. %97县名称号码、纬度、经度
  3.  
  4. %LATLON57.txt：
  5. 57554 294 1101
  6. 57562 296 1114
  7. 57574 294 1124
  8. 57584 294 1130
  9. 57682 287 1136
  10. 57662 289 1116
  11. 57671 289 1124
  12. 57687 282 1128
  13. 57649 283 1097
  14. 57669 284 1112
  15. 57655 258 1104
  16. 57745 275 1096
  17. 57774 275 1122
  18. 57766 273 1114
  19. 57776 273 1128
  20. 57872 269 1126
  21. 57853 267 1106
  22. 57845 262 1098
  23. 57866 262 1116
  24. 57972 258 1130
  25. 57965 255 1116
  26. 57544 295 1095
  27. 57643 290 1098
  28. 57642 287 1096
  29. 57640 286 1095
  30. 57646 286 1100
  31. 57657 283 1102
  32. 57740 280 1096
  33. 57558 291 1105
  34. 57564 294 1111
  35. 57565 297 1118
  36. 57566 295 1117
  37. 57577 294 1122
  38. 57661 289 1115
  39. 57663 289 1120
  40. 57674 286 1124
  41. 57666 285 1121
  42. 57575 295 1126
  43. 57680 288 1131
  44. 57585 295 1135
  45. 57673 287 1129
  46. 57679 282 1130
  47. 57678 283 1126
  48. 57688 282 1126
  49. 57773 279 1128
  50. 57771 279 1125
  51. 57772 278 1125
  52. 57780 279 1132
  53. 57781 277 1135
  54. 57779 270 1134
  55. 57882 268 1136
  56. 57886 265 1138
  57. 57746 276 1100
  58. 57658 280 1102
  59. 57743 279 1098
  60. 57752 279 1106
  61. 57744 274 1092
  62. 57754 274 1102
  63. 57842 269 1097
  64. 57841 266 1097
  65. 57763 277 1120
  66. 57761 278 1113
  67. 57760 277 1114
  68. 57762 277 1117
  69. 57860 270 1113
  70. 57768 273 1115
  71. 57769 272 1117
  72. 57758 271 1116
  73. 57767 271 1120
  74. 57846 266 1102
  75. 57857 264 1103
  76. 57851 265 1109
  77. 57871 270 1124
  78. 57777 272 1129
  79. 57778 271 1130
  80. 57875 269 1125
  81. 57870 268 1121
  82. 57874 264 1124
  83. 57876 264 1129
  84. 57868 266 1119
  85. 57867 264 1113
  86. 57962 260 1117
  87. 57971 259 1122
  88. 57966 256 1120
  89. 57975 254 1122
  90. 57969 253 1113
  91. 59063 252 1115
  92. 57881 267 1133
  93. 57887 261 1131
  94. 57889 261 1140
  95. 57981 260 1134
  96. 57973 258 1127
  97. 57974 256 1124
  98. 57978 253 1126
  99. 57976 254 1129
  100. 57985 256 1137
  101. 57865 264 1116
  102.  
  103. %边界经纬度
  104. %hunan.txt：
  105. 59
  106. 113.5 29.8
  107. 113.7 29.5
  108. 113.7 29.1
  109. 113.8 29.1
  110. 114.1 28.8
  111. 114.1 28.5
  112. 114.3 28.3
  113. 113.6 27.5
  114. 113.7 27.3
  115. 113.9 27.3
  116. 113.9 26.7
  117. 114.1 26.6
  118. 113.9 26.2
  119. 114.2 26.2
  120. 114.2 26.1
  121. 114.0 26.0
  122. 114.0 25.6
  123. 113.8 25.4
  124. 113.3 25.5
  125. 112.9 25.4
  126. 112.9 25.3
  127. 113.0 25.3
  128. 113.0 25.0
  129. 112.8 25.0
  130. 112.7 25.2
  131. 112.2 25.3
  132. 112.2 24.8
  133. 111.5 24.6
  134. 111.4 25.2
  135. 111.3 25.2
  136. 111.2 25.0
  137. 111.1 25.0
  138. 111.1 25.2
  139. 111.4 25.6
  140. 111.3 26.3
  141. 110.8 26.3
  142. 110.4 26.0
  143. 110.2 26.1
  144. 110.2 26.3
  145. 110.0 25.9
  146. 109.8 25.9
  147. 109.8 26.0
  148. 109.4 26.3
  149. 109.6 27.1
  150. 109.0 27.0
  151. 108.9 27.1
  152. 109.4 27.6
  153. 109.3 27.8
  154. 109.2 29.1
  155. 109.2 29.5
  156. 109.9 29.8
  157. 110.3 29.8
  158. 110.5 29.7
  159. 110.7 29.8
  160. 110.6 30.1
  161. 111.2 30.0
  162. 111.9 29.8
  163. 112.3 29.5
  164. 112.8 29.8
  165.  
  166. %97县温度最高值、最低值
  167. %etw00100.txt：
  168. 2001 8 1
  169. 57554 22.6 12.0
  170. 57562 24.3 10.6
  171. 57574 24.7 10.0
  172. 57584 26.2 19.2
  173. 57682 25.2 4.2
  174. 57662 24.7 9.8
  175. 57671 24.4 0.4
  176. 57687 25.0 16.5
  177. 57649 22.2 46.2
  178. 57669 21.7 2.7
  179. 57655 22.7 23.8
  180. 57745 23.3 2.8
  181. 57774 24.9 0.0
  182. 57766 25.5 0
  183. 57776 17.8 0.2
  184. 57872 25.7 0.1
  185. 57853 24.7 1.1
  186. 57845 23.5 1.1
  187. 57866 25.0 5.1
  188. 57972 24.3 37.7
  189. 57965 24.6 22.0
  190. 57544 22.1 59.5
  191. 57643 22.5 43.6
  192. 57642 23.6 49.8
  193. 57640 23.2 33.4
  194. 57646 23.7 27.9
  195. 57657 23.1 13.7
  196. 57740 22.0 13.5
  197. 57558 22.7 20.5
  198. 57564 24.0 22.0
  199. 57565 24.8 4.2
  200. 57566 0 16.1
  201. 57577 0 0.9
  202. 57661 23.3 6.7
  203. 57663 23.9 0.9
  204. 57674 25.3 0.0
  205. 57666 23.9 0.0
  206. 57575 0 8.4
  207. 57680 0 0.2
  208. 57585 23.6 13.7
  209. 57673 25.0 0.3
  210. 57679 0 0
  211. 57678 24.4 0.1
  212. 57688 26.8 0.0
  213. 57773 25.2 2.1
  214. 57771 24.8 2.9
  215. 57772 25.4 1.3
  216. 57780 25.1 1.3
  217. 57781 25.8 0.0
  218. 57779 25.4 0.0
  219. 57882 25.8 0.1
  220. 57886 23.2 53.6
  221. 57746 23.4 6.5
  222. 57658 23.1 14.5
  223. 57743 0 8.2
  224. 57752 22.3 21.6
  225. 57744 24.0 0.8
  226. 57754 24.0 0.9
  227. 57842 25.1 0.2
  228. 57841 24.2 2.1
  229. 57763 25.2 0.0
  230. 57761 0 0.0
  231. 57760 23.6 0.0
  232. 57762 24.3 0.6
  233. 57860 24.5 5.0
  234. 57768 0 0.0
  235. 57769 24.7 0.0
  236. 57758 25.2 0.2
  237. 57767 25.4 0.3
  238. 57846 24.4 0.0
  239. 57857 23.8 0.1
  240. 57851 24.6 9.3
  241. 57871 25.5 0.9
  242. 57777 25.4 0.0
  243. 57778 0 0.0
  244. 57875 0 0.0
  245. 57870 25.0 15.5
  246. 57874 25.2 0.4
  247. 57876 24.4 74.2
  248. 57868 26.5 7.0
  249. 57867 25.0 42.4
  250. 57962 25.1 56.0
  251. 57971 24.3 16.1
  252. 57966 24.2 57.6
  253. 57975 24.0 9.2
  254. 57969 23.4 39.0
  255. 59063 23.6 18.2
  256. 57881 25.1 9.4
  257. 57887 23.9 44.2
  258. 57889 22.0 17.4
  259. 57981 23.9 0
  260. 57973 23.6 84.6
  261. 57974 24.2 37.0
  262. 57978 23.9 0
  263. 57976 23.7 23.2
  264. 57985 22.0 21.0
  265. 57865 0 57.4


 

 

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