生命游戏
游戏的宇宙是无限可扩展的二维矩形网格，群体是那些标注为存活的网格的集合。群体可以依照称为代的离散时间步距进化。在每一步中，每个网格的命运由它周围最近的8个网格邻居的活度决定，规则如下：
如果一个存活的网格有两个（或三个）存活的邻居，或者一个死亡的网格周围有三个存活的邻居，则它在下一步中也是存活的。
滑翔机
滑翔机五网格群体初始结构，每一步进化过程中，都有两个网格死亡，并有两个新网格出生。
访问周围网格，并计算存活数目
n = size(X,1);p = [1 1:n-1];q = [2:n n];% Count how many of the eight neighbors are alive.Y = X(:,p) + X(:,q) + X(p,:) + X(q,:) + ...X(p,p) + X(q,q) + X(p,q) + X(q,p);% A live cell with two live neighbors, or any cell with% three live neigbhors, is alive at the next step.X = (X & (Y == 2)) | (Y == 3);[loop,buttons] = query_buttons(buttons,pop);t = t + 1;
产生随机的初始群体
% Generate a random initial populationX = sparse(50,50);X(21:30,21:30) = (rand(10,10) > .75);p0 = nnz(X);
循环100代
% Loop over 100 generations.for t = 1:100spy(X)title(num2str(t))drawnow% Whether cells stay alive, die, or generate new cells depends% upon how many of their eight possible neighbors are alive.% Index vectors increase or decrease the centered index by one.n = size(X,1);p = [1 1:n-1];q = [2:n n];% Count how many of the eight neighbors are alive.Y = X(:,p) + X(:,q) + X(p,:) + X(q,:) + ...X(p,p) + X(q,q) + X(p,q) + X(q,p);% A live cell with two live neighbors, or any cell with% three live neigbhors, is alive at the next step.X = (X & (Y == 2)) | (Y == 3);endp100 = nnz(X);fprintf('%5d %5d %8.3f\n',p0,p100,p100/p0)
循环
function lifex(varargin)lex = open_lex('lexicon.txt');
pop = population(lex,varargin{:});repeat = true;
while repeat% Place the initial population in the universe, X.[lex,pop] = read_lexicon(lex,pop);X = populate(pop);[plothandle,buttons] = initial_plot(size(X),pop);title(pop.name)t = 0;loop = true;while loop% Expand the universe if necessary to avoid the boundary.X = expand(X,pop);% Update the plot.[i,j] = find(X);set(plothandle,'xdata',j,'ydata',i);caption(t,nnz(X))drawnow% Whether cells stay alive, die, or generate new cells depends% upon how many of their eight possible neighbors are alive.% Index vectors increase or decrease the centered index by one.n = size(X,1);p = [1 1:n-1];q = [2:n n];% Count how many of the eight neighbors are alive.Y = X(:,p) + X(:,q) + X(p,:) + X(q,:) + ...X(p,p) + X(q,q) + X(p,q) + X(q,p);% A live cell with two live neighbors, or any cell with% three live neigbhors, is alive at the next step.X = (X & (Y == 2)) | (Y == 3);[loop,buttons] = query_buttons(buttons,pop);t = t + 1;end[repeat,pop] = what_next(buttons,lex,pop);end
fclose(lex.fid);
set(buttons(1:6),'vis','off')
set(buttons(7),'value',0,'string','close','call','close(gcf)')% ------------------------function lex = open_lex(filename)
% lex = file_open(filename)
% lex.fid = file identifier
% lex.len = number of entries
% lex.index = index of current entrylex.fid = fopen(filename);
if lex.fid < 3error(['Cannot open "' filename '"'])
end
% Count number of usable entries,
lex.index = 0;
lex.len = 0;
while ~feof(lex.fid)% Look for a line with two colons, ':name:'.line = fgetl(lex.fid);if sum(line == ':') >= 2% name = line(2:find(line(2:end) == ':',1));% Look for an empty line or a line starting with a tab.tab = char(9);task = [tab '*'];tdot = [tab '.'];while ~feof(lex.fid)line = fgetl(lex.fid);if isempty(line)breakelseif strncmp(line(1:2),task,2) || strncmp(line(1:2),tdot,2)lex.len = lex.len + 1;% fprintf('%d: %s\n',lex.len,name)breakendendend
end
frewind(lex.fid);% ------------------------function pop = population(varargin)
% pop = population(varargin)
% pop.index = index within lexicon of population
% pop.name = name of population
% pop.all = logical flag for slide show of populations
% pop.b = border width, default = 20
% pop.S = sparse matrix representation of populationlex = varargin{1};
pop.index = 0;
pop.name = ' ';
pop.all = false;
pop.b = 20;
pop.S = [];
if nargin < 2pop.index = ceil(rand*lex.len);
elseif ischar(varargin{2}) && isequal(varargin{2},'all')pop.all = true;pop.b = 10;
elseif ischar(varargin{2})pop.name = varargin{2};pop.index = -1;
elseif min(size(varargin{2})) > 1pop.S = sparse(varargin{2});pop.name = sprintf('my %d-by-%d matrix',size(pop.S));
elsepop.index = varargin{2};
end
if nargin == 3pop.b = varargin{3};
end% ------------------------function [lex,pop] = read_lexicon(lex,pop)
% [lex,pop] = read_lexicon(lex,pop)
% Update lex and pop to new populationif pop.all || pop.index > lex.lenpop.index = mod(pop.index,lex.len)+1;
end
if pop.index < lex.indexfrewind(lex.fid)lex.index = 0;
end
while lex.index ~= pop.index% Look for a line with two colons, ':name:'.line = fgetl(lex.fid);if sum(line == ':') >= 2name = line(2:find(line(2:end) == ':',1));% Look for an empty line or a line starting with a tab.tab = char(9);task = [tab '*'];tdot = [tab '.'];while ~feof(lex.fid) && lex.index <= lex.lenline = fgetl(lex.fid);if isempty(line)breakelseif strncmp(line(1:2),task,2) || strncmp(line(1:2),tdot,2)lex.index = lex.index + 1;if lex.index == pop.index || ...strncmpi(name,pop.name,length(pop.name))pop.index = lex.index;if pop.allpop.name = [name ', index = ' int2str(pop.index)];elsepop.name = name;end% Form sparse matrix by rows from '.' and '*'.S = sparse(0,0);m = 0;while ~isempty(line) && (line(1)==tab)row = sparse(line(2:end) == '*');m = m+1;n = length(row);S(m,n) = 0;S(m,1:n) = row;line = fgetl(lex.fid);endpop.S = S;elseif lex.index == lex.lenerror('Population name is not in lexicon.')endbreakendendend
end% ------------------------function [plothandle,buttons] = initial_plot(sizex,pop)
% [plothandle,buttons] = initial_plot(size(X),pop)
% plothandle = handle to customized "spy" plot
% buttons = array of handles to toggle buttonsclf
shg
m = max(sizex);
ms = max(10-ceil(m/10),2);
plothandle = plot(0,0,'o','markersize',ms,'markerfacecolor','blue');
set(gca,'xlim',[0.5 m+0.5],'ylim',[0.5 m+0.5],'ydir','rev', ...'xtick',[],'ytick',[],'plotboxaspectratio',[m+2 m+2 1])
buttons = zeros(7,1);
bstrings = {'step','slow','fast','redo','next','random','quit'};
for k = 1:7buttons(k) = uicontrol('style','toggle','units','normal', ...'position',[.10+.12*(k-1) .005 .10 .045],'string',bstrings{k});
end
set(buttons(1),'userdata',0);
if pop.all, set(buttons(1:6),'vis','off'), end% ------------------------function X = populate(pop)
% X = populate(pop);
% X = sparse matrix universe with centered initial population[p,q] = size(pop.S);
n = max(p,q) + 2*pop.b;
X = sparse(n,n);
i = floor((n-p)/2)+(1:p);
j = floor((n-q)/2)+(1:q);
X(i,j) = pop.S;% ------------------------function X = expand(X,pop)
% X = expand(X);
% Expand size if necessary to keep zeros around the boundary.
% Border width b avoids doing this every time step.
n = size(X,1);
b = max(pop.b,1);
if any(X(:,n-1) ~= 0) || any(X(n-1,:) ~= 0)X = [X sparse(n,b); sparse(b,n+b)];n = n + b;
end
if any(X(2,:) ~= 0) || any(X(:,2) ~= 0)X = [sparse(b,n+b); sparse(n,b) X];xlim = get(gca,'xlim')+b;ylim = get(gca,'ylim')+b;set(gca,'xlim',xlim,'ylim',ylim)
end% ------------------------function [loop,buttons] = query_buttons(buttons,pop)
% [loop,buttons] = query_buttons(buttons);
% loop = true: continue time stepping
% loop = false: restartif pop.allpause(1)loop = false;
elsebv = cell2mat(get(buttons,'value'));bk = get(buttons(1),'userdata');if bk == 0 || sum(bv==1) ~= 1while all(bv == 0)drawnowbv = cell2mat(get(buttons,'value'));endif sum(bv==1) > 1bv(bk) = 0;endbk = find(bv == 1);endset(buttons([1:bk-1 bk+1:7]),'value',0)switch bkcase 1 % stepset(buttons(1),'value',0);bk = 0;loop = true;case 2 % slowpause(.5)loop = true;case 3 % fastpause(.05)loop = true;otherwise, loop = false;end% Remember button numberset(buttons(1),'userdata',bk);
end% ------------------------function [repeat,pop] = what_next(buttons,lex,pop)
% [next,pop] = what_next(buttons,lex,pop);
% repeat = true: start with a new population
% repeat = false: exitbv = cell2mat(get(buttons,'value'));
bk = find(bv == 1);
set(buttons,'value',0)
repeat = true;
if ~isempty(bk)switch bkcase 4 % redopop.index = lex.index;case 5 % nextpop.index = mod(lex.index,lex.len)+1;case 6 % randompop.index = ceil(rand*lex.len);case 7 % quitrepeat = false;end
end% ------------------------function caption(t,nnz)
% caption(t,nnz(X))
% Print time step count and population size on the x-label.
s = sprintf('t=%3d, pop=%3d',t,nnz);
fs = get(0,'defaulttextfontsize')+2;
xlabel(s,'fontname','courier','fontsize',fs);
曼德勃罗集
曼德勃罗集是一个包含z0值的复平面区域，其轨迹定义为
z0=0.25-0.54i z=0 z=z^2+z0
数组运算
%% Define the region.定义复平面域的分割方式x = 0: 0.05: 0.8;y = x';%% Create the two-dimensional complex grid using Tony's indexing trick.n = length(x);e = ones(n,1);z0 = x(e,:) + i*y(:,e);%% Or, do the same thing with meshgrid.[X,Y] = meshgrid(x,y);z0 = X + i*Y;%% Initialize the iterates and counts arrays.z = zeros(n,n);c = zeros(n,n);%% Here is the Mandelbrot iteration.depth = 32;for k = 1:depth %循环次数z = z.^3 + z0;c(abs(z) < 2) = k;end%% Create an image from the counts.c %直接显示矩阵cimage(c)axis image%% Colorscolormap(flipud(jet(depth)))
