原文地址：http://www.bfcat.com/index.php/2012/11/speed-up-app/
这篇文章原文是matlab网站上的，我把它翻译过来同时自己也学习一下。原文见这里
这篇文章主要使用到了如下几种加速方法：
这篇文章原文是matlab网站上的，我把它翻译过来同时自己也学习一下。原文见这里
这篇文章主要使用到了如下几种加速方法：
- 预分配空间
- 向量化
- 移除重复运算
我们要加速的程序是这样的。代码首先生成一个 x1 x2为横纵坐标的2D网格. 这个程序是要循环遍历所有初始和终止点的组合。给定一组位置，程序计算一个指数，如果这个指数小于阈值gausThresh，那么这个值就用于计算 out. subs变量保存所有点的位置坐标。
最初的程序如下：
|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 |
% Initialize the grid and initial and final points
nx1 = 10; nx2 = 10; x1l = 0; x1u = 100; x2l = 0; x2u = 100; x1 = linspace(x1l,x1u,nx1+1); x2 = linspace(x2l,x2u,nx2+1); limsf1 = 1:nx1+1; limsf2 = 1:nx2+1; % Initalize other variables t = 1; sigmax1 = 0.5; sigmax2 = 1; sigma = t * [sigmax1^2 0; 0 sigmax2^2]; invSig = inv(sigma); detSig = det(sigma); expF = [1 0; 0 1]; n = size (expF, 1); gausThresh = 10; small = 0; subs = []; vals = []; % Iterate through all possible initial % and final positions and calculate % the values of exponent and out % if exponent > gausThresh. for i1 = 1:nx1+1 for i2 = 1:nx2+1 for f1 = limsf1 for f2 = limsf2 % Initial and final position xi = [x1(i1) x2(i2)]'; xf = [x1(f1) x2(f2)]'; exponent = 0.5 * (xf - expF * xi)'... * invSig * (xf - expF * xi); if exponent > gausThresh small = small + 1; else out = 1 / (sqrt((2 * pi)^n * detSig))... * exp(-exponent); subs = [subs; i1 i2 f1 f2]; vals = [vals; out]; end end end end end |
下面是一个图形表示的可视化（nx1=nx2=100）. 因为数据非常稠密，所以我们只显示其中一部分。红线链接的就是指数计算结果小于阈值的部分，线的粗细反应了数值大小。
这个程序在一个T60 Lenovo dual-core laptop 上面，开启了多线程以后，运行时间如下
|
1
2 3 4 |
displayRunTimes(1)
nx1 nx2 time 50 50 296 seconds 100 100 9826 seconds |
第一步，也是最简单的一步就是根据 M-Lint 的建议修改, 这是matlab自带的静态代码分析工具。在editor里面就可以看到这些建议内容，不过也可以自己写一个函数让建议内容更加集中的显示一下。例如
|
1
2 |
output = mlint('initial.m');
displayMlint(output); |
'subs' might be growing inside a loop. Consider preallocating for speed.
'vals' might be growing inside a loop. Consider preallocating for speed.
根据建议，第一步是要给一些数组预分配空间。这样做是因为matlab使用的内存中连续的块，因此，如果在循环里不断改变数组大小，matlab就要不断寻找合适大小的内存片段并把数据移动过去。如果分配了一个比较大的空间，matlab就可以一直在这个连续空间工作。
但是，目前我们不知道 subs和vars的个数，如果要预分配，我们得分配最大可能的空间。那就是100^4，我们来试试：
|
1
2 3 4 5 |
try
zeros(1,100^4) catch ME end display(ME.message) |
Out of memory. Type HELP MEMORY for your options.
看来不行啊。太大了。
因此，我们只能通过分块分配空间来实现，每次分配一个可以接受的大小，并设置一个计数器，当这块空间满了的时候，再分配一个块。这样，内存移动的次数大大得到了降低。
（bfcat注： 这种不知道数组大小的时候，还有一个方法就是使用cell。我没有仔细分析cell的原理，但是我觉得它像是一个链表，因此cell里面的每一个元素不需要在连续的内存空间。因此，当我们执行类似 M{end+1} = m 的时候，matlab也不需要将M 中已有的元素都拷贝一次。这样，虽然Mlint还会提示让我们为cell预先分配空间，但是没关系，不分配对速度影响也不大。当循环结束以后，执行类似 M = cat(1, M{:}) 这样的语句就可以将其变回数组了。）
预分配空间后的代码|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 |
% Initialization
nx1 = 10; nx2 = 10; [x1,x2,limsf1,limsf2,expF,gausThresh,small,invSig,detSig,n]= ... initialize(nx1,nx2); % Initial guess for preallocation mm = min((nx1+1)^2*(nx2+1)^2, 10^6); subs = zeros(mm,4); vals = zeros(mm,1); counter = 0; % Iterate through all possible initial % and final positions for i1 = 1:nx1+1 for i2 = 1:nx2+1 for f1 = limsf1 for f2 = limsf2 xi = [x1(i1) x2(i2)]'; %% Initial position xf = [x1(f1) x2(f2)]'; %% Final position exponent = 0.5 * (xf - expF * xi)'... * invSig * (xf - expF * xi); % Increase preallocation if necessary if counter == length(vals) subs = [subs; zeros(mm, 4)]; vals = [vals; zeros(mm, 1)]; end if exponent > gausThresh small = small + 1; else % Counter introduced counter=counter + 1; out = 1 / (sqrt((2 * pi)^n * detSig))... * exp(-exponent); subs(counter,:) = [i1 i2 f1 f2]; vals(counter) = out; end end end end end % Remove zero components that came from preallocation vals = vals(vals > 0); subs = subs(vals > 0); |
|
1
2 3 4 |
displayRunTimes(2)
nx1 nx2 time 50 50 267 seconds 100 100 4228 seconds |
运行速度变快了一些，但是还是不够理想。
向量化因为matlab是基于矩阵的语言，因此，我们最好尽量用向量代替循环。尤其是多重循环嵌套的时候更要注意速度问题。对于这个代码，我们主要进行以下两种改动
- 向量化里面的两个循环
- 向量化里面的三个循环
尝试1: 向量化两个内循环
|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 |
% Initialization
nx1 = 10; nx2 = 10; [x1,x2,limsf1,limsf2,expF,gausThresh,small,invSig,detSig,n]= ... initialize(nx1,nx2); vals = cell(nx1+1,nx2+1); % Cell preallocation subs = cell(nx1+1,nx2+1); % Cell preallocation [xind,yind] = meshgrid(limsf1,limsf2); xyindices = [xind(:)' ; yind(:)']; [x,y] = meshgrid(x1(limsf1),x2(limsf2)); xyfinal = [x(:)' ; y(:)']; exptotal = zeros(length(xyfinal),1); % Loop over all possible combinations of positions for i1 = 1:nx1+1 for i2 = 1:nx2+1 xyinitial = repmat([x1(i1);x2(i2)],1,length(xyfinal)); expa = 0.5 * (xyfinal - expF * xyinitial); expb = invSig * (xyfinal - expF * xyinitial); exptotal(:,1) = expa(1,:).*expb(1,:)+expa(2,:).*expb(2,:); index = find(exptotal < gausThresh); expreduced = exptotal(exptotal < gausThresh); out = 1 / (sqrt((2 * pi)^n * detSig)) * exp(-(expreduced)); vals{i1,i2} = out; subs{i1,i2} = [i1*ones(1,length(index)) ; ... i2*ones(1,length(index)); xyindices(1,index); ... xyindices(2,index)]' ; end end % Reshape and convert output so it is in a % simple matrix format vals = cell2mat(vals(:)); subs = cell2mat(subs(:)); small = ((nx1+1)^2*(nx2+1)^2)-length(subs); |
这个向量化效果非常明显
|
1
2 3 4 |
displayRunTimes(3)
nx1 nx2 time 50 50 1.51 seconds 100 100 19.28 seconds |
这里主要使用了下面几个向量化的手段
可以参考这个页面获得更多向量化技巧：http://www.mathworks.com/help/matlab/matlab_prog/techniques-for-improving-performance.html?s_tid=doc_12b#f8-783589
尝试2: 向量化三个内循环
|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 |
% Initialization
nx1 = 10; nx2 = 10; [x1,x2,limsf1,limsf2,expF,gausThresh,small,invSig,detSig,n]= ... initialize(nx1,nx2); limsi1 = limsf1; limsi2 = limsf2; % ndgrid gives a matrix of all the possible combinations [aind,bind,cind] = ndgrid(limsi2,limsf1,limsf2); [a,b,c] = ndgrid(x2,x1,x2); vals = cell(nx1+1,nx2+1); % Cell preallocation subs = cell(nx1+1,nx2+1); % Cell preallocation % Convert grids to single vector to use in a single loop b = b(:); aind = aind(:); bind = bind(:); cind = cind(:); expac = a(:)-c(:); % Calculate x2-x1 % Iterate through initial x1 positions (i1) for i1 = limsi1 exbx1= b-x1(i1); expaux = invSig(2)*exbx1.*expac; exponent = 0.5*(invSig(1)*exbx1.*exbx1+expaux); index = find(exponent < gausThresh); expreduced = exponent(exponent < gausThresh); vals{i1} = 1 / (sqrt((2 * pi)^n * detSig))... .*exp(-expreduced); subs{i1} = [i1*ones(1,length(index)); aind(index)' ; bind(index)';... cind(index)']'; end vals = cell2mat(vals(:)); subs = cell2mat(subs(:)); small = ((nx1+1)^2*(nx2+1)^2)-length(subs); |
现在运行时间更短了：
|
1
2 3 4 |
displayRunTimes(4)
nx1 nx2 time 50 50 0.658 seconds 100 100 8.77 seconds |
|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 |
nx1 = 100; nx2 = 100;
x1l = 0; x1u = 100; x2l = 0; x2u = 100; x1 = linspace(x1l,x1u,nx1+1); x2 = linspace(x2l,x2u,nx2+1); limsi1 = 1:nx1+1; limsi2 = 1:nx2+1; limsf1 = 1:nx1+1; limsf2 = 1:nx2+1; t = 1; sigmax1 = 0.5; sigmax2 = 1; sigma = t * [sigmax1^2 sigmax2^2]; detSig = sigma(1)*sigma(2); invSig = [1/sigma(1) 1/sigma(2)]; gausThresh = 10; n=3; const=1 / (sqrt((2 * pi)^n * detSig)); % ndgrid gives a matrix of all the possible combinations % of position, except limsi1 which we iterate over [aind,bind,cind] = ndgrid(limsi2,limsf1,limsf2); [a,b,c] = ndgrid(x2,x1,x2); vals = cell(nx1+1,nx2+1); % Cell preallocation subs = cell(nx1+1,nx2+1); % Cell preallocation % Convert grids to single vector to % use in a single for-loop b = b(:); aind = aind(:); bind = bind(:); cind = cind(:); expac= a(:)-c(:); expaux = invSig(2)*expac.*expac; % Iterate through initial x1 positions for i1 = limsi1 expbx1= b-x1(i1); exponent = 0.5*(invSig(1)*expbx1.*expbx1+expaux); % Find indices where exponent < gausThresh index = find(exponent < gausThresh); % Find and keep values where exp < gausThresh expreduced = exponent(exponent < gausThresh); vals{i1} = const.*exp(-expreduced); subs{i1} = [i1*ones(1,length(index)); aind(index)' ; bind(index)';... cind(index)']'; end vals = cell2mat(vals(:)); subs = cell2mat(subs(:)); small = ((nx1+1)^2*(nx2+1)^2)-length(subs); |
最终的运行时间
|
1
2 3 4 |
displayRunTimes(5)
nx1 nx2 time 50 50 0.568 seconds 100 100 8.36 seconds |
- 预分配空间
- 向量化
- 移除重复运算
我们要加速的程序是这样的。代码首先生成一个 x1 x2为横纵坐标的2D网格. 这个程序是要循环遍历所有初始和终止点的组合。给定一组位置，程序计算一个指数，如果这个指数小于阈值gausThresh，那么这个值就用于计算 out. subs变量保存所有点的位置坐标。
最初的程序如下：
|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 |
% Initialize the grid and initial and final points
nx1 = 10; nx2 = 10; x1l = 0; x1u = 100; x2l = 0; x2u = 100; x1 = linspace(x1l,x1u,nx1+1); x2 = linspace(x2l,x2u,nx2+1); limsf1 = 1:nx1+1; limsf2 = 1:nx2+1; % Initalize other variables t = 1; sigmax1 = 0.5; sigmax2 = 1; sigma = t * [sigmax1^2 0; 0 sigmax2^2]; invSig = inv(sigma); detSig = det(sigma); expF = [1 0; 0 1]; n = size (expF, 1); gausThresh = 10; small = 0; subs = []; vals = []; % Iterate through all possible initial % and final positions and calculate % the values of exponent and out % if exponent > gausThresh. for i1 = 1:nx1+1 for i2 = 1:nx2+1 for f1 = limsf1 for f2 = limsf2 % Initial and final position xi = [x1(i1) x2(i2)]'; xf = [x1(f1) x2(f2)]'; exponent = 0.5 * (xf - expF * xi)'... * invSig * (xf - expF * xi); if exponent > gausThresh small = small + 1; else out = 1 / (sqrt((2 * pi)^n * detSig))... * exp(-exponent); subs = [subs; i1 i2 f1 f2]; vals = [vals; out]; end end end end end |
下面是一个图形表示的可视化（nx1=nx2=100）. 因为数据非常稠密，所以我们只显示其中一部分。红线链接的就是指数计算结果小于阈值的部分，线的粗细反应了数值大小。
这个程序在一个T60 Lenovo dual-core laptop 上面，开启了多线程以后，运行时间如下
|
1
2 3 4 |
displayRunTimes(1)
nx1 nx2 time 50 50 296 seconds 100 100 9826 seconds |
第一步，也是最简单的一步就是根据 M-Lint 的建议修改, 这是matlab自带的静态代码分析工具。在editor里面就可以看到这些建议内容，不过也可以自己写一个函数让建议内容更加集中的显示一下。例如
|
1
2 |
output = mlint('initial.m');
displayMlint(output); |
'subs' might be growing inside a loop. Consider preallocating for speed.
'vals' might be growing inside a loop. Consider preallocating for speed.
根据建议，第一步是要给一些数组预分配空间。这样做是因为matlab使用的内存中连续的块，因此，如果在循环里不断改变数组大小，matlab就要不断寻找合适大小的内存片段并把数据移动过去。如果分配了一个比较大的空间，matlab就可以一直在这个连续空间工作。
但是，目前我们不知道 subs和vars的个数，如果要预分配，我们得分配最大可能的空间。那就是100^4，我们来试试：
|
1
2 3 4 5 |
try
zeros(1,100^4) catch ME end display(ME.message) |
Out of memory. Type HELP MEMORY for your options.
看来不行啊。太大了。
因此，我们只能通过分块分配空间来实现，每次分配一个可以接受的大小，并设置一个计数器，当这块空间满了的时候，再分配一个块。这样，内存移动的次数大大得到了降低。
（bfcat注： 这种不知道数组大小的时候，还有一个方法就是使用cell。我没有仔细分析cell的原理，但是我觉得它像是一个链表，因此cell里面的每一个元素不需要在连续的内存空间。因此，当我们执行类似 M{end+1} = m 的时候，matlab也不需要将M 中已有的元素都拷贝一次。这样，虽然Mlint还会提示让我们为cell预先分配空间，但是没关系，不分配对速度影响也不大。当循环结束以后，执行类似 M = cat(1, M{:}) 这样的语句就可以将其变回数组了。）
预分配空间后的代码|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 |
% Initialization
nx1 = 10; nx2 = 10; [x1,x2,limsf1,limsf2,expF,gausThresh,small,invSig,detSig,n]= ... initialize(nx1,nx2); % Initial guess for preallocation mm = min((nx1+1)^2*(nx2+1)^2, 10^6); subs = zeros(mm,4); vals = zeros(mm,1); counter = 0; % Iterate through all possible initial % and final positions for i1 = 1:nx1+1 for i2 = 1:nx2+1 for f1 = limsf1 for f2 = limsf2 xi = [x1(i1) x2(i2)]'; %% Initial position xf = [x1(f1) x2(f2)]'; %% Final position exponent = 0.5 * (xf - expF * xi)'... * invSig * (xf - expF * xi); % Increase preallocation if necessary if counter == length(vals) subs = [subs; zeros(mm, 4)]; vals = [vals; zeros(mm, 1)]; end if exponent > gausThresh small = small + 1; else % Counter introduced counter=counter + 1; out = 1 / (sqrt((2 * pi)^n * detSig))... * exp(-exponent); subs(counter,:) = [i1 i2 f1 f2]; vals(counter) = out; end end end end end % Remove zero components that came from preallocation vals = vals(vals > 0); subs = subs(vals > 0); |
|
1
2 3 4 |
displayRunTimes(2)
nx1 nx2 time 50 50 267 seconds 100 100 4228 seconds |
运行速度变快了一些，但是还是不够理想。
向量化因为matlab是基于矩阵的语言，因此，我们最好尽量用向量代替循环。尤其是多重循环嵌套的时候更要注意速度问题。对于这个代码，我们主要进行以下两种改动
- 向量化里面的两个循环
- 向量化里面的三个循环
尝试1: 向量化两个内循环
|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 |
% Initialization
nx1 = 10; nx2 = 10; [x1,x2,limsf1,limsf2,expF,gausThresh,small,invSig,detSig,n]= ... initialize(nx1,nx2); vals = cell(nx1+1,nx2+1); % Cell preallocation subs = cell(nx1+1,nx2+1); % Cell preallocation [xind,yind] = meshgrid(limsf1,limsf2); xyindices = [xind(:)' ; yind(:)']; [x,y] = meshgrid(x1(limsf1),x2(limsf2)); xyfinal = [x(:)' ; y(:)']; exptotal = zeros(length(xyfinal),1); % Loop over all possible combinations of positions for i1 = 1:nx1+1 for i2 = 1:nx2+1 xyinitial = repmat([x1(i1);x2(i2)],1,length(xyfinal)); expa = 0.5 * (xyfinal - expF * xyinitial); expb = invSig * (xyfinal - expF * xyinitial); exptotal(:,1) = expa(1,:).*expb(1,:)+expa(2,:).*expb(2,:); index = find(exptotal < gausThresh); expreduced = exptotal(exptotal < gausThresh); out = 1 / (sqrt((2 * pi)^n * detSig)) * exp(-(expreduced)); vals{i1,i2} = out; subs{i1,i2} = [i1*ones(1,length(index)) ; ... i2*ones(1,length(index)); xyindices(1,index); ... xyindices(2,index)]' ; end end % Reshape and convert output so it is in a % simple matrix format vals = cell2mat(vals(:)); subs = cell2mat(subs(:)); small = ((nx1+1)^2*(nx2+1)^2)-length(subs); |
这个向量化效果非常明显
|
1
2 3 4 |
displayRunTimes(3)
nx1 nx2 time 50 50 1.51 seconds 100 100 19.28 seconds |
这里主要使用了下面几个向量化的手段
可以参考这个页面获得更多向量化技巧：http://www.mathworks.com/help/matlab/matlab_prog/techniques-for-improving-performance.html?s_tid=doc_12b#f8-783589
尝试2: 向量化三个内循环
|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 |
% Initialization
nx1 = 10; nx2 = 10; [x1,x2,limsf1,limsf2,expF,gausThresh,small,invSig,detSig,n]= ... initialize(nx1,nx2); limsi1 = limsf1; limsi2 = limsf2; % ndgrid gives a matrix of all the possible combinations [aind,bind,cind] = ndgrid(limsi2,limsf1,limsf2); [a,b,c] = ndgrid(x2,x1,x2); vals = cell(nx1+1,nx2+1); % Cell preallocation subs = cell(nx1+1,nx2+1); % Cell preallocation % Convert grids to single vector to use in a single loop b = b(:); aind = aind(:); bind = bind(:); cind = cind(:); expac = a(:)-c(:); % Calculate x2-x1 % Iterate through initial x1 positions (i1) for i1 = limsi1 exbx1= b-x1(i1); expaux = invSig(2)*exbx1.*expac; exponent = 0.5*(invSig(1)*exbx1.*exbx1+expaux); index = find(exponent < gausThresh); expreduced = exponent(exponent < gausThresh); vals{i1} = 1 / (sqrt((2 * pi)^n * detSig))... .*exp(-expreduced); subs{i1} = [i1*ones(1,length(index)); aind(index)' ; bind(index)';... cind(index)']'; end vals = cell2mat(vals(:)); subs = cell2mat(subs(:)); small = ((nx1+1)^2*(nx2+1)^2)-length(subs); |
现在运行时间更短了：
|
1
2 3 4 |
displayRunTimes(4)
nx1 nx2 time 50 50 0.658 seconds 100 100 8.77 seconds |
|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 |
nx1 = 100; nx2 = 100;
x1l = 0; x1u = 100; x2l = 0; x2u = 100; x1 = linspace(x1l,x1u,nx1+1); x2 = linspace(x2l,x2u,nx2+1); limsi1 = 1:nx1+1; limsi2 = 1:nx2+1; limsf1 = 1:nx1+1; limsf2 = 1:nx2+1; t = 1; sigmax1 = 0.5; sigmax2 = 1; sigma = t * [sigmax1^2 sigmax2^2]; detSig = sigma(1)*sigma(2); invSig = [1/sigma(1) 1/sigma(2)]; gausThresh = 10; n=3; const=1 / (sqrt((2 * pi)^n * detSig)); % ndgrid gives a matrix of all the possible combinations % of position, except limsi1 which we iterate over [aind,bind,cind] = ndgrid(limsi2,limsf1,limsf2); [a,b,c] = ndgrid(x2,x1,x2); vals = cell(nx1+1,nx2+1); % Cell preallocation subs = cell(nx1+1,nx2+1); % Cell preallocation % Convert grids to single vector to % use in a single for-loop b = b(:); aind = aind(:); bind = bind(:); cind = cind(:); expac= a(:)-c(:); expaux = invSig(2)*expac.*expac; % Iterate through initial x1 positions for i1 = limsi1 expbx1= b-x1(i1); exponent = 0.5*(invSig(1)*expbx1.*expbx1+expaux); % Find indices where exponent < gausThresh index = find(exponent < gausThresh); % Find and keep values where exp < gausThresh expreduced = exponent(exponent < gausThresh); vals{i1} = const.*exp(-expreduced); subs{i1} = [i1*ones(1,length(index)); aind(index)' ; bind(index)';... cind(index)']'; end vals = cell2mat(vals(:)); subs = cell2mat(subs(:)); small = ((nx1+1)^2*(nx2+1)^2)-length(subs); |
最终的运行时间
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1
2 3 4 |
displayRunTimes(5)
nx1 nx2 time 50 50 0.568 seconds 100 100 8.36 seconds |
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