I have this function below and I need to calculate the minimum and maximum of this function U, and also plotting the maximum and minimum value in 3D graph together with the function.
How can I write the code?
[x,y]=meshgrid(0:0.1:pi,0:0.1:2*pi);% x-theta,y-phi
a=90.7;b=36.2;c=12.9;
E=1.44;
U=-E.^2*(a.*sin(x).^2.*cos(y).^2+b.*sin(x).^2.*sin(y).^2+c.*cos(x).^2);
meshc(x,y,U)
xlabel('\theta')
ylabel('\Phi ')
zlabel('U')
I tired this way to find max but I don't know if i did it correct
max(U,[],1) %max row-wise
max(U,[],2) %max column-wise
and for the minimum it didn't work the same idea, also I didn't get the exact value of the maximum
As stated above, to simply find the maximum/minimum of your sampled function, use m = min(U(:)); M = max(U(:)). To be able to plot them, what you are missing are the coordinates that give you those values. For this, you will need the second output of the min/max functions, which gives you the index where the extreme happens.
A possible implementation (possibly not the best one) would be (might not work perfectly, I don't have matlab at hand):
[Ms,I] = max(U,[],1); %row-wise maximum and their indexes
[M,j] = max(Ms); %maximum among all rows and its index
Now i = I(j) is the location of the maximum. You can finally do plot3(x(i,j),y(i,j),U(i,j),'ro') to plot a big red circle in the maximums location, or whatever you like.
Note: I might have it backwards and it might be x(j,i), and so on. Just check. Of course you can do the same thing for min().
EDIT: I just remembered the ind2sub function , which solves all your problems. Following the syntax used above:
[M,ind] = max(U(:));
[i,j] = ind2sub(size(U),ind)
The rest holds the unchanged.
You can simply use something like
max(max(U))
this will find the maximum for your 2D matrix.
For the minimum you just have to replace max with min.
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I think everyone would agree that the MATLAB language is not pretty, or particularly consistent. But nevermind! We still have to use it to get things done.
What are your favourite tricks for making things easier? Let's have one per answer so people can vote them up if they agree. Also, try to illustrate your answer with an example.
Using the built-in profiler to see where the hot parts of my code are:
profile on
% some lines of code
profile off
profile viewer
or just using the built in tic and toc to get quick timings:
tic;
% some lines of code
toc;
Directly extracting the elements of a matrix that satisfy a particular condition, using logical arrays:
x = rand(1,50) .* 100;
xpart = x( x > 20 & x < 35);
Now xpart contains only those elements of x which lie in the specified range.
Provide quick access to other function documentation by adding a "SEE ALSO" line to the help comments. First, you must include the name of the function in all caps as the first comment line. Do your usual comment header stuff, then put SEE ALSO with a comma separated list of other related functions.
function y = transmog(x)
%TRANSMOG Transmogrifies a matrix X using reverse orthogonal eigenvectors
%
% Usage:
% y = transmog(x)
%
% SEE ALSO
% UNTRANSMOG, TRANSMOG2
When you type "help transmog" at the command line, you will see all the comments in this comment header, with hyperlinks to the comment headers for the other functions listed.
Turn a matrix into a vector using a single colon.
x = rand(4,4);
x(:)
Vectorizing loops. There are lots of ways to do this, and it is entertaining to look for loops in your code and see how they can be vectorized. The performance is astonishingly faster with vector operations!
Anonymous functions, for a few reasons:
to make a quick function for one-off uses, like 3x^2+2x+7. (see listing below) This is useful for functions like quad and fminbnd that take functions as arguments. It's also convenient in scripts (.m files that don't start with a function header) since unlike true functions you can't include subfunctions.
for closures -- although anonymous functions are a little limiting as there doesn't seem to be a way to have assignment within them to mutate state.
.
% quick functions
f = #(x) 3*x.^2 + 2*x + 7;
t = (0:0.001:1);
plot(t,f(t),t,f(2*t),t,f(3*t));
% closures (linfunc below is a function that returns a function,
% and the outer functions arguments are held for the lifetime
% of the returned function.
linfunc = #(m,b) #(x) m*x+b;
C2F = linfunc(9/5, 32);
F2C = linfunc(5/9, -32*5/9);
Matlab's bsxfun, arrayfun, cellfun, and structfun are quite interesting and often save a loop.
M = rand(1000, 1000);
v = rand(1000, 1);
c = bsxfun(#plus, M, v);
This code, for instance, adds column-vector v to each column of matrix M.
Though, in performance critical parts of your application you should benchmark these functions versus the trivial for-loop because often loops are still faster.
LaTeX mode for formulas in graphs: In one of the recent releases (R2006?) you add the additional arguments ,'Interpreter','latex' at the end of a function call and it will use LaTeX rendering. Here's an example:
t=(0:0.001:1);
plot(t,sin(2*pi*[t ; t+0.25]));
xlabel('t');
ylabel('$\hat{y}_k=sin 2\pi (t+{k \over 4})$','Interpreter','latex');
legend({'$\hat{y}_0$','$\hat{y}_1$'},'Interpreter','latex');
Not sure when they added it, but it works with R2006b in the text(), title(), xlabel(), ylabel(), zlabel(), and even legend() functions. Just make sure the syntax you are using is not ambiguous (so with legend() you need to specify the strings as a cell array).
Using xlim and ylim to draw vertical and horizontal lines. Examples:
Draw a horizontal line at y=10:
line(xlim, [10 10])
Draw vertical line at x=5:
line([5 5], ylim)
Here's a quick example:
I find the comma separated list syntax quite useful for building function calls:
% Build a list of args, like so:
args = {'a', 1, 'b', 2};
% Then expand this into arguments:
output = func(args{:})
Here's a bunch of nonobvious functions that are useful from time to time:
mfilename (returns the name of the currently running MATLAB script)
dbstack (gives you access to the names & line numbers of the matlab function stack)
keyboard (stops execution and yields control to the debugging prompt; this is why there's a K in the debug prompt K>>
dbstop error (automatically puts you in debug mode stopped at the line that triggers an error)
I like using function handles for lots of reasons. For one, they are the closest thing I've found in MATLAB to pointers, so you can create reference-like behavior for objects. There are a few neat (and simpler) things you can do with them, too. For example, replacing a switch statement:
switch number,
case 1,
outargs = fcn1(inargs);
case 2,
outargs = fcn2(inargs);
...
end
%
%can be turned into
%
fcnArray = {#fcn1, #fcn2, ...};
outargs = fcnArray{number}(inargs);
I just think little things like that are cool.
Using nargin to set default values for optional arguments and using nargout to set optional output arguments. Quick example
function hLine=myplot(x,y,plotColor,markerType)
% set defaults for optional paramters
if nargin<4, markerType='none'; end
if nargin<3, plotColor='k'; end
hL = plot(x,y,'linetype','-', ...
'color',plotColor, ...
'marker',markerType, ...
'markerFaceColor',plotColor,'markerEdgeColor',plotColor);
% return handle of plot object if required
if nargout>0, hLine = hL; end
Invoking Java code from Matlab
cellfun and arrayfun for automated for loops.
Oh, and reverse an array
v = 1:10;
v_reverse = v(length(v):-1:1);
conditional arguments in the left-hand side of an assignment:
t = (0:0.005:10)';
x = sin(2*pi*t);
x(x>0.5 & t<5) = 0.5;
% This limits all values of x to a maximum of 0.5, where t<5
plot(t,x);
Know your axis properties! There are all sorts of things you can set to tweak the default plotting properties to do what you want:
set(gca,'fontsize',8,'linestyleorder','-','linewidth',0.3,'xtick',1:2:9);
(as an example, sets the fontsize to 8pt, linestyles of all new lines to all be solid and their width 0.3pt, and the xtick points to be [1 3 5 7 9])
Line and figure properties are also useful, but I find myself using axis properties the most.
Be strict with specifying dimensions when using aggregation functions like min, max, mean, diff, sum, any, all,...
For instance the line:
reldiff = diff(a) ./ a(1:end-1)
might work well to compute relative differences of elements in a vector, however in case the vector degenerates to just one element the computation fails:
>> a=rand(1,7);
>> diff(a) ./ a(1:end-1)
ans =
-0.5822 -0.9935 224.2015 0.2708 -0.3328 0.0458
>> a=1;
>> diff(a) ./ a(1:end-1)
??? Error using ==> rdivide
Matrix dimensions must agree.
If you specify the correct dimensions to your functions, this line returns an empty 1-by-0 matrix, which is correct:
>> diff(a, [], 2) ./ a(1, 1:end-1)
ans =
Empty matrix: 1-by-0
>>
The same goes for a min-function which usually computes minimums over columns on a matrix, until the matrix only consists of one row. - Then it will return the minimum over the row unless the dimension parameter states otherwise, and probably break your application.
I can almost guarantee you that consequently setting the dimensions of these aggregation functions will save you quite some debugging work later on.
At least that would have been the case for me. :)
The colon operator for the manipulation of arrays.
#ScottieT812, mentions one: flattening an array, but there's all the other variants of selecting bits of an array:
x=rand(10,10);
flattened=x(:);
Acolumn=x(:,10);
Arow=x(10,:);
y=rand(100);
firstSix=y(1:6);
lastSix=y(end-5:end);
alternate=y(1:2:end);
In order to be able to quickly test a function, I use nargin like so:
function result = multiply(a, b)
if nargin == 0 %no inputs provided, run using defaults for a and b
clc;
disp('RUNNING IN TEST MODE')
a = 1;
b = 2;
end
result = a*b;
Later on, I add a unit test script to test the function for different input conditions.
Using ismember() to merge data organized by text identfiers. Useful when you are analyzing differing periods when entries, in my case company symbols, come and go.
%Merge B into A based on Text identifiers
UniverseA = {'A','B','C','D'};
UniverseB = {'A','C','D'};
DataA = [20 40 60 80];
DataB = [30 50 70];
MergeData = NaN(length(UniverseA),2);
MergeData(:,1) = DataA;
[tf, loc] = ismember(UniverseA, UniverseB);
MergeData(tf,2) = DataB(loc(tf));
MergeData =
20 30
40 NaN
60 50
80 70
Asking 'why' (useful for jarring me out of a Matlab runtime-fail debugging trance at 3am...)
Executing a Simulink model directly from a script (rather than interactively) using the sim command. You can do things like take parameters from a workspace variable, and repeatedly run sim in a loop to simulate something while varying the parameter to see how the behavior changes, and graph the results with whatever graphical commands you like. Much easier than trying to do this interactively, and it gives you much more flexibility than the Simulink "oscilloscope" blocks when visualizing the results. (although you can't use it to see what's going on in realtime while the simulation is running)
A really important thing to know is the DstWorkspace and SrcWorkspace options of the simset command. These control where the "To Workspace" and "From Workspace" blocks get and put their results. Dstworkspace defaults to the current workspace (e.g. if you call sim from inside a function the "To Workspace" blocks will show up as variables accessible from within that same function) but SrcWorkspace defaults to the base workspace and if you want to encapsulate your call to sim you'll want to set SrcWorkspace to current so there is a clean interface to providing/retrieving simulation input parameters and outputs. For example:
function Y=run_my_sim(t,input1,params)
% runs "my_sim.mdl"
% with a From Workspace block referencing I1 as an input signal
% and parameters referenced as fields of the "params" structure
% and output retrieved from a To Workspace block with name O1.
opt = simset('SrcWorkspace','current','DstWorkspace','current');
I1 = struct('time',t,'signals',struct('values',input1,'dimensions',1));
Y = struct;
Y.t = sim('my_sim',t,opt);
Y.output1 = O1.signals.values;
Contour plots with [c,h]=contour and clabel(c,h,'fontsize',fontsize). I usually use the fontsize parameter to reduce the font size so the numbers don't run into each other. This is great for viewing the value of 2-D functions without having to muck around with 3D graphs.
Vectorization:
function iNeedle = findClosest(hay,needle)
%FINDCLOSEST find the indicies of the closest elements in an array.
% Given two vectors [A,B], findClosest will find the indicies of the values
% in vector A closest to the values in vector B.
[hay iOrgHay] = sort(hay(:)'); %#ok must have row vector
% Use histogram to find indices of elements in hay closest to elements in
% needle. The bins are centered on values in hay, with the edges on the
% midpoint between elements.
[iNeedle iNeedle] = histc(needle,[-inf hay+[diff(hay)/2 inf]]); %#ok
% Reversing the sorting.
iNeedle = iOrgHay(iNeedle);
Using persistent (static) variables when running an online algorithm. It may speed up the code in areas like Bayesian machine learning where the model is trained iteratively for the new samples. For example, for computing the independent loglikelihoods, I compute the loglikelihood initially from scratch and update it by summing this previously computed loglikelihood and the additional loglikelihood.
Instead of giving a more specialized machine learning problem, let me give a general online averaging code which I took from here:
function av = runningAverage(x)
% The number of values entered so far - declared persistent.
persistent n;
% The sum of values entered so far - declared persistent.
persistent sumOfX;
if x == 'reset' % Initialise the persistent variables.
n = 0;
sumOfX = 0;
av = 0;
else % A data value has been added.
n = n + 1;
sumOfX = sumOfX + x;
av = sumOfX / n; % Update the running average.
end
Then, the calls will give the following results
runningAverage('reset')
ans = 0
>> runningAverage(5)
ans = 5
>> runningAverage(10)
ans = 7.5000
>> runningAverage(3)
ans = 6
>> runningAverage('reset')
ans = 0
>> runningAverage(8)
ans = 8
I'm surprised that while people mentioned the logical array approach of indexing an array, nobody mentioned the find command.
e.g. if x is an NxMxO array
x(x>20) works by generating an NxMxO logical array and using it to index x (which can be bad if you have large arrays and are looking for a small subset
x(find(x>20)) works by generating list (i.e. 1xwhatever) of indices of x that satisfy x>20, and indexing x by it. "find" should be used more than it is, in my experience.
More what I would call 'tricks'
you can grow/append to arrays and cell arrays if you don't know the size you'll need, by using end + 1 (works with higher dimensions too, so long as the dimensions of the slice match -- so you'll have to initialize x to something other than [] in that case). Not good for numerics but for small dynamic lists of things (or cell arrays), e.g. parsing files.
e.g.
>> x=[1,2,3]
x = 1 2 3
>> x(end+1)=4
x = 1 2 3 4
Another think many people don't know is that for works on any dim 1 array, so to continue the example
>> for n = x;disp(n);end
1
2
3
4
Which means if all you need is the members of x you don't need to index them.
This also works with cell arrays but it's a bit annoying because as it walks them the element is still wrapped in a cell:
>> for el = {1,2,3,4};disp(el);end
[1]
[2]
[3]
[4]
So to get at the elements you have to subscript them
>> for el = {1,2,3,4};disp(el{1});end
1
2
3
4
I can't remember if there is a nicer way around that.
-You can make a Matlab shortcut to an initialization file called startup.m. Here, I define formatting, precision of the output, and plot parameters for my Matlab session (for example, I use a larger plot axis/font size so that .fig's can be seen plainly when I put them in presentations.) See a good blog post from one of the developers about it http://blogs.mathworks.com/loren/2009/03/03/whats-in-your-startupm/ .
-You can load an entire numerical ascii file using the "load" function. This isn't particularly fast, but gets the job done quickly for prototyping (shouldn't that be the Matlab motto?)
-As mentioned, the colon operator and vectorization are lifesavers. Screw loops.
x=repmat([1:10],3,1); % say, x is an example array of data
l=x>=3; % l is a logical vector (1s/0s) to highlight those elements in the array that would meet a certain condition.
N=sum(sum(l));% N is the number of elements that meet that given condition.
cheers -- happy scripting!
I want to apply a function to all columns in a matrix with MATLAB. For example, I'd like to be able to call smooth on every column of a matrix, instead of having smooth treat the matrix as a vector (which is the default behaviour if you call smooth(matrix)).
I'm sure there must be a more idiomatic way to do this, but I can't find it, so I've defined a map_column function:
function result = map_column(m, func)
result = m;
for col = 1:size(m,2)
result(:,col) = func(m(:,col));
end
end
which I can call with:
smoothed = map_column(input, #(c) (smooth(c, 9)));
Is there anything wrong with this code? How could I improve it?
The MATLAB "for" statement actually loops over the columns of whatever's supplied - normally, this just results in a sequence of scalars since the vector passed into for (as in your example above) is a row vector. This means that you can rewrite the above code like this:
function result = map_column(m, func)
result = [];
for m_col = m
result = horzcat(result, func(m_col));
end
If func does not return a column vector, then you can add something like
f = func(m_col);
result = horzcat(result, f(:));
to force it into a column.
Your solution is fine.
Note that horizcat exacts a substantial performance penalty for large matrices. It makes the code be O(N^2) instead of O(N). For a 100x10,000 matrix, your implementation takes 2.6s on my machine, the horizcat one takes 64.5s. For a 100x5000 matrix, the horizcat implementation takes 15.7s.
If you wanted, you could generalize your function a little and make it be able to iterate over the final dimension or even over arbitrary dimensions (not just columns).
Maybe you could always transform the matrix with the ' operator and then transform the result back.
smoothed = smooth(input', 9)';
That at least works with the fft function.
A way to cause an implicit loop across the columns of a matrix is to use cellfun. That is, you must first convert the matrix to a cell array, each cell will hold one column. Then call cellfun. For example:
A = randn(10,5);
See that here I've computed the standard deviation for each column.
cellfun(#std,mat2cell(A,size(A,1),ones(1,size(A,2))))
ans =
0.78681 1.1473 0.89789 0.66635 1.3482
Of course, many functions in MATLAB are already set up to work on rows or columns of an array as the user indicates. This is true of std of course, but this is a convenient way to test that cellfun worked successfully.
std(A,[],1)
ans =
0.78681 1.1473 0.89789 0.66635 1.3482
Don't forget to preallocate the result matrix if you are dealing with large matrices. Otherwise your CPU will spend lots of cycles repeatedly re-allocating the matrix every time it adds a new row/column.
If this is a common use-case for your function, it would perhaps be a good idea to make the function iterate through the columns automatically if the input is not a vector.
This doesn't exactly solve your problem but it would simplify the functions' usage. In that case, the output should be a matrix, too.
You can also transform the matrix to one long column by using m(:,:) = m(:). However, it depends on your function if this would make sense.