Good evening everyone,
I want to create a function
f(x) = [f1(x), f2(x), ... , fn(x)]
in MatLab, with an arbitrary form and number for the fi. In my current case they are meant to be basis elements for a finite-dimensional function space, so for example a number of multi variable polynomials. I want to able to be able to set form (e.g. hermite/lagrange polynomials, ...) and number via arguments in some sort of "function creating" function, so I would like to solve this for arbitrary functions fi.
Assume for now that the fi are fi:R^d -> R, so vector input to scalar output. This means the result from f should be a n-dim vector containing the output of all n functions. The number of functions n could be fairly large, as there is permutation involved. I also need to evaluate the resulting function very often, so I hope to do it as efficiently as possible.
Currently I see two ways to do this:
Create a cell with each fi using a loop, using something like
funcell{i}=matlabFunction(createpoly(degree, x),'vars',{x})
and one of the functions from the symbolic toolbox and a symbolic x (vector). It is then possible to create the desired function with cellfun, e.g.
f=#(x) cellfun(#(v) v(x), funcell)
This is relatively short, easy and what can be found when doing searches. It even allows extension to vector output using 'UniformOutput',false and cell2mat. On the downside it is very inefficient, first during creation because of matlabFunction and then during evaluation because of cellfun.
The other idea I had is to create a string and use eval. One way to do this would be
stringcell{i}=[char(createpoly(degree, x)),';']
and then use strjoin. In theory this should yield an efficient function. There are two problems however. The first is the use of eval (mostly on principle), the second is inserting the correct arguments. The symbolic toolbox does not allow symbols of the form x(i), so the resulting string will not contain them either. The only remedy I have so far is some sort of string replacement on the xi that are allowed, but this is also far from elegant.
So I do have ways to do what I need right now, but I would appreciate any ideas for a better solution.
From my understanding of the problem, you could do the straightforward:
Initialization step:
my_fns = cell(n, 1); %where n is number of functions
my_fns{1} = #f1; % Assuming f1 is defined in f1.m etc...
my_fns{2} = #f2;
Evaluation at x:
z = zeros(n, 1);
for i=1:n,
z(i) = my_fns{i}(x)
end
For example if you put it in my_evaluate.m:
function z = my_evaluate(my_fns, x)
z = zeros(n, 1);
for i=1:n,
z(i) = my_fns{i}(x)
end
How might this possibly be sped up?
Depends on if you have special structure than can be exploited.
Are there calculations common to some subset of f1 through fn that need not be repeated with each function call? Eg. if the common calculation step is costly, you could do y = f_helper(x) and z(i) = fi(x, y).
Can the functions f1...fn be vector / matrix friendly, allowing evaluation of multiple points with each function call?
The big issue is how fast your function calls f1 through fn are, not how you collect the results from those calls in a vector.
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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!