I have been asked to obtain the first 15 triplets according to this series and this code ought to work. However, it does only produce a table (15*3) filled with zero rather than the 15 Pythagorean triplets? Any help will be welcome.
A = zeros(15, 3);
ii = 1;
for c = 5:120
c2=c^2;
for a=1:c-1
a2=a^2;
for b=a:c-1
if c2-(a2+b^2) == 0
A(ii,1) = a;
A(ii,2) = b;
A(ii,3) = c;
ii=ii+1;
if A(15, 1) ~= 0
flag = 1;
break
end
end
end
if flag == 1
break
end
end
if flag == 1
break
end
end
T1 = array2table(A);
disp(T1)
So, the code generated a correct table on application-restart before failing on all subsequent attempts. And, now I notice that the code runs successfully only for the first time after every relaunch of the application. (Resolved, thanks Dan Pollard.)
Also, interested in knowing if there is any way to not write an upper limit (120) into the code.
I don't think your if statement is ever satisfied. For example, for c=5, you'd expect a=3, b=4 to be a triplet. But you're only letting a and b go up to floor(sqrt(c-1)), which is 2.
Do you mean to let a and b go up to floor(sqrt(c2-1))?
Edit As the question has changed.
When you run the code, Matlab creates all the variables which you assign, and stores them in the workspace. This can be useful, but here it's hurting you as you have the variable flag which is stored as 1. This means that when the code runs, it checks if flag==1 after the first run through b, which it is, so the code ends. Resolve this by placing clear; at the beginning of your script.
There isn't a practical way to remove the upper limit on c. Matlab has the built-in variable Inf but at best Matlab won't let you use it in that context. Realistically you could just replace the 120 with a really large number, but this will take more time and more memory as the number gets bigger. Computers have a finite RAM to store matlab arrays in though, and there are infinitely many pythagorean triples, so doing the calculation without an upper limit will fail in some way.
Related
In a MATLAB-function the following code is used:
function stuff()
if a == 2
do1();
else
do2();
end
end
This code is placed inside a simulation-loop and gets called 1000 times or more per second. The if-statement does only matter in the first call of the function, after that either do1 or do2 are used, the variable a will not change any more.
How do I prevent to waste processing time with this if-statement? Basically, how do I tell Matlab, to not check the if-statement any more, and just call the one function, that gets selected in the first call to stuff?
Contrary to your beliefs this is not a problem, the compiler (should) automatically does this optimization for you. See e.g. Loop-invariant code motion.
What you can do to help the compiler is to move the calculation of the check outside as a flag, e.g.
flag = a==2;
for i = 1:100
stuff(flag)
end
Then you only have to do the calculation once and it is clear to the compiler that the value does not change.
NOTE: Obviously, if your check really is a==2, this would not make much of a difference.
EDIT: I have not been able to definitely verify that MATLAB does this automatically. However, this is only the first layer of optimization that is done for you. All modern processors use what is called a Branch predictor, see e.g. this brilliant answer Why is processing a sorted array faster than processing an unsorted array?, or this wiki page. In short, the processor guesses the result of the if-statement, if it is correct, everything goes faster. I think it is fair to say that the processor guesses correctly in all of your cases.
TLDR: Do not wory about it.
Given the comments above, it seems what you are actually looking for is a way to dynamically chose the function to be run in your simulation. This choice should be dynamic (you do not know which function to use at runtime) but the choice should only be done once. This is easily achievable using function handles: https://www.mathworks.com/help/matlab/function-handles.html
Here is an example:
function dynamicSimulation()
if ( rand() > 0.5 ) % determine which function should be called dynamically
sim=#func1;
else
sim=#func2;
end
other_params = [];
for k = 1:5 % run the simulation
sim( k, other_params );
end
end
function func1( index, other_params )
fprintf( 'Index=%d: Simulating using function 1\n', index );
end
function func2( index, other_params )
fprintf( 'Index=%d: Simulating using function 2\n', index );
end
If you run this several times you will notice that the (random) choice of func1 or func2 will mean you do not get the same function being run each time, although the same one is used for the entire simulation.
I reckon you don't waste much time on checking the validity that if statement. However, since you specifically mention it only checks for the first iteration: why not get that out? So instead of:
for ii = 1:10
if ii == 1
k = 1;
else
k = k + 1;
end
end
You could do
k = 1;
for ii = 2:10
k = k + 1;
end
Thus eliminating the check.
NB: this scales badly of course, but as it is only a single iteration here I consider it a good option.
The MATLAB documentation describes the break keyword thus:
break terminates the execution of a for or while loop. Statements in the loop after the break statement do not execute.
In nested loops, break exits only from the loop in which it occurs. Control passes to the statement that follows the end of that loop.
(my emphasis)
What if you want to exit from multiple nested loops? Other languages, such as Java, offer labelled breaks, which allow you to specify where the flow of control is to be transferred, but MATLAB lacks such a mechanism.
Consider the following example:
% assume A to be a 2D array
% nested 'for' loops
for j = 1 : n
for i = 1 : m
if f(A(i, j)) % where f is a predicate
break; % if want to break from both loops, not just the inner one
else
% do something interesting with A
end
end
% <--- the break transfers control to here...
end
% <--- ... but I want to transfer control to here
What is an idiomatic way (in MATLAB) of exiting from both loops?
I would say for your original specific example, rather use linear indexing and a single loop:
%// sample-data generation
m = 4;
n = 5;
A = rand(m, n);
temp = 0;
for k = 1:numel(A)
if A(k) > 0.8 %// note that if you had switched your inner and outer loops you would have had to transpose A first as Matlab uses column-major indexing
break;
else
temp = temp + A(k);
end
end
Or the practically identical (but with less branching):
for k = 1:numel(A)
if A(k) <= 0.8 %// note that if you had switched your inner and outer loops you would have had to transpose A first as Matlab uses column-major indexing
temp = temp + A(k);
end
end
I would think that this answer will vary from case to case and there is no general one size fits all idiomatically correct solution but I would approach it in the following way depending on your problem (note these all assumes that a vectorized solution is not practical as that is the obvious first choice)
Reduce the dimensions of the nesting and use either no breaks or just one single break (i.e. as shown above).
Don't use break at all because unless the calculation of your predicate is expensive and your loop has very many iterations, those extra iterations at the end should be practically free.
Failing that set a flag and break at each level.
Or finally wrap your loop into a function and call return instead of break.
As far as I know there are no such functionality built in. However, in most cases matlab does not need nested loops due to its support for vectorization. In those cases where vectorization does not work, the loops are mostly long and complicated and thus multiple breaks would not hinder readability significantly. As noted in a comment, you would not really need a nested loop here. Vectorization would do the trick,
m = 5;
n=4;
x = rand(m,n);
tmp = find(x>0.8, 1, 'first');
if (isempty(tmp))
tmp = m*n+1;
end
tmp = tmp-1;
tot = sum(x(1:tmp));
There might of course be people claiming that for loops are not necessarily slow anymore, but the fact remains that Matlab is column heavy and using more than one loop will in most cases include looping over non optimal dimensions. Vectorized solutions does not require that since they can use smart methods avoiding such loops (which of course does not hold if the input is a row vector, so avoiding this is also good).
The best idiomatic way to use Python (or poison of your choice) and forget all this but that's another story. Also I don't agree with the vectorization claims of the other answers anymore. Recent matlab versions handle for loops pretty quickly. You might be surprised.
My personal preference goes to raising an exception deliberately and cradling it within a try and catch block.
% assume A to be a 2D array
A = rand(10) - 0.5;
A(3,2) = 0;
wreaker = MException('Loop:breaker','Breaking the law');
try
for j = 1 : size(A,1)
% forloop number 1
for i = 1 : size(A,2)
% forloop number 2
for k = 1:10
% forloop number 3
if k == 5 && j == 3 && i == 6
mycurrentval = 5;
throw(wreaker)
end
end
end
end
catch
return % I don't remember the do nothing keyword for matlab apparently
end
You can change the location of your try catch indentation to fall back to the loop of your choice. Also by slaying kittens, you can write your own exceptions such that they label the exception depending on the nest count and then you can listen for them. There is no end to ugliness though still prettier than having counters or custom variables with if clauses in my opinion.
Note that, this is exactly why matlab drives many people crazy. It silently throws exceptions in a pretty similar way and you get a nonsensical error for the last randomly chosen function while passing by, such as size mismatch in some differential equation solver so on. I actually learned all this stuff after reading a lot matlab toolbox source codes.
I have to iterate a process where I have an initial guess for the Mach number (M0). This initial guess will give me another guess for the Mach number by using two equations (Mn). Eventually, i want to iterate this process untill the error between M0 and Mn is small. I have the following piece of code and it actually works well with a while loop.
However, I am afraid that the while loop will take many iterations and computational time for certain inputs since this will be part of a bigger code which most likely will give unfeasible inputs for the while loop.
Therefore my question is the following. How can I iterate this process within Matlab without consulting a while loop? The code that I am implementing now is the following:
%% Input
gamma = 1.4;
theta = atan(0.315);
cpi = -0.732;
%% Loop
M0 = 0.2; %initial guess
Err = 100;
iterations = 0;
while Err > 0.5E-3
B = (1-(M0^2)*(1-M0*cpi))^0.5;
Mn = (((gamma+1)/2) * ((B+((1-cpi)^0.5)*sec(theta)-1)^2/(B^2 + (tan(theta))^2)) - ((gamma-1)/2) )^-0.5;
Err = abs(M0 - Mn);
M0 = Mn;
iterations=iterations+1;
end
disp(iterations) disp(Mn)
Many thanks
Since M0 is calculated in each iteration and you have trigonometric functions, you cannot use another way than iteration structures (i.e. while).
If you had a specific increase or decrease at M0, then you could initialize a vector of M0 and do vector calculations for B and Err.
But, with sec and tan this is not possible.
Another wat would be to use the parallel processing. But, since you change the M0 at each iteration then you cannot use the parfor loop.
As for a for loop, in MATLAB you need an array for for "command" argument (e.g. 1:10 or 1:length(x) or i = A, where A = 1:10 or A = [1:10;11:20]). Since you evaluate a condition and depending on the result of the evaluation you judge if you continue the execution or not, it seems that the while loop (or do while in another language) is the only way to go.
I think you need to clarify the issue. If it the issue you want to solve is that some inputs take a long time to calculate, it is not the while loop that takes the time, it is the execution of the code multiple times that causes it. Any method that loops through will be restricted by the time the block of code takes to execute multiplied by the number of iterations required to converge.
You can introduce something to stop at a certain number of iterationtions, conceptually:
While ((err > tolerance) && (numIterations < limit))
If you want an answer which does not require iterating over the code, this is akin to finding a closed form solution, and I suspect this does not exist.
Edit to add: by not exist I mean in a practical form which can be implemented in a more efficient way then iterating to a solution.
Im implementing the k-means algorithm on matlab without using the k-means built-in function, The stopping criteria is when the new centroids doesn't change by new iterations, but i cannot implement it in matlab , can anybody help?
Thanks
Setting no change as a stopping criteria is a bad idea. There are a few main reasons you shouldn't use a 0 change condition
even for a well behaved function the difference between 0 change and a very small change (say 1e-5 perhaps)could be 1000+ iterations, so you are wasting time trying to get them to be exactly the same. Especially because computers usually keep far more digits than we are interested in. IF you only need 1 digit accuracy, why wait for the computer to find an answer within 1e-31?
computers have floating point errors everywhere. Try doing some easily reversible matrix operations like a = rand(3,3); b = a*a*inv(a); a-b theoretically this should be 0 but you will see it isn't. So these errors alone could prevent your program from ever stopping
dithering. lets say we have a 1d k means problem with 3 numbers and we want to split them into 2 groups. One iteration the grouping can be a,b vs c. the next iteration could be a vs b,c the next could be a,b vs c the next.... This is of course a simplified example, but there can be instances where a few data points can dither between clusters, and you will end up with a never ending algorithm. Since those few points are reassigned, the change will never be 0
the solution is to use a delta threshold. basically you subtract the current values from the previous and if they are less than a threshold you are done. This on its own is powerful, but as with any loop, you need a backup escape plan. And that is setting a max_iterations variable. Look at matlabs documentation for kmeans, even they have a MaxIter variable (default is 100) so even if your kmeans doesn't converge, at least it wont run endlessly. Something like this might work
%problem specific
max_iter = 100;
%choose a small number appropriate to your problem
thresh = 1e-3;
%ensures it runs the first time
delta_mu = thresh + 1;
num_iter = 0;
%do your kmeans in the loop
while (delta_mu > thresh && num_iter < max_iter)
%save these right away
old_mu = curr_mu;
%calculate new means and variances, this is the standard kmeans iteration
%then store the values in a variable called curr_mu
curr_mu = newly_calculate_values;
%use the two norm to find the delta as a single number. no matter what
%the original dimensionality of mu was. If old_mu -new_mu was
% 0 the norm is still 0. so it behaves well as a distance measure.
delta_mu = norm(old_mu - curr_mu,2);
num_ter = num_iter + 1;
end
edit
if you don't know the 2 norm is essentially the euclidean distance
I have recently started learning MatLab, and wrote the following script today as part of my practice to see how we can generate a vector:
x = [];
n = 4;
for i = i:n
x = [x,i^2];
end
x
When I run this script I get what I expect, namely the following vector:
x = 0 1 4 9 16
However, if I run the script a second time right afterwards I only get the following output:
x = 16
What is the reason for this? How come I only get the last vector entry as output the second time I run the script, and not the vector in its entirety? If anyone can explain this to me, I would greatly appreciate it.
Beginning with a fresh workspace, i will simply be the complex number 1i (as in x^2=-1). I imagine you got this warning on the first run:
Warning: Colon operands must be real scalars.
So the for statement basically loops over for i = real(1i):4. Note that real(1i)=0.
When you rerun the script again with the variables already initialized (assuming you didn't clear the workspace), i will refer to a variable containing the last value of 4, shadowing the builtin function i with the same name, and the for-loop executes:
x=[];
for i=4:4
x = [x, i^2]
end
which iterates only one time, thus you end up with x=16
you forget to initialize i.
after first execution i is 4 and remains 4.
then you initialize x as an empty vector but because i is 4 the loop runs only once.
clear your workspace and inspect it before and after first execution.
Is it possibly a simple typo?
for i = i:n
and should actually mean
for i = 1:n
as i is (probably) uninitialized in the first run, and therefore 0, it works just fine.
The second time, i is still n (=4), and only runs once.
Also, as a performance-tip: in every iteration of your loop you increase the size of your vector, the more efficient (and more matlaboid) way would be to create the vector with the basevalues first, for example with
x = 1:n
and then square each value by
x = x^2
In Matlab, using vector-operations (or matrix-operations on higher dimensions) should be prefered over iterative loop approaches, as it gives matlab the opportunity to do optimised operations. It is also often more readable that way.