I am interested in writing some matrix elements as functions which can take value as per my convenience and then the required matrix operations can be applied on top of that. More precisely, I am trying to integrate over x, the trace of a matrix which has matrix elements as functions of x (which are unknown analytically as they come through products of matrices dependent on x) .
When I try to write the matrix elements as functions, then I obviously get the error- Conversion to double from function_handle is not possible. Is there an easy way to write the matrix elements as functions ?
Thanks. Please ask if my question is not clear.
For example, its something like this:-
N_k = 10;
M_sigma = cell(N_k,1);
M_rho = cell(N_k,1);
for ii = 1:N_k
M_sigma {ii}(1,2) = #(sigma) sigma; %this kind of thing is not allowed in matlab
M_sigma {ii}(2,1) = #(sigma) -conj(sigma);
M_sigma {ii}(1,1) = 0;
M_sigma {ii}(2,2) = 0;
end
for ii = 1:N_k
M_rho {ii}(1,2) = #(rho) rho;
M_rho {ii}(2,1) = #(rho) -conj(rho);
M_rho {ii}(1,1) = 0;
M_rho {ii}(2,2) = 0;
end
M_tau = cell(N_k,1);
for ii = 1:N_k
M_tau {ii} = exp(M_sigma{ii})*exp(M_rho{ii});
end
% the following statement is wrong but I want to do something like
%:-write M_tau as a function of sigma and sum(integrate) the trace of M_tau for all values of sigma
integral(#(sigma) M_tau{1}(sigma), {0,1})
I don't think that you would be able to accomplish that with function handles. I think the best way you could do it would be to define a function as follows :
function x = myFunction(rowindex,colindex)
x = x * rowindex + colindex;
end
You would replace this function with your algorithm and then you could iterate over it doing the following:
for a=1:10
for b=1:10
x(a,b)=myFunction(a,b);
end
end
Related
The following MATLAB code loops through all elements of a matrix with size 2IJ x 2IJ.
for i=1:(I-2)
for j=1:(J-2)
ij1 = i*J+j+1; % row
ij2 = i*J+j+1 + I*J; % col
D1(ij1,ij1) = 2;
D1(ij1,ij2) = -1;
end
end
Is there any way I can parallelize it use MATLAB's parfor command? You can assume any element not defined is 0. So this matrix ends up being sparse (mostly 0s).
Before using parfor it is recommended to read the guidelines related to decide when to use parfor. Specially this:
Generally, if you want to make code run faster, first try to vectorize it.
Here vectorization can be used effectively to compute indices of the nonzero elements. Those indices are used in function sparse. For it you need to define one of i or j to be a column vector and another a row vector. Implicit expansion takes effect and indices are computed.
I = 300;
J = 300;
i = (1:I-2).';
j = 1:J-2;
ij1 = i*J+j+1;
ij2 = i*J+j+1 + I*J;
D1 = sparse(ij1, ij1, 2, 2*I*J, 2*I*J) + sparse(ij1, ij2, -1, 2*I*J, 2*I*J);
However for the comparison this can be a way of using parfor (not tested):
D1 = sparse (2*I*J, 2*I*J);
parfor i=1:(I-2)
for j=1:(J-2)
ij1 = i*J+j+1;
ij2 = i*J+j+1 + I*J;
D1 = D1 + sparse([ij1;ij1], [ij1;ij2], [2;-1], 2*I*J, 2*I*J) ;
end
end
Here D1 used as reduction variable.
I have the following piece of code, which runs through two nested for-loops and has an if condition in the middle:
N=1e4;
cond_array = [0:(N/2-1) -N/2+(0:(N/2-1))]/(N);
condition = 0.1;
arr_one = zeros(N, 1);
arr_two = zeros(N, 1);
for m=1:N
for k=1:N
if(abs(cond_array(k)) <= condition)
arr_one(m) = arr_one(m) + m*k;
else
arr_two(m) = arr_two(m) + m*k;
end
end
end
I'd like to optimize this code as I potentially need to use very large N (>1e4) and from my own experience, for-loops in MATLAB are often very CPU-consuming and not very efficient.
Is there a way to optimize this piece of code, maybe by using vectorized functions that work on entire arrays?
Here is a much faster (and readable) way to get the same result:
N=1e4;
cond_array = [0:(N/2-1) -N/2+(0:(N/2-1))]/(N);
condition = 0.1;
% this is so you don't check the same condition multiple times:
cond = abs(cond_array)<=condition;
% a vector of the positions in 'arr_one' and 'arr_two':
pos = (1:N).';
% instead of m*k(1)+m*k(2)... use: m*sum(k):
k1 = sum(pos(cond)); % for arr_one
k2 = sum(pos(~cond));% for arr_two
% the final result:
arr_one = pos.*k1;
arr_two = pos.*k2;
For the second case you mention in the comments, where m*k becomes exp((m-1)*(k-1)), we can again compute the sum of exp((m(1)-1)*(k(1)-1)) +...+ exp((m(1)-1)*(k(N)-1))... using vectorization, and then use some minimal loop to go over all ms:
% we define new vectors of constants:
k1 = pos(cond);
k2 = pos(~cond);
% define new functions for each of the arrays:
ek1 = #(m) sum(exp((m-1).*(k1-1)));
ek2 = #(m) sum(exp((m-1).*(k2-1)));
% and we use 'arrayfun' for the final result:
arr_one = arrayfun(ek1,1:N).';
arr_two = arrayfun(ek2,1:N).';
arrayfun is not faster than a for loop, just more compact. Using a for loop will be like this:
arr_one = zeros(N,1);
arr_two = zeros(N,1);
for k = 1:N
arr_one(k) = ek1(k);
arr_two(k) = ek2(k);
end
And here is another, more general, option using bsxfun:
ek = #(m,k) exp((m-1).*(k-1));
arr_one = sum(bsxfun(ek,1:N,k1)).';
arr_two = sum(bsxfun(ek,1:N,k2)).';
Look into the use of logical indexing.
Without understanding your code too well, I'd imagine that you can remove the loops through array functions similar to the below.
arr_one(abs(cond_array)<condition)
This will remove the need for the comparison function, which will allow you to use an anonymous array function in order to create your computation.
However, if I understand your code correctly, you're just adding the product of the position of the condition index and the position of the array index to the array.
If you do that, it will be much easier to just do something similar to the below.
position=(1:N)'
arr_one(abs(cond_array)<condition)=arr_one(abs(cond_array)<condition)+position(abs(cond_array)<condition)
In this statement, you're looking for all the items where the cond_array is less than the condition, and adding the position (effectively a proxy for the m and k in your previous example).
In terms of runtime, I think you have to be 1 to 3 orders of magnitude larger than 1e4 for this code to make a significant difference in runtimes.
I want to write a function that approximates integrals with the trapezoidal rule.
I first defined a function in one file:
function[y] = integrand(x)
y = x*exp(-x^2); %This will be integrand I want to approximate
end
Then I wrote my function that approximates definite integrals with lower bound a and upper bound b (also in another file):
function [result] = trapez(integrand,a,b,k)
sum = 0;
h = (b-a)/k; %split up the interval in equidistant spaces
for j = 1:k
x_j = a + j*h; %this are the points in the interval
sum = sum + ((x_j - x_(j-1))/2) * (integrand(x_(j-1)) + integrand(x_j));
end
result = sum
end
But when I want to call this function from the command window, using result = trapez(integrand,0,1,10) for example, I always get an error 'not enough input arguments'. I don't know what I'm doing wrong?
There are numerous issues with your code:
x_(j-1) is not defined, and is not really a valid Matlab syntax (assuming you want that to be a variable).
By calling trapez(integrand,0,1,10) you're actually calling integrand function with no input arguments. If you want to pass a handle, use #integrand instead. But in this case there's no need to pass it at all.
You should avoid variable names that coincide with Matlab functions, such as sum. This can easily lead to issues which are difficult to debug, if you also try to use sum as a function.
Here's a working version (note also a better code style):
function res = trapez(a, b, k)
res = 0;
h = (b-a)/k; % split up the interval in equidistant spaces
for j = 1:k
x_j1 = a + (j-1)*h;
x_j = a + j*h; % this are the points in the interval
res = res+ ((x_j - x_j1)/2) * (integrand(x_j1) + integrand(x_j));
end
end
function y = integrand(x)
y = x*exp(-x^2); % This will be integrand I want to approximate
end
And the way to call it is: result = trapez(0, 1, 10);
Your integrandfunction requires an input argument x, which you are not supplying in your command line function call
I have a set of three vectors (stored into a 3xN matrix) which are 'entangled' (e.g. some value in the second row should be in the third row and vice versa). This 'entanglement' is based on looking at the figure in which alpha2 is plotted. To separate the vector I use a difference based approach where I calculate the difference of one value with respect the three next values (e.g. comparing (1,i) with (:,i+1)). Then I take the minimum and store that. The method works to separate two of the three vectors, but not for the last.
I was wondering if you guys can share your ideas with me how to solve this problem (if possible). I have added my coded below.
Thanks in advance!
Problem in figures:
clear all; close all; clc;
%%
alpha2 = [-23.32 -23.05 -22.24 -20.91 -19.06 -16.70 -13.83 -10.49 -6.70;
-0.46 -0.33 0.19 2.38 5.44 9.36 14.15 19.80 26.32;
-1.58 -1.13 0.06 0.70 1.61 2.78 4.23 5.99 8.09];
%%% Original
figure()
hold on
plot(alpha2(1,:))
plot(alpha2(2,:))
plot(alpha2(3,:))
%%% Store start values
store1(1,1) = alpha2(1,1);
store2(1,1) = alpha2(2,1);
store3(1,1) = alpha2(3,1);
for i=1:size(alpha2,2)-1
for j=1:size(alpha2,1)
Alpha1(j,i) = abs(store1(1,i)-alpha2(j,i+1));
Alpha2(j,i) = abs(store2(1,i)-alpha2(j,i+1));
Alpha3(j,i) = abs(store3(1,i)-alpha2(j,i+1));
[~, I] = min(Alpha1(:,i));
store1(1,i+1) = alpha2(I,i+1);
[~, I] = min(Alpha2(:,i));
store2(1,i+1) = alpha2(I,i+1);
[~, I] = min(Alpha3(:,i));
store3(1,i+1) = alpha2(I,i+1);
end
end
%%% Plot to see if separation worked
figure()
hold on
plot(store1)
plot(store2)
plot(store3)
Solution using extrapolation via polyfit:
The idea is pretty simple: Iterate over all positions i and use polyfit to fit polynomials of degree d to the d+1 values from F(:,i-(d+1)) up to F(:,i). Use those polynomials to extrapolate the function values F(:,i+1). Then compute the permutation of the real values F(:,i+1) that fits those extrapolations best. This should work quite well, if there are only a few functions involved. There is certainly some room for improvement, but for your simple setting it should suffice.
function F = untangle(F, maxExtrapolationDegree)
%// UNTANGLE(F) untangles the functions F(i,:) via extrapolation.
if nargin<2
maxExtrapolationDegree = 4;
end
extrapolate = #(f) polyval(polyfit(1:length(f),f,length(f)-1),length(f)+1);
extrapolateAll = #(F) cellfun(extrapolate, num2cell(F,2));
fitCriterion = #(X,Y) norm(X(:)-Y(:),1);
nFuncs = size(F,1);
nPoints = size(F,2);
swaps = perms(1:nFuncs);
errorOfFit = zeros(1,size(swaps,1));
for i = 1:nPoints-1
nextValues = extrapolateAll(F(:,max(1,i-(maxExtrapolationDegree+1)):i));
for j = 1:size(swaps,1)
errorOfFit(j) = fitCriterion(nextValues, F(swaps(j,:),i+1));
end
[~,j_bestSwap] = min(errorOfFit);
F(:,i+1) = F(swaps(j_bestSwap,:),i+1);
end
Initial solution: (not that pretty - Skip this part)
This is a similar solution that tries to minimize the sum of the derivatives up to some degree of the vector valued function F = #(j) alpha2(:,j). It does so by stepping through the positions i and checks all possible permutations of the coordinates of i to get a minimal seminorm of the function F(1:i).
(I'm actually wondering right now if there is any canonical mathematical way to define the seminorm so we get our expected results... I initially was going for the H^1 and H^2 seminorms, but they didn't quite work...)
function F = untangle(F)
nFuncs = size(F,1);
nPoints = size(F,2);
seminorm = #(x,i) sum(sum(abs(diff(x(:,1:i),1,2)))) + ...
sum(sum(abs(diff(x(:,1:i),2,2)))) + ...
sum(sum(abs(diff(x(:,1:i),3,2)))) + ...
sum(sum(abs(diff(x(:,1:i),4,2))));
doSwap = #(x,swap,i) [x(:,1:i-1), x(swap,i:end)];
swaps = perms(1:nFuncs);
normOfSwap = zeros(1,size(swaps,1));
for i = 2:nPoints
for j = 1:size(swaps,1)
normOfSwap(j) = seminorm(doSwap(F,swaps(j,:),i),i);
end
[~,j_bestSwap] = min(normOfSwap);
F = doSwap(F,swaps(j_bestSwap,:),i);
end
Usage:
The command alpha2 = untangle(alpha2); will untangle your functions:
It should even work for more complicated data, like these shuffled sine-waves:
nPoints = 100;
nFuncs = 5;
t = linspace(0, 2*pi, nPoints);
F = bsxfun(#(a,b) sin(a*b), (1:nFuncs).', t);
for i = 1:nPoints
F(:,i) = F(randperm(nFuncs),i);
end
Remark: I guess if you already know that your functions will be quadratic or some other special form, RANSAC would be a better idea for larger number of functions. This could also be useful if the functions are not given with the same x-value spacing.
I've just started learning Matlab(a few days ago) and I have the following homework that I just don't know how to code it:
Write a script that creates a graphic using the positions of the roots of the polynomial function: p(z)=z^n-1 in the complex plan for various values for n-natural number(nth roots of unity)
so I am assuming the function you are using is p(z) = (z^n) - 1 where n is some integer value.
you can the find the roots of this equation by simply plugging into the roots function. The array passed to roots are the coefficients of the input function.
Example
f = 5x^2-2x-6 -> Coefficients are [5,-2,-6]
To get roots enter roots([5,-2,-6]). This will find all points at which x will cause the function to be equal to 0.
so in your case you would enter funcRoots = roots([1,zeros(1,n-1),-1]);
You can then plot these values however you want, but a simple plot like plot(funcRoots) would likely suffice.
To do in a loop use the following. Mind you, if there are multiple roots that are the same, there may be some overlap so you may not be able to see certain values.
minN = 1;
maxN = 10;
nVals = minN:maxN;
figure; hold on;
colors = hsv(numel(nVals));
legendLabels = cell(1,numel(nVals));
iter = 1;
markers = {'x','o','s','+','d','^','v'};
for n = nVals
funcRoots = roots([1,zeros(1,n-1),-1]);
plot(real(funcRoots),imag(funcRoots),...
markers{mod(iter,numel(markers))+1},...
'Color',colors(iter,:),'MarkerSize',10,'LineWidth',2)
legendLabels{iter} = [num2str(n),'-order'];
iter = iter+1;
end
hold off;
xlabel('Real Value')
ylabel('Imaginary Value')
title('Unity Roots')
axis([-1,1,-1,1])
legend(legendLabels)