I am new to Matlab and I have to use fixed point iteration to find the x value for the intersection between y = x and y = sqrt(10/x+4), which after graphing it, looks to be around 1.4. I'm using an initial guess of x1 = 0. This is my current Matlab code:
f = #(x)sqrt(10./(x+4));
x1 = 0;
xArray(10) = [];
for i = 1:10
x2 = f(x1);
xArray(i) = x2;
x1 = x1 + 1;
end
plot(xArray);
fprintf('%15.8e\n',xArray);
Now when I run this it seems like my x is approaching 0.8. Can anyone tell me what I am doing wrong?
Well done. You've made a decent start at this.
Lets look at the graphical solution. BTW, this is how I'd have done the graphical part:
ezplot(#(x) x,[-1 3])
hold on
ezplot(#(x) sqrt(10./(x+4)),[-1 3])
grid on
Or, I might subtract the two functions, then looking for a zero of the difference, so where it crosses the x axis.
This is what the fixed point iteration does anyway, trying to solve for x, such that
x = sqrt(10/(x+4))
So how would I change your code to fix it? First of all, I'd want to use more descriptive names for the variables. You don't get charged by the character, and making your code easier to read & follow will pay off greatly in the future for you.
There were a couple of code issues. To initialize a vector, use a form like one of these:
xArray = zeros(1,10);
xArray(1,10) = 0;
Note that if xArray was ALREADY defined because you have been working on this problem, the latter form will only zero out that single element. So the first form is best by a large margin. It affirmatively creates an array, or overwrites an existing array if it is already present in your workspace.
Finally, I like to initialize an array like this with something special, rather than zero, so we can see when an element was overwritten. NaNs are good for this.
Next, there was no need to add one to x1 in your code. Again, I'd strongly suggest using better variable names. It is also a good idea to use comments. Be liberal.
I'd suggest the idea of a convergence tolerance. You can also have an iteration counter.
f = #(x)sqrt(10./(x+4));
% starting value
xcurrent = 0;
% count the iterations, setting a maximum in maxiter, here 25
iter = 0;
maxiter = 25;
% initialize the array to store our iterations
xArray = NaN(1,maxiter);
% convergence tolerance
xtol = 1e-8;
% before we start, the error is set to be BIG. this
% just lets our while loop get through that first iteration
xerr = inf;
% the while will stop if either criterion fails
while (iter < maxiter) && (xerr > xtol)
iter = iter + 1;
xnew = f(xcurrent);
% save each iteration
xArray(iter) = xnew;
% compute the difference between successive iterations
xerr = abs(xnew - xcurrent);
xcurrent = xnew;
end
% retain only the elements of xArray that we actually generated
xArray = xArray(1:iter);
plot(xArray);
fprintf('%15.8e\n',xArray);
What was the result?
1.58113883e+00
1.33856229e+00
1.36863563e+00
1.36479692e+00
1.36528512e+00
1.36522300e+00
1.36523091e+00
1.36522990e+00
1.36523003e+00
1.36523001e+00
1.36523001e+00
For a little more accuracy to see how well we did...
format long g
xcurrent
xcurrent =
1.36523001364783
f(xcurrent)
ans =
1.36523001338436
By the way, it is a good idea to know why the loop terminated. Did it stop for insufficient iterations?
The point of my response here was NOT to do your homework, since you were close to getting it right anyway. The point is to show some considerations on how you might improve your code for future work.
There is no need to add 1 to x1. your output from each iteration is input for next iteration. So, x2 from output of f(x1) should be the new x1. The corrected code would be
for i = 1:10
x2 = f(x1);
xArray(i) = x2;
x1 = x2;
end
f(x)x^3+4*x^2-10 in [1,2] find an approximate root
Related
I am getting confused on how to properly set up this equation. To find a value of V(i,j). The end result would be plotting V over time. I understand that there needs to be loops to allow this equation to work, however I am lost when it comes to setting it up. Basically I am trying to take the sum from n=1 to infinity of (1-(-1)^n)/(n^4 *pi^4)*sin((n*pi*c*j)/L)*sin((n*pi*i)/L)
I originally thought that I should make it a while loop to increment n by 1 until I reach say 10 or so just to get an idea of what the output would look like. All of the variables were unknown and values were added again to see what the plot would look like.
I have down another code where the equation is just dependent on i and j. However with this n term, I am thrown off. Any advice would be great as to setting up the equation. Thank you.
L=10;
x=linspace(0,L,30);
t1= 50;
X=30;
p=1
c=t1/1000;
V=zeros(X,t1);
V(1,:)=0;
V(30,:)=0;
R=((4*p*L^3)/c);
n=1;
t=1:50;
while n < 10
for i=1:31
for j=1:50
V(i,j)=R*sum((1-(-1)^n)/(n^4 *pi^4)*sin((n*pi*c*j)/L)*sin((n*pi*i)/L));
end
end
n=n+1;
end
figure(1)
plot(V(i,j),t)
Various ways of doing so:
1) Computing the sum up to one Nmax in one shot:
Nmax = 30;
Vijn = #(i,j,n) R*((1-(-1)^n)/(n^4 *pi^4)*sin((n*pi*c*j)/L)*sin((n*pi*i)/L));
i = 1:31;
j = 1:50;
n = 1:Nmax;
[I,J,N] = ndgrid(i,j,n);
V = arrayfun(Vijn,I,J,N);
Vc = cumsum(V,3);
% now Vc(:,:,k) is sum_n=1^{k+1} V(i,j,n)
figure(1);clf;imagesc(Vc(:,:,end));
2) Looping indefinitely
n = 1;
V = 0;
i = 1:31;
j = 1:50;
[I,J] = meshgrid(i,j);
while true
V = V + R*((1-(-1)^n)/(n^4 *pi^4)*sin((n*pi*c*J)/L).*sin((n*pi*I)/L));
n = n + 1;
figure(1);clf;
imagesc(V);
title(sprintf('N = %d',n))
drawnow;
pause(0.25);
end
Note that in your example you won't need many terms, since:
Every second term is zero (for even n, the term 1-(-1)^n is zero).
The terms decay with 1/n^4. In norms: n=1 contributes ~2e4, n=3 contributes ~4e2, n=5 contributes 5e1, n=7 contributes ~14, etc. Visually, there is a small difference between n=1 and n=1+n=3 but barely a noticeable one for n=1+n=3+n=5.
Given that so few terms are needed, the first approach is probably the better one. Also, skip the even indices, as you don't need them.
I have a mathematical equation that describes a dynamical system as
The parameters are defined as follows
k1=1; S=1; Kd=1; p=2; tau=10; k2=1; ET=1; Km=1;
I coded the system as
y(1) = 1; % based on the y-axes starting point in the last figure
y(2) = y(1) + k1*S*Kd^p/(Kd^p + y(1)^p) - k2*ET*y(1)/(Km + y(1)); % to avoid errors
for t=1:100
y(t+1) = y(t+1) + (k1*S*Kd^p/(Kd^p + y(t)^p) - k2*ET*y(t+1)/(Km + y(t+1)));
end
plot(y);
Note that I did not use tau=10 for simplicity and instead used a delayed version by 1 instead of 10 (because I am not sure how to insert a delay of 10)
And obtained the following result
However, I need to obtain this
Can anyone help me rectify the mistake in my code?
Thanking you in advance.
If we assume that for Y(t) = 0 for t < 0 then you're code could be modified to produce a similar plot. However, it looks like the plot you are looking to generate uses different initial conditions. If you're just looking to measure Tc then it appears that the signal stabilizes with the period you're looking for.
k1=1; S=1; Kd=1; p=2; tau=10; k2=1; ET=1; Km=1;
% time step size (tau MUST be divisible by dt to ensure proper array indexing)
dt = 0.01;
% time series
t = -10:dt:100;
% initialize y to all zeros so that y(t)=0 for all t<0 (initial condition)
y = zeros(size(t));
% Find starting and ending indexes to iterate from t=0 to t=100-dt
idx0 = find(t == 0);
idx1 = numel(t)-1;
% initial condition y(0) = 1
y(idx0) = 1;
for n = idx0:idx1
% The indexing used here ensures the following equivalences.
% y(n+1) = y(t+dt)
% y(n) = y(t)
% y(n - round(tau/dt)) = y(t-tau)
%
% Note that (y(t+dt)-y(t))/dt is approximately y'(t)
% Solving for y(t+dt) we get the following formula
y(n+1) = y(n) + dt*((k1*S*Kd^p/(Kd^p + y(n - round(tau/dt))^p) - k2*ET*y(n)/(Km + y(n))));
end
% plot y(t) for t > 0
plot(t(t>0),y(t>0));
Result
Seeing as things stabilize we can take the values in one of the periods and use those for the initial conditions and we get.
Edit: To elaborate, the function contains a delay of 10 which means that instead of just a single initial condition at y(0), we also need to initialize all values from t=-10 to 0. In the code posted in this answer I arbitrarily assumed that y(t) = 0 for t < 0 and y(0) = 1 because I don't know otherwise. Once we run the code and see that the signal becomes periodic we can borrow the values from one of these periods to use those as the initial conditions.
From the diagram you posted we can use our intuition to guess that, before time 0, the signal probably looks something like the region highlighted in the figure below.
If, rather than using zero to initialize y at y < 0, we copy the values in the red highlighted region, then we get a plot that is more like what you desire.
To get the plot shown above I ran the script once, then found the indices in y for the part I wanted to use as initial conditions, then copied those into a new array.
init_cond = y(7004:8004);
Then I changed script to use this array as the initial condition and changed the initial y values to
y = zeros(size(t));
y(1:1001) = init_cond;
and ran the modified script again.
Edit 2: The built-in function dde23 appears to be applicable for your problem. To see an example run the command edit ddex1 in the command window.
I have a 10000x1 matrix. I need to find the percentage of information contained in each case. The method of doing so is to make another matrix that contains the sum of remaining cells.
Example:
% info is a 10000x1
% info_n contains percentages of information.
info_n = zeros([10000 1]);
info_n(1)= info(1) /sum(info);
info_n(2)=(info(1)+info(2)) /sum(info);
info_n(3)=(info(1)+info(2)+info(3))/sum(info);
I need to do this but all the way to info_n(10000).
So far, I tried this:
for i = 1:10000
info_n(i)=(info(i))/sum(info);
end
but I can't think of a way to add previous information. Thank you for the help!
You can use cumsum ("cumulative sum") for this task:
info_n = cumsum(info)/sum(info);
EDIT: #LuisMendo's comment sparked my curiosity. It seems that you actually gain something by using his method if the length N of the vector is below about 2^15 but the advantage after that drops. This is because cumsum probably needs O(N^2) but sum only needs O(N) time.
On the left you see the times as well as the ratio of the times of the two methods plotted against the logarithm of the size. On the right you see the actual absolute time differences, which varies a lot depending on what else is currently running:
Testing script:
N = 2.^(1:28);
y1 = zeros(size(N));
y2 = zeros(size(N));
vec = 1:numel(N);
for k=vec;
disp(k)
info = zeros(N(k),1);
% flawr's suggestion
tic
info_n = cumsum(info);
info_n = info_n / info_n(end);
y1(k) = toc;
% LuisMendo's suggestion
tic
info_m = cumsum(info)/sum(info);
y2(k) = toc;
end
subplot(1,2,1)
semilogy(vec,y1,vec,y2,vec,y1./y2,vec,vec.^0);
legend('LuisMendo','flawr','LM/f','Location','SouthEast');
subplot(1,2,2)
plot(vec,y1-y2)
I need help finding an integral of a function using trapezoidal sums.
The program should take successive trapezoidal sums with n = 1, 2, 3, ...
subintervals until there are two neighouring values of n that differ by less than a given tolerance. I want at least one FOR loop within a WHILE loop and I don't want to use the trapz function. The program takes four inputs:
f: A function handle for a function of x.
a: A real number.
b: A real number larger than a.
tolerance: A real number that is positive and very small
The problem I have is trying to implement the formula for trapezoidal sums which is
Δx/2[y0 + 2y1 + 2y2 + … + 2yn-1 + yn]
Here is my code, and the area I'm stuck in is the "sum" part within the FOR loop. I'm trying to sum up 2y2 + 2y3....2yn-1 since I already accounted for 2y1. I get an answer, but it isn't as accurate as it should be. For example, I get 6.071717974723753 instead of 6.101605982576467.
Thanks for any help!
function t=trapintegral(f,a,b,tol)
format compact; format long;
syms x;
oldtrap = ((b-a)/2)*(f(a)+f(b));
n = 2;
h = (b-a)/n;
newtrap = (h/2)*(f(a)+(2*f(a+h))+f(b));
while (abs(newtrap-oldtrap)>=tol)
oldtrap = newtrap;
for i=[3:n]
dx = (b-a)/n;
trapezoidsum = (dx/2)*(f(x) + (2*sum(f(a+(3:n-1))))+f(b));
newtrap = trapezoidsum;
end
end
t = newtrap;
end
The reason why this code isn't working is because there are two slight errors in your summation for the trapezoidal rule. What I am precisely referring to is this statement:
trapezoidsum = (dx/2)*(f(x) + (2*sum(f(a+(3:n-1))))+f(b));
Recall the equation for the trapezoidal integration rule:
Source: Wikipedia
For the first error, f(x) should be f(a) as you are including the starting point, and shouldn't be left as symbolic. In fact, you should simply get rid of the syms x statement as it is not useful in your script. a corresponds to x1 by consulting the above equation.
The next error is the second term. You actually need to multiply your index values (3:n-1) by dx. Also, this should actually go from (1:n-1) and I'll explain later. The equation above goes from 2 to N, but for our purposes, we are going to go from 1 to N-1 as you have your code set up like that.
Remember, in the trapezoidal rule, you are subdividing the finite interval into n pieces. The ith piece is defined as:
x_i = a + dx*i; ,
where i goes from 1 up to N-1. Note that this starts at 1 and not 3. The reason why is because the first piece is already taken into account by f(a), and we only count up to N-1 as piece N is accounted by f(b). For the equation, this goes from 2 to N and by modifying the code this way, this is precisely what we are doing in the end.
Therefore, your statement actually needs to be:
trapezoidsum = (dx/2)*(f(a) + (2*sum(f(a+dx*(1:n-1))))+f(b));
Try this and let me know if you get the right answer. FWIW, MATLAB already implements trapezoidal integration by doing trapz as #ADonda already pointed out. However, you need to properly structure what your x and y values are before you set this up. In other words, you would need to set up your dx before hand, then calculate your x points using the x_i equation that I specified above, then use these to generate your y values. You then use trapz to calculate the area. In other words:
dx = (b-a) / n;
x = a + dx*(0:n);
y = f(x);
trapezoidsum = trapz(x,y);
You can use the above code as a reference to see if you are implementing the trapezoidal rule correctly. Your implementation and using the above code should generate the same results. All you have to do is change the value of n, then run this code to generate the approximation of the area for different subdivisions underneath your curve.
Edit - August 17th, 2014
I figured out why your code isn't working. Here are the reasons why:
The for loop is unnecessary. Take a look at the for loop iteration. You have a loop going from i = [3:n] yet you don't reference the i variable at all in your loop. As such, you don't need this at all.
You are not computing successive intervals properly. What you need to do is when you compute the trapezoidal sum for the nth subinterval, you then increment this value of n, then compute the trapezoidal rule again. This value is not being incremented properly in your while loop, which is why your area is never improving.
You need to save the previous area inside the while loop, then when you compute the next area, that's when you determine whether or not the difference between the areas is less than the tolerance. We can also get rid of that code at the beginning that tries and compute the area for n = 2. That's not needed, as we can place this inside your while loop. As such, this is what your code should look like:
function t=trapintegral(f,a,b,tol)
format long; %// Got rid of format compact. Useless
%// n starts at 2 - Also removed syms x - Useless statement
n = 2;
newtrap = ((b-a)/2)*(f(a) + f(b)); %// Initialize
oldtrap = 0; %// Initialize to 0
while (abs(newtrap-oldtrap)>=tol)
oldtrap = newtrap; %//Save the old area from the previous iteration
dx = (b-a)/n; %//Compute width
%//Determine sum
trapezoidsum = (dx/2)*(f(a) + (2*sum(f(a+dx*(1:n-1))))+f(b));
newtrap = trapezoidsum; % //This is the new sum
n = n + 1; % //Go to the next value of n
end
t = newtrap;
end
By running your code, this is what I get:
trapezoidsum = trapintegral(#(x) (x+x.^2).^(1/3),1,4,0.00001)
trapezoidsum =
6.111776299189033
Caveat
Look at the way I defined your function. You must use element-by-element operations as the sum command inside the loop will be vectorized. Take a look at the ^ operations specifically. You need to prepend a dot to the operations. Once you do this, I get the right answer.
Edit #2 - August 18th, 2014
You said you want at least one for loop. This is highly inefficient, and whoever specified having one for loop in the code really doesn't know how MATLAB works. Nevertheless, you can use the for loop to accumulate the sum term. As such:
function t=trapintegral(f,a,b,tol)
format long; %// Got rid of format compact. Useless
%// n starts at 3 - Also removed syms x - Useless statement
n = 3;
%// Compute for n = 2 first, then proceed if we don't get a better
%// difference tolerance
newtrap = ((b-a)/2)*(f(a) + f(b)); %// Initialize
oldtrap = 0; %// Initialize to 0
while (abs(newtrap-oldtrap)>=tol)
oldtrap = newtrap; %//Save the old area from the previous iteration
dx = (b-a)/n; %//Compute width
%//Determine sum
%// Initialize
trapezoidsum = (dx/2)*(f(a) + f(b));
%// Accumulate sum terms
%// Note that we multiply each term by (dx/2), but because of the
%// factor of 2 for each of these terms, these cancel and we thus have dx
for n2 = 1 : n-1
trapezoidsum = trapezoidsum + dx*f(a + dx*n2);
end
newtrap = trapezoidsum; % //This is the new sum
n = n + 1; % //Go to the next value of n
end
t = newtrap;
end
Good luck!
I have a code which works perfectly, and I'm looking to make it more efficient.
t = -1:.001:1;
t_for_y = -50:.01:50;
x = zeros(size(t));
x(1001:end) = exp(-3 * t(1001:end));
h = zeros(size(t));
h(1001:end) = exp(-2 * t(1001:end)); % FIXED TYPO
for k = 1:length(t_for_y)
X(k)=trapz(t,x.*exp(-1i*t*t_for_y(k)));
H(k)=trapz(t,h.*exp(-1i*t*t_for_y(k)));
end
Y = X.*H;
for k = 1:length(t)
y(k) = (1/(2*pi))*trapz(t_for_y,Y.*exp(1i*t(k)*t_for_y));
end
plot(t,real(y));grid on;
I'd like to only use one for-loop or no for loops is this possible?
Is there a way of using doing this faster?
The trapz function can take a matrix as the second input (see help trapz for more info). This means that your first column can be replaced by the following:
t_i = 1i*t';
exp_t = bsxfun(#times,t_i,t_for_y); % Precompute for speed
xexp = bsxfun(#times,x',exp_t);
hexp = bsxfun(#times,h',exp_t);
% NOTE: As you've got it, X and H are identical - I assume this is a typo
X = trapz(t,xexp,1);
H = trapz(t,xexp,1);
Be aware that this will generate some fairly large matrices (~2000 X 10000), which can eat up your memory if you're not careful.
The second loop can be linearised in a similar manner:
% Using exp_t from the previous loop
yexp = bsxfun(#times,Y,exp_t);
% NOTE: As you've got it, X and H are identical - I assume this is a typo
y = trapz(t_for_y,xexp,2);
Again, this will use a lot of memory. You may find that you will save memory by using sparse matrices.
If memory is at a premium for you, then your original code is better (though you should preallocate X, H and y for a slight speed boost), as the time saved by linearising it is not really enough to justify the extra memory. If you've got memory aplenty, then this method is slightly faster.