I am searching for a way to get rid of the following loop (over theta):
for i=1:1:length(theta)
V2_ = kV2*cos(theta(i));
X = X0+V2_;
Y = Y0-V2_*(k1-k2);
Z = sqrt(X.^2-Z0-4*V2_.*(k.^2*D1+k1));
pktheta(:,i)=exp(-t/2*V2_).*(cosh(t/2*Z)+...
Y./((k1+k2)*Z).*sinh(t/2*Z));
end
where X0,Y0,Z0 and kV2 are dependent on the vector k (same size). t, D1, k1 and k2 are numbers. Since I have to go through this loop several times, how can I speed it up?
Thanks
Try this -
N = numel(theta);
V2_ = kV2*cos(theta(1:N));
X0 = repmat(X0,[1 N]);
Y0 = repmat(Y0,[1 N]);
Z0 = repmat(Z0,[1 N]);
X = X0 + V2_;
Y = Y0-V2_*(k1-k2);
Z = sqrt(X.^2-Z0-4.*V2_ .* repmat(((1:N).^2)*D1 + k1.*ones(1,N),[size(X0,1) 1]));
pktheta = exp(-t/2*V2_).*(cosh(t/2*Z) + Y./((k1+k2)*Z).*sinh(t/2*Z));
Definitely BSXFUN must be faster, if someone could post with it.
Related
I'm having some issues getting my RK2 algorithm to work for a certain second-order linear differential equation. I have posted my current code (with the provided parameters) below. For some reason, the value of y1 deviates from the true value by a wider margin each iteration. Any input would be greatly appreciated. Thanks!
Code:
f = #(x,y1,y2) [y2; (1+y2)/x];
a = 1;
b = 2;
alpha = 0;
beta = 1;
n = 21;
h = (b-a)/(n-1);
yexact = #(x) 2*log(x)/log(2) - x +1;
ye = yexact((a:h:b)');
s = (beta - alpha)/(b - a);
y0 = [alpha;s];
[y1, y2] = RungeKuttaTwo2D(f, a, b, h, y0);
error = abs(ye - y1);
function [y1, y2] = RungeKuttaTwo2D(f, a, b, h, y0)
n = floor((b-a)/h);
y1 = zeros(n+1,1); y2 = y1;
y1(1) = y0(1); y2(1) = y0(2);
for i=1:n-1
ti = a+(i-1)*h;
fvalue1 = f(ti,y1(i),y2(i));
k1 = h*fvalue1;
fvalue2 = f(ti+h/2,y1(i)+k1(1)/2,y2(i)+k1(2)/2);
k2 = h*fvalue2;
y1(i+1) = y1(i) + k2(1);
y2(i+1) = y2(i) + k2(2);
end
end
Your exact solution is wrong. It is possible that your differential equation is missing a minus sign.
y2'=(1+y2)/x has as its solution y2(x)=C*x-1 and as y1'=y2 then y1(x)=0.5*C*x^2-x+D.
If the sign in the y2 equation were flipped, y2'=-(1+y2)/x, one would get y2(x)=C/x-1 with integral y1(x)=C*log(x)-x+D, which contains the given exact solution.
0=y1(1) = -1+D ==> D=1
1=y1(2) = C*log(2)-1 == C=1/log(2)
Additionally, the arrays in the integration loop have length n+1, so that the loop has to be from i=1 to n. Else the last element remains zero, which gives wrong residuals for the second boundary condition.
Correcting that and enlarging the computation to one secant step finds the correct solution for the discretization, as the ODE is linear. The error to the exact solution is bounded by 0.000285, which is reasonable for a second order method with step size 0.05.
t(1)= 0;
y0= 3;
y(1)= y0;
h=0.1;
for n = 1:11
k1 = 2*y(n)+(t(n))^2;
k2 = 2*(y(n)+h/2*k1)+(t(n)+h/2^2);
k3 = 2*(y(n)+h/2*k2)+(t(n)+h/2)^2;
k4 = 2*(y(n)+h*k3)+(t(n)+h)^2;
t(n+1) = t(n)+h;
y(n+1) = y(n)+h/6*(k1+2*k2+2*k3+k4);
end
t1(1)= 0;
y1(1)= y0;
h1=0.01;
for n = 1:101
k1 = 2*y1(n)+(t1(n))^2;
k2 = 2*(y1(n)+h1/2*k1)+(t1(n)+h1/2^2);
k3 = 2*(y1(n)+h1/2*k2)+(t1(n)+h1/2)^2;
k4 = 2*(y1(n)+h1*k3)+(t1(n)+h1)^2;
t1 (n+1) = t1(n)+h1;
y1 (n+1) = y1(n)+h1/6*(k1+2*k2+2*k3+k4);
end
plot (t,y,t1,y1) will show very similar curves, In this plot MAX difference may be very small, such as 0.001, 0.0001 or whatever. Actually, I want more clearly the difference, and plot (t,y,t1,y1) doesn't work so good. I think long format for k1,..,k4 and y is appropriate. Also, I tried
z (1) = 0;
for j = 2:11
z = [ y (j) - y ( (j-1)*10*1 ) (11) ];
end
What about plotting the difference explicitly. Interpolate to get y at t1:
y_t1 = interp1(t,y,t1);
plot(t1,y_t1-y1)
On an unrelated note, you are growing a vector inside a loop. No doubt mlint has told you this is bad. Rather initialize your vectors like this:
t = zeros(1,11); % i.e. don't do this: t(1)= 0;
Assume we have three equations:
eq1 = x1 + (x1 - x2) * t - X == 0;
eq2 = z1 + (z1 - z2) * t - Z == 0;
eq3 = ((X-x1)/a)^2 + ((Z-z1)/b)^2 - 1 == 0;
while six of known variables are:
a = 42 ;
b = 12 ;
x1 = 316190;
z1 = 234070;
x2 = 316190;
z2 = 234070;
So we are looking for three unknown variables that are:
X , Z and t
I wrote two method to solve it. But, since I need to run these code for 5.7 million data, it become really slow.
Method one (using "solve"):
tic
S = solve( eq1 , eq2 , eq3 , X , Z , t ,...
'ReturnConditions', true, 'Real', true);
toc
X = double(S.X(1))
Z = double(S.Z(1))
t = double(S.t(1))
results of method one:
X = 316190;
Z = 234060;
t = -2.9280;
Elapsed time is 0.770429 seconds.
Method two (using "fsolve"):
coeffs = [a,b,x1,x2,z1,z2]; % Known parameters
x0 = [ x2 ; z2 ; 1 ].'; % Initial values for iterations
f_d = #(x0) myfunc(x0,coeffs); % f_d considers x0 as variables
options = optimoptions('fsolve','Display','none');
tic
M = fsolve(f_d,x0,options);
toc
results of method two:
X = 316190; % X = M(1)
Z = 234060; % Z = M(2)
t = -2.9280; % t = M(3)
Elapsed time is 0.014 seconds.
Although, the second method is faster, but it still needs to be improved. Please let me know if you have a better solution for that. Thanks
* extra information:
if you are interested to know what those 3 equations are, the first two are equations of a line in 2D and the third equation is an ellipse equation. I need to find the intersection of the line with the ellipse. Obviously, we have two points as result. But, let's forget about the second answer for simplicity.
My suggestion it's to use the second approce,which it's the recommended by matlab for nonlinear equation system.
Declare a M-function
function Y=mysistem(X)
%X(1) = X
%X(2) = t
%X(3) = Z
a = 42 ;
b = 12 ;
x1 = 316190;
z1 = 234070;
x2 = 316190;
z2 = 234070;
Y(1,1) = x1 + (x1 - x2) * X(2) - X(1);
Y(2,1) = z1 + (z1 - z2) * X(2) - X(3);
Y(3,1) = ((X-x1)/a)^2 + ((Z-z1)/b)^2 - 1;
end
Then for solving use
x0 = [ x2 , z2 , 1 ];
M = fsolve(#mysistem,x0,options);
If you may want to reduce the default precision by changing StepTolerance (default 1e-6).
Also for more increare you may want to use the jacobian matrix for greater efficencies.
For more reference take a look in official documentation:
fsolve Nonlinear Equations with Analytic Jacobian
Basically giving the solver the Jacobian matrix of the system(and special options) you can increase method efficency.
With the following variables:
m = 1:4; n = 1:32;
phi = linspace(0, 2*pi, 100);
theta = linspace(-pi, pi, 50);
S_mn = <a 4x32 coefficient matrix, corresponding to m and n>;
how do I compute the sum over m and n of S_mn*exp(1i*(m*theta + n*phi)), i.e.
I've thought of things like
[m, n] = meshgrid(m,n);
[theta, phi] = meshgrid(theta,phi);
r_mn = S_mn.*exp(1i*(m.*theta + n.*phi));
thesum = sum(r_mn(:));
but that requires theta and phi to have the same number of elements as m and n, and it gives me just one element in return - I want a matrix the the size of meshgrid(theta,phi), regardless of the sizes of theta and phi (i.e. I want to be able to evaluate the sum as a function of theta and phi).
How do I do this calculation in matlab?
Since I don't know what S is...
S = randn(4,32);
[m,n] = ndgrid(1:4,1:32);
fun = #(theta,phi) sum(sum(S.*exp(sqrt(-1)*(m*theta + n*phi))));
Works fine for me.
fun(pi,3*pi/2)
ans =
-15.8643373238676 - 1.45785698818839i
If you now wish to do this for a large set of values phi and theta, a pair of loops now are the trivial solution. Or, you can do it all in one computation, although the arrays will get larger. Still not hard. WTP?
You do realize that both meshgrid and ndgrid take more than just two arguments? So it is time to learn how to use bsxfun, and then squeeze.
[m,n,theta,phi] = ndgrid(1:4,1:32,linspace(-pi, pi, 50),linspace(0, 2*pi, 100));
res = bsxfun(#times,S,exp(sqrt(-1)*(m.*theta + n.*phi)));
res = squeeze(sum(sum(res,1),2));
Or do this, which will be a bit faster. The previous computation took my machine .07 seconds. This last one took .05, so some savings by using bsxfun heavily.
m = (1:4)';
n = 1:32;
[theta,phi] = ndgrid(linspace(-pi, pi, 50),linspace(0, 2*pi, 100));
theta = reshape(theta,[1,1,size(theta)]);
phi = reshape(phi,size(theta));
res = bsxfun(#plus,bsxfun(#times,m,theta*sqrt(-1)),bsxfun(#times,n,phi*sqrt(-1)));
res = bsxfun(#times,S,exp(res));
res = squeeze(sum(sum(res,1),2));
If you need to do the above 2000 times, so it should take 100 seconds to do. WTP? Get some coffee and relax.
First save the size of each variable:
size_m = size(m);
size_n = size(n);
size_theta = size(theta);
size_phi = size(phi);
Use ngrid function like this:
[theta, phi, m, n] = ngrid(theta, phi, m, n)
This will give you an array with 4 dimensions (one for each of your variables: theta, phi, m, n). Now you can calculate this:
m.*theta + n.*phi
Now you need to make S_mn have 4 dimensions with sizes size_theta, size_phi, size_m, size_n like this:
S_tpmn = repmat(S_mn, [size_theta size_phi size_m size_n]);
Now you can calculate your sum like this:
aux_sum = S_tpmn.*exp(1i*(m.*theta + n.*phi));
Finally you can sum along the last 2 dimensions (m and n) to get an array with 2 dimensions with size size_theta by size_phi:
final_sum = sum(sum(aux_sum, 4), 3);
Note: I don't have access to Matlab right now, so I can't test if this actually works.
There are several ways you could go about this.
One way is to create a function(-handle) that returns the sum as a function of theta and phi, and then use arrayfun to do the sums. Another is to fully vectorize the computation, though that will use more memory.
The arrayfun version:
[m, n] = meshgrid(m,n);
sumHandle = #(theta,phi)sum(reshape(...
S_mn.*exp(1i(m*theta + n*phi)),...
[],1))
[theta, phi] = meshgrid(theta,phi);
sumAsFunOfThetaPhi = arrayfun(sumHandle,theta,phi);
The vectorized version:
[m, n] = meshgrid(m,n);
m = permute(m(:),[2 4 1 3]); %# vector along dim 3
n = permute(n(:),[2 3 4 1]); %# vector along dim 4
S_mn = repmat( permute(S_mn,[3 4 1 2]), length(theta),length(phi));
theta = theta(:); %# vector along dim 1 (phi is along dim 2 b/c of linspace)
fullSum = S_mn.* exp( 1i*(...
bsxfun(#plus,...
bsxfun(#times, m, theta),...
bsxfun(#times, n, phi),...
)));
sumAsFunOfThetaPhi = sum(sum( fullSum, 3),4);
I made a Wiener filter simulation in matlab but it seem that I made a mistake in the code as the result is incorrect. I would be grateful if somebody glanced at the code and pointed out the error.
Here is the code:
k = 1:100;
sigma = 0.1;
s1 = randn(1,100);
n1 = randn(1,100) * sigma;
Pnum = [1 0.1];
Pdenum = [1 0.9];
Gnum = [1 0.9];
Gdenum = [1 0.1];
n = filter(Pnum, Pdenum, n1);
s = filter(Gnum, Gdenum, s1);
x = n + s;
rxx = xcorr(x);
rxs = xcorr(x,s);
toeplitz_rxx = toeplitz(rxx);
wopt = inv(toeplitz_rxx) * rxs;
wopt = inv(toeplitz_rxx) * transpose(rxs);
s_hat = filter(wopt,1,x);
error = x - s_hat; %this is where the wrong result is being calculated
plot(k,error, 'r',k,s_hat, 'b', k, x, 'g');
The problem is that the error calculated is exactly a mirror image of the s_hat.
Surely this cant be right.
Thanks in advance.
No problem.
I have perfectly a void idea of what you are trying to do. But if what you are trying to do is to get some signal to noise ratio, and trying to mimic those effects through some linear 1st order systems, you should correct your P and G polynomials, which are totally wrong, if that is the case. Type help filter and see the A and B definitions.
This is the correct code:
k = 1:100;
sigma = 0.1;
s1 = randn(1,100);
n1 = randn(1,100) * sigma;
Pnum = [0 0.1];
Pdenum = [1 -0.9];
Gnum = [0 0.9];
Gdenum = [1 -0.1];
n = filter(Pnum, Pdenum, n1);
s = filter(Gnum, Gdenum, s1);
x = n + s;
rxx = xcorr(x);
rxs = xcorr(x,s);
toeplitz_rxx = toeplitz(rxx);
wopt = inv(toeplitz_rxx) * transpose(rxs);
s_hat = filter(wopt,1,x);
error = x - s_hat; %this is where the wrong result is being calculated
plot(k,error, 'r',k,s_hat, 'b', k, x, 'g');
There are still an ill-conditioning on the toeplitz. I leave it that way, for you still have fun for dealing with. This is easy to solve too.
THe guys are true. StackOverflow is NOT for solve your homework, but for helping you in finding the light.