I have a function X_t which I have defined anonymously to take as variables t which is a scalar time and z which is a vector of potentially arbitrary dimension.
That is,
X_t = #(t,z) fun(t,z).
I want to find the integral L2 norm of this object over the space of z's. That is, I want to find
X_t_norm = #(t) integral(#(z) abs(X_t(t,z))^2, -infinity,infinity).
Now clearly two things are a problem here.
Firstly, taking the limits as infinite isn't going to work, but I think I should be fine to take the limits as just large numbers (100 would certainly be enough for my purposes).
However, the real problem comes with taking this integral. My variable z is a vector of somewhat arbitrary (even) dimension and as a result I've gotten stuck figuring out how to pass z into the integral in such a way that I can compute this n-dimensional integral. (Basically I want to vary the length of z for different cases and compute the L2 norm for each of these cases).
In particular, I'm not sure how to tell the integral function to compute the integral over every component of z between the limits discussed above.
Any help would be greatly appreciated!
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I have a Matlab function G(x,y,z). At each given (x,y,z), G(x,y,z) is a scalar. x=(x1,x2,...,xK) is a Kx1 vector.
Let us fix y,z at some given values. I would like your help to understand how to compute the derivative of G with respect to xk evaluated at a certain x.
For example, suppose K=3
function f= G(x1,x2,x3,y,z)
f=3*x1*sin(z)*cos(y)+3*x2*sin(z)*cos(y)+3*x3*sin(z)*cos(y);
end
How do I compute the derivative of G(x1,x2,x3,4,3) wrto x2 and then evaluate it at x=(1,2,6)?
You're looking for the partial derivative of dG/dx2
So the first thing would be getting rid of your fixed variables
G2 = #(x2) G(1,x2,6,4,3);
The numerical derivatives are finite differences, you need to choose an step h for your finite difference, and an appropriate method
The simplest one is
(G2(x2+h)-G2(x2))/h
You can make h as small as your numeric precision allows you to. At the limit h -> 0 the finite difference is the partial derivative
I have the following problem. I have a N x N real matrix called Z(x; t), where x and t might be vectors in general. I have N_s observations (x_k, Z_k), k=1,..., N_s and I'd like to find the vector of parameters t that better approximates the data in the least square sense, which means I want t that minimizes
S(t) = \sum_{k=1}^{N_s} \sum_{i=1}^{N} \sum_{j=1}^N (Z_{k, i j} - Z(x_k; t))^2
This is in general a non-linear fitting of a matrix function. I'm only finding examples in which one has to fit scalar functions which are not immediately generalizable to a matrix function (nor a vector function). I tried using the scipy.optimize.leastsq function, the package symfit and lmfit, but still I don't manage to find a solution. Eventually, I'm ending up writing my own code...any help is appreciated!
You can do curve-fitting with multi-dimensional data. As far as I am aware, none of the low-level algorithms explicitly support multidimensional data, but they do minimize a one-dimensional array in the least-squares sense. And the fitting methods do not really care about the "independent variable(s)" x except in that they help you calculate the array to be minimized - perhaps to calculate a model function to match to y data.
That is to say: if you can write a function that would take the parameter values and calculate the matrix to be minimized, just flatten that 2-d (on n-d) array to one dimension. The fit will not mind.
Can someone please explain why the following symmetric function cannot pass a certain limit of negative values?
D = 0.1; l = 4;
c = #(x,v) (v/D).*exp(-v*x/D)./(1-exp(-v*l/D));
v_vec = -25:0.01:25;
figure(2)
hold on
plot(v_vec,c(l,v_vec),'b')
plot(v_vec,c(0,v_vec),'r')
Notice at the figure where the blue line chops, this is where I get inf/nan values.
It seems that Matlab is trying to compute a result that is too large, outputs +inf, and then operates on that, which yields +/- inf and NaNs.
For instance, at v=-25, part of the function computes exp(-(-25)*4/0.1), which is exp(1000), and that outputs +inf. (larger than the largest representable double precision float).
You can potentially solve that problem by rewriting your function to avoid operating of such very large (or very small) numbers, say by reorganising the fraction containing exp() functions.
I did encounter the same hurdle using exp() with arguments triggering overflow. Sometimes it is difficult to trace back numeric imprecision or convergence errors. In principle the function definition using exp() only create intermediate issues as your purpose as a transition function. The intention I guess was to provide a continuous function.
My solution to this problem is to divide the argument into regions and provide in each region an approximation function. In your case zero for negative x and proportional to x for positive x. In between you can use the orginal function. Care should be taken to match the approximation at the borders of the regions and the number of continuous differentiations which is important for convergence in loops.
I have an integration function which does not have indefinite integral expression.
Specifically, the function is f(y)=h(y)+integral(#(x) exp(-x-1/x),0,y) where h(y) is a simple function.
Matlab numerically computes f(y) well, but I want to compute the following function.
g(w)=w*integral(1-f(y).^(1/w),0,inf) where w is a real number in [0,1].
The problem for computing g(w) is handling f(y).^(1/w) numerically.
How can I calculate g(w) with MATLAB? Is it impossible?
Expressions containing e^(-1/x) are generally difficult to compute near x = 0. Actually, I am surprised that Matlab computes f(y) well in the first place. I'd suggest trying to compute g(w)=w*integral(1-f(y).^(1/w),epsilon,inf) for epsilon greater than zero, then gradually decreasing epsilon toward 0 to check if you can get numerical convergence at all. Convergence is certainly not guaranteed!
You can calculate g(w) using the functions you have, but you need to add the (ArrayValued,true) name-value pair.
The option allows you to specify a vector-valued w and allows the nested integral call to receive a vector of y values, which is how integral naturally works.
f = #(y) h(y)+integral(#(x) exp(-x-1/x),0,y,'ArrayValued',true);
g = #(w) w .* integral(1-f(y).^(1./w),0,Inf,'ArrayValued',true);
At least, that works on my R2014b installation.
Note: While h(y) may be simple, if it's integral over the positive real line does not converge, g(w) will more than likely not converge (I don't think I need to qualify that, but I'll hedge my bets).
I'm not too familiar with MATLAB or computational mathematics so I was wondering how I might solve an equation involving the sum of squares, where each term involves two vectors- one known and one unknown. This formula is supposed to represent the error and I need to minimize the error. I think I'm supposed to use least squares but I don't know too much about it and I'm wondering what function is best for doing that and what arguments would represent my equation. My teacher also mentioned something about taking derivatives and he formed a matrix using derivatives which confused me even more- am I required to take derivatives?
The problem that you must be trying to solve is
Min u'u = min \sum_i u_i^2, u=y-Xbeta, where u is the error, y is the vector of dependent variables you are trying to explain, X is a matrix of independent variables and beta is the vector you want to estimate.
Since sum u_i^2 is diferentiable (and convex), you can evaluate the minimal of this expression calculating its derivative and making it equal to zero.
If you do that, you find that beta=inv(X'X)X'y. This maybe calculated using the matlab function regress http://www.mathworks.com/help/stats/regress.html or writing this formula in Matlab. However, you should be careful how to evaluate the inverse (X'X) see Most efficient matrix inversion in MATLAB