I have a matrix that is a function of some parameter A=A(x). I would like to find the points x where this matrix becomes singular. Example (I have a large matrix though):
syms x
A=[x sin(x); cos(x^2) 2.5];
So far I have been symbolically computing the determinant of the matrix and then used fzero or newtzero to find the roots of that characteristic equation. I.e.
detA = det(A);
fzero(matlabFunction(detA),startingGuess)
Then I found this: How to find out if a matrix is singular?, where it is advocated to not use the determinant under any circumstances.
Indeed the symbolic determinant calculation is terribly slow. However I tried to use rank(A) instead as suggested in the link and it does not seem to work for symbolic matrices.
Is there any way to implement the suggestions in the link for finding the roots of a characteristic equation of a matrix that is given symbolically?
A possible approach would be the following: a square matrix A is singular if and only if the homogeneous linear (with respect to the vector y) system A*y = 0 has nontrivial solutions y <> 0 (which is equivalent to det(A) = 0 and rank(A) = 0 among others. So a more or less standard, as I recall from the past, technique to compute such points x is to solve the nonlinear system
A(x)*y = 0 (1)
||y|| = 1 (2)
This way you can compute a point x* and a vector y* such that A(x*) is singular and y* is an eigenvector corresponding to the zero eigenvalue of A(x*).
If I remember correctly, you can also solve the somewhat easier system
A(x)*y = 0 (1)
<y,c> = 1 (2a)
where c is "almost" any nonzero random vector (normalize it to 1 to avoid numerical problems).
As a matter of fact there is an enormous bibliography on the subject - you can look for saddle-node bifurcation computations (in case A(x) is the Jacobian of a vector field), or for "distance to instability".
From a discussion with Ander Biguri it seems that the determinant is actually a perfectly fine method of approaching this problem. The problem seems to be to solve the final equation in a stable manner, which would be a different question.
I have the following matrix
R=(A-C)*inv(A+B-C-C')*(A-C');
where A and B are n by n matrices. I want to find n*n matrix C such that the determinant of R is minimized, SO:
C=arg min (det(R));
Is there any function in MATLAB that can handle this problem?
It seems like you are trying to find the minimum of an unconstrained multivariable function. This can probably be achieved with fminunc
fun = #(x)x(1)*exp(-(x(1)^2 + x(2)^2)) + (x(1)^2 + x(2)^2)/20;
x0 = [1,2];
[x,fval] = fminunc(fun,x0)
Note that there are no examples in the documentation where a matrix is used, this is probably because horrendous performance could be expected when trying to solve this problem for a matrix of any nontiny size. (This is not because of matlab, but because of the nature of the problem).
It is also good to realize that this method does not (cannot) guarantee an optimum, only a local optimum.
I have the following equation:
I want to do a exponential curve fitting using MATLAB for the above equation, where y = f(u,a). y is my output while (u,a) are my inputs. I want to find the coefficients A,B for a set of provided data.
I know how to do this for simple polynomials by defining states. As an example, if states= (ones(size(u)), u u.^2), this will give me L+Mu+Nu^2, with L, M and N being regression coefficients.
However, this is not the case for the above equation. How could I do this in MATLAB?
Building on what #eigenchris said, simply take the natural logarithm (log in MATLAB) of both sides of the equation. If we do this, we would in fact be linearizing the equation in log space. In other words, given your original equation:
We get:
However, this isn't exactly polynomial regression. This is more of a least squares fitting of your points. Specifically, what you would do is given a set of y and set pair of (u,a) points, you would build a system of equations and solve for this system via least squares. In other words, given the set y = (y_0, y_1, y_2,...y_N), and (u,a) = ((u_0, a_0), (u_1, a_1), ..., (u_N, a_N)), where N is the number of points that you have, you would build your system of equations like so:
This can be written in matrix form:
To solve for A and B, you simply need to find the least-squares solution. You can see that it's in the form of:
Y = AX
To solve for X, we use what is called the pseudoinverse. As such:
X = A^{*} * Y
A^{*} is the pseudoinverse. This can eloquently be done in MATLAB using the \ or mldivide operator. All you have to do is build a vector of y values with the log taken, as well as building the matrix of u and a values. Therefore, if your points (u,a) are stored in U and A respectively, as well as the values of y stored in Y, you would simply do this:
x = [u.^2 a.^3] \ log(y);
x(1) will contain the coefficient for A, while x(2) will contain the coefficient for B. As A. Donda has noted in his answer (which I embarrassingly forgot about), the values of A and B are obtained assuming that the errors with respect to the exact curve you are trying to fit to are normally (Gaussian) distributed with a constant variance. The errors also need to be additive. If this is not the case, then your parameters achieved may not represent the best fit possible.
See this Wikipedia page for more details on what assumptions least-squares fitting takes:
http://en.wikipedia.org/wiki/Least_squares#Least_squares.2C_regression_analysis_and_statistics
One approach is to use a linear regression of log(y) with respect to u² and a³:
Assuming that u, a, and y are column vectors of the same length:
AB = [u .^ 2, a .^ 3] \ log(y)
After this, AB(1) is the fit value for A and AB(2) is the fit value for B. The computation uses Matlab's mldivide operator; an alternative would be to use the pseudo-inverse.
The fit values found this way are Maximum Likelihood estimates of the parameters under the assumption that deviations from the exact equation are constant-variance normally distributed errors additive to A u² + B a³. If the actual source of deviations differs from this, these estimates may not be optimal.
I want to solve multiple measurement vector (MMV) sparse representation problem with CVX toolbox. I have a N*L matrix X. matrix X have only a few nonzero rows. I have system of equations Y=A*X. Y is M*L matrix of measurements (M
min Relax(X)
subject to Y=A*X
Realx(.) is a function that applies norm 1 to the vector t. (N*1)vector t consists of the norm 2 of each rows of matrix X. i.e. Relax(X)= norm_1(t) and t(i)=norm_2(X(i,:))
I can’t transform my objective function into a language that CVX can understand and solve.
Please tell me how I should change the problem objective and constraints that CVX could solve it.
'norms' is the cvx command you're looking for. Suppose sigma is some known parameter that allows Y to be only approximately equal to A*X (e.g. I tried it with sigma=10e-6). Then you could use this code:
cvx_begin separable
variable X(n,n)
minimize( norms(X,2,1) )
subject to
norm(Y - A*X,2)<= 10*sigma
cvx_end
I need to solve the following SOCP in Matlab:
argmin_x ||R*x||_2 s.t. s^H * x = 1 and ||x||_2 < d,
where x is an Nx1 vector and R is an MxN matrix.
CVX can solve this type of problem. However, CVX requires me to give R and does not allow me to instead give a function handle that will return R*x. This is a problem for me since once R becomes large, computing R*x directly takes too long. There exists an efficient algorithm for computing R*x that I would like to take advantage of, so I am hoping that there is another SOCP solver that I could use.