My questions concerns the Mixed integer programming (MIP) in Scip:
I have the following code:
$\min trace(X)$
subject to
$$(A+D)^TX+X(A+D)=I\\
d_i \in \left\{0,1\right\} \mbox{ for } i=1,\ldots,n$$
where A is a n*n matrix and $D=diag(d_1,\ldots,d_n)$ is a diagonal matrix.
Since the matrix constraints are linear the equation can be transformed to a system of linear equations (via Kronecker product and vectorize operation), but this is limited to small n. Is it possible to solve the matrix equation directly with Scip? Is there a way to embed an external solver? Or do I have to write my own solver for the continuous lyapunov matrix equation?
You could try using the pip file format used for polynomial constraints and objective. See http://polip.zib.de/ and http://polip.zib.de/pipformat.php
You would have to do the matrix operations yourself or use ZIMPL
Matrix equations cannot be handled in SCIP. You would need to transform them into linear equations. Also, all the data has to be loaded into an LP solver at some time and needs to be formulated as usual constraints here as well. So even if SCIP itself would be able to handle matrix equations you are sooner or later to required to expand the problem.
Related
I'm trying to solve simultaneously the same ODE at different point (each point n is an independent vector of shape m) using the scipy BDF solver. In other world, i have a matrix n x m, and i want to solve n points (by solving, I mean make them advance in time with a while loop ), knowing that each point n are independant from each other.
Obviously you can loop on the different points, but this method takes too much time. Is there any way to make this faster and use it as a vectorized function?
I also tried to reshape my matrix to a 1D vector, but it looks like the solver compute the jacobian matrix of the complete vector, which takes too much time and is useless as the points along n are independent.
Maybe there is a way to specify that the derivatives of points n-m are zeros to speed up the jacobian computation ?
Thanks in advance for the answer
Edit:
Thanks for your answer #Lutz Lehmann. I was able to sped up the computation a little using jac_sparcity, that avoid computing a lot of unnecessary points.
The other improvement I can imagine is regarding the rate of progress h_abs : each independent ODE should have its own h_abs. Using the 1D vector method implies that all the ODE's are advancing at the same rate of progress h_abs i.e. the most restricting one. I don't know if there is anyway of doing this.
I am already using a vectorized atol built as an n x m matrix and reshaped, the same way as the complete set of ODE to make sure that the good tolerances are applied for each variable. I've never used jumba so far, but I will definitely have a look.
I have a piece of code in Fortran 90 in which I have to solve both a non-linear (for which I have to invert the Jacobian matrix) and a linear system of equations. When I say small I mean n unknowns for both operations, with n<=4. Unfortunately, n is not known a priori. What do you think is the best option?
I thought of writing explicit formulas for cases with n=1,2 and using other methods for n=3,4 (e.g. some functions of the Intel MKL libraries), for the sake of performance. Is this sensible or should I write explicit formulas for the inverse matrix also for n=3,4?
I need to compute the determinant of complex matrix which is symmetric. Size of matrix ranges from 500*500 to 2000*2000. Is there any subroutine for me to call? By the way, I use ifort to compile.
The easiest way would be to do an LU-decomposition as described here. I would suggest using LAPACK for this task...
This article has some code in C doing that for a real-valued symmetric matrix, so you need to exchange dspsv by zspsv to handle double-precision complex matrices.
I have some experimental data and I would like to fit them to obtain my parameters using the least-square method (Levenberg-Marquardt).
I am using two non-linear equations and I am using some computational programs (Origin and Matlab).
The first is:
y=A+B*(((2*pi*x)^2+Alfa4^2)*((2*pi*x)^2+Alfa5^2))/(((2*pi*x)^2+Alfa1^2)*((2*pi*x)^2+Alfa2^2)*((2*pi*x)^2+Alfa3^2));
Non-linear equation with the parameters (Alfa1,Alfa2,Alfa3,Alfa4,Alfa5)
And the second fitting equation is:
y=((T2^2+Lc^2*(2*pi*x)^2)/(((2*pi*x)^2*(Lc^2*(2*pi*x)^2+T8))+T6^2))*A1+G;
Rational function, i.e. quadratic function on the numerator and a 4th polynomial function on the denominator
I want to fit using this two equations, but I dont know how to do it. If someone want the experimental data I can post here.
Thank you very much,
Eduardo
What is the algorithm that Excel uses to calculate a 2nd-order polynomial regression (curve fitting)? Is there sample code or pseudo-code available?
I found a solution that returns the same formula that Excel gives:
Put together an augmented matrix of values used in a Least-Squares Parabola. See the sum equations in http://www.efunda.com/math/leastsquares/lstsqr2dcurve.cfm
Use Gaussian elimination to solve the matrix. Here is C# code that will do that http://www.codeproject.com/Tips/388179/Linear-Equation-Solver-Gaussian-Elimination-Csharp
After running that, the left-over values in the matrix (M) will equal the coefficients given in Excel.
Maybe I can find the R^2 somehow, but I don't need it for my purposes.
The polynomial trendlines in charts use least squares based on a QR decomposition method like the LINEST worksheet function ( http://support.microsoft.com/kb/828533 ). A second order or quadratic trend for given (x,y) data could be calculated using =LINEST(y,x^{1,2}).
You can call worksheet formulas from C# using the Worksheet.Evaluate method.
It depends, because there are a lot of ways to do such a thing depending on the data you supply and how important it is to have the curve pass through those points.
I'm guessing that you have many more points than you do coefficients in the polynomial (e.g. more than three points for a 2nd order curve).
If that's true, then the best you can do is least square fitting, which calculates the coefficients that minimize the mean square error between all the points and the resulting curve.
Since this is second order, my recommendation would be just create the damn second order terms and do a linear regression.
Ex. If you are doing z~second_order(x,y), it is equivalent to doing z~first_order(x,y,x^2,y^2, xy).