I'm trying to inverse a matrix with version Boost boost_1_37_0 and MTL mtl4-alpha-1-r6418. I can't seem to locate the matrix inversion code. I've googled for examples and they seem to reference lu.h that seems to be missing in the above release(s). Any hints?
#Matt suggested copying lu.h, but that seems to be from MTL2 rather than MTL4. I'm having trouble compiling with MTL2 with VS05 or higher.
So, any idea how to do a matrix inversion in MTL4?
Update: I think I understand Matt better and I'm heading down this ITL path.
Looks like you use lu_factor, and then lu_inverse. I don't remember what you have to do with the pivots, though. From the documentation.
And yeah, like you said, it looks like their documentations says you need lu.h, somehow:
How do I invert a matrix?
The first question you should ask
yourself is whether you want to really
compute the inverse of a matrix or if
you really want to solve a linear
system. For solving a linear system of
equations, it is not necessary to
explicitly compute the matrix inverse.
Rather, it is more efficient to
compute triangular factors of the
matrix and then perform forward and
backward triangular solves with the
factors. More about solving linear
systems is given below. If you really
want to invert a matrix, there is a
function lu_inverse() in mtl/lu.h.
If nothing else, you can look at lu.h on their site.
I've never used boost or MTL for matrix math but I have used JAMA/TNT.
This page http://wiki.cs.princeton.edu/index.php/TNT shows how to take a matrix inverse. The basic method is library-independent:
factor matrix M into XY where X and Y are appropriate factorizations (LU would be OK but for numerical stability I would think you would want to use QR or maybe SVD).
solve I = MN = (XY)N for N with the prerequisite that M has been factored; the library should have a routine for this.
In MTL4 use this:
mtl::matrix::inv(Matrix const &A, MatrixOut &Inv);
Here is a link to the api.
Related
I'm trying to use Eigen::CholmodSupernodalLLT for Cholesky decomposition, however, it seems that I could not get matrixL() and matrixU(). How can I extract matrixL() and matrixU() from Eigen::CholmodSupernodalLLT for future use?
A partial answer to integrate what others have said.
Consider Y ~ MultivariateNormal(0, A). One may want to (1) evaluate the (log-)likelihood (a multivariate normal density), (2) sample from such density.
For (1), it is necessary to solve Ax = b where A is symmetric positive-definite, and compute its log-determinant. (2) requires L such that A = L * L.transpose() since Y ~ MultivariateNormal(0, A) can be found as Y = L u where u ~ MultivariateNormal(0, I).
A Cholesky LLT or LDLT decomposition is useful because chol(A) can be used for both purposes. Solving Ax=b is easy given the decomposition, andthe (log)determinant can be easily derived from the (sum)product of the (log-)components of D or the diagonal of L. By definition L can then be used for sampling.
So, in Eigen one can use:
Eigen::SimplicialLDLT solver(A) (or Eigen::SimplicialLLT), when solver.solve(b) and calculate the determinant using solver.vectorD().diag(). Useful because if A is a covariance matrix, then solver can be used for likelihood evaluations, and matrixL() for sampling.
Eigen::CholmodDecomposition does not give access to matrixL() or vectorD() but exposes .logDeterminant() to achieve the (1) goal but not (2).
Eigen::PardisoLDLT does not give access to matrixL() or vectorD() and does not expose a way to get the determinant.
In some applications, step (2) - sampling - can be done at a later stage so Eigen::CholmodDecomposition is enough. At least in my configuration, Eigen::CholmodDecomposition works 2 to 5 times faster than Eigen::SimplicialLDLT (I guess because of the permutations done under the hood to facilitate parallelization)
Example: in Bayesian spatial Gaussian process regression, the spatial random effects can be integrated out and do not need to be sampled. So MCMC can proceed swiftly with Eigen::CholmodDecomposition to achieve convergence for the uknown parameters. The spatial random effects can then be recovered in parallel using Eigen::SimplicialLDLT. Typically this is only a small part of the computations but having matrixL() directly from CholmodDecomposition would simplify them a bit.
You cannot do this using the given class. The class you are referencing is equotation solver (which indeed uses cholesky decomposition). To decompose your matrix you should rather use Eigen::LLT. Code example from their website:
MatrixXd A(3,3);
A << 4,-1,2, -1,6,0, 2,0,5;
LLT<MatrixXd> lltOfA(A);
MatrixXd L = lltOfA.matrixL();
MatrixXd U = lltOfA.matrixU();
As reported somewhere else, e.g., it cannot be done easily.
I am copying a possible recommendation (answered by Gael Guennebaud himself), even if somewhat old:
If you really need access to the factor to do your own cooking, then
better use the built-in SimplicialL{D}LT<> class. Extracting the
factors from the supernodal internal represations of Cholmod/Pardiso
is indeed not straightforward and very rarely needed. We have to
check, but if Cholmod/Pardiso provide routines to manipulate the
factors, like applying it to a vector, then we could let
matrix{L,U}() return a pseudo expression wrapping these routines.
Developing code for extracting this is likely beyond SO, and probably a topic for a feature request.
Of course, the solution with LLT is at hand (but not the topic of the OP).
I have a sparse matrix whose shape is 570000*3000. I tried nima to do NMF (using the default nmf method, and set max_iter to 65). However, I found nimfa very slow. Have anyone used a faster library(can be used by Python/R) or software to do NMF?
This question (Sparse matrix factorization with Nimfa is very slow with implicit zeros) shows that you might want to try the .todense() version of the code perhaps?
First, please excuse my ignorance in this field, I'm a programmer by trade but have been stuck in a situation a little beyond my expertise (in math and signals processing).
I have a Matlab script that I need to port to a C++ program (without compiling the matlab code into a DLL). It uses the hilbert() function with one argument. I'm trying to find a way to implement the same thing in C++ (i.e. have a function that also takes only one argument, and returns the same values).
I have read up on ways of using FFT and IFFT to build it, but can't seem to get anything as simple as the Matlab version. The main thing is that I need it to work on a 128*2000 matrix, and nothing I've found in my search has showed me how to do that.
I would be OK with either a complex value returned, or just the absolute value. The simpler it is to integrate into the code, the better.
Thank you.
The MatLab function hilbert() does actually not compute the Hilbert transform directly but instead it computes the analytical signal, which is the thing one needs in most cases.
It does it by taking the FFT, deleting the negative frequencies (setting the upper half of the array to zero) and applying the inverse FFT. It would be straight forward in C/C++ (three lines of code) if you've got a decent FFT implementation.
This looks pretty good, as long as you can deal with the GPL license. Part of a much larger numerical computing resource.
Simple code below. (Note: this was part of a bigger project). The value for L is based on the your determination of your order, N. With N = 2L-1. Round N to an odd number. xbar below is based on the signal you define as the input to your designed system. This was implemented in MATLAB.
L = 40;
n = -L:L; % index n from [-40,-39,....,-1,0,1,...,39,40];
h = (1 - (-1).^n)./(pi*n); %impulse response of Hilbert Transform
h(41) = 0; %Corresponds to the 0/0 term (for 41st term, 0, in n vector above)
xhat = conv(h,xbar); %resultant from Hilbert Transform H(w);
plot(abs(xhat))
Not a true answer to your question but maybe a way of making you sleep better. I believe that you won't be able to be much faster than Matlab in the particular case of what is basically ffts on a matrix. That is where Matlab excels!
Matlab FFTs are computed using FFTW, the de-facto fastest FFT algorithm written in C which seem to be also parallelized by Matlab. On top of that, quoting from http://www.mathworks.com/help/matlab/ref/fftw.html:
For FFT dimensions that are powers of 2, between 214 and 222, MATLAB
software uses special preloaded information in its internal database
to optimize the FFT computation.
So don't feel bad if your code is slightly slower...
My question is an extension of the discussion How to fit the 2D scatter data with a line with C++. Now I want to extend my question further: when estimating the line that fits 2D scatter data, it would be better if we can treat each 2D scatter data differently. That is to say, if the scatter point is far away from the line, we can give the point a low weighting, and vice versa. Therefore, the question then becomes: given an array of 2D scatter points as well as their weighting factors, how can we estimate the linear line that passes them? A good implementation of this method can be found in this article (weighted least regression). However, the implementation of the algorithm in that article is too complicated as it involves matrix calculation. I am therefore trying to find a method without matrix calculation. The algorithm is an extension of simple linear regression, and in order to illustrate the algorithm, I wrote the following MATLAB codes:
function line = weighted_least_squre_for_line(x,y,weighting);
part1 = sum(weighting.*x.*y)*sum(weighting(:));
part2 = sum((weighting.*x))*sum((weighting.*y));
part3 = sum( x.^2.*weighting)*sum(weighting(:));
part4 = sum(weighting.*x).^2;
beta = (part1-part2)/(part3-part4);
alpha = (sum(weighting.*y)-beta*sum(weighting.*x))/sum(weighting);
a = beta;
c = alpha;
b = -1;
line = [a b c];
In the above codes, x,y,weighting represent the x-coordinate, y-coordinate and the weighting factor respectively. I test the algorithm with several examples but still not sure whether it is right or not as this method gets a different result with Polyfit, which relies on matrix calculation. I am now posting the implementation here and for your advice. Do you think it is a right implementation? Thanks!
If you think it is a good idea to downweight points that are far from the line, you might be attracted by http://en.wikipedia.org/wiki/Least_absolute_deviations, because one way of calculating this is via http://en.wikipedia.org/wiki/Iteratively_re-weighted_least_squares, which will give less weight to points far from the line.
If you think all your points are "good data", then it would be a mistake to weight them naively according to their distance from your initial fit. However, it's a fairly common practice to discard "outliers": if a few data points are implausibly far from the fit, and you have reason to believe that there's an error mechanism that could generate a small subset of "bad" datapoints, you could simply remove the implausible points from the dataset to get a better fit.
As far as the math is concerned, I would recommend biting the bullet and trying to figure out the matrix math. Perhaps you could find a different article, or a book which has a better presentation. I will not comment on your Matlab code, except to say that it looks like you will have some precision problems when subtracting part4 from part3, and probably part2 from part1 as well.
Not exactly what you are asking for, but you should look into robust regression. MATLAB has the function robustfit (requires Statistics Toolbox).
There is even an interactive demo you can play with to compare regular linear regression vs. robust regression:
>> robustdemo
This shows that in the presence of outliers, robust regression tends to give better results.
My program tries to solve a system of linear equations. In order to do that, it assembles matrix coeff_matrix and vector value_vector, and uses Eigen to solve them like:
Eigen::VectorXd sol_vector = coeff_matrix
.colPivHouseholderQr().solve(value_vector);
The problem is that the system can be both over- and under-determined. In the former case, Eigen either gives a correct or uncorrect solution, and I check the solution using coeff_matrix * sol_vector - value_vector.
However, please consider the following system of equations:
a + b - c = 0
c - d = 0
c = 11
- c + d = 0
In this particular case, Eigen solves the three latter equations correctly but also gives solutions for a and b.
What I would like to achieve is that only the equations which have only one solution would be solved, and the remaining ones (the first equation here) would be retained in the system.
In other words, I'm looking for a method to find out which equations can be solved in a given system of equations at the time, and which cannot because there will be more than one solution.
Could you suggest any good way of achieving that?
Edit: please note that in most cases the matrix won't be square. I've added one more row here just to note that over-determination can happen too.
I think what you want to is the singular value decomposition (SVD), which will give you exact what you want. After SVD, "the equations which have only one solution will be solved", and the solution is pseudoinverse. It will also give you the null space (where infinite solutions come from) and left null space (where inconsistency comes from, i.e. no solution).
Based on the SVD comment, I was able to do something like this:
Eigen::FullPivLU<Eigen::MatrixXd> lu = coeff_matrix.fullPivLu();
Eigen::VectorXd sol_vector = lu.solve(value_vector);
Eigen::VectorXd null_vector = lu.kernel().rowwise().sum();
AFAICS, the null_vector rows corresponding to single solutions are 0s while the ones corresponding to non-determinate solutions are 1s. I can reproduce this throughout all my examples with the default treshold Eigen has.
However, I'm not sure if I'm doing something correct or just noticed a random pattern.
What you need is to calculate the determinant of your system. If the determinant is 0, then you have an infinite number of solutions. If the determinant is very small, the solution exists, but I wouldn't trust the solution found by a computer (it will lead to numerical instabilities).
Here is a link to what is the determinant and how to calculate it: http://en.wikipedia.org/wiki/Determinant
Note that Gaussian elimination should also work: http://en.wikipedia.org/wiki/Gaussian_elimination
With this method, you end up with lines of 0s if there are an infinite number of solutions.
Edit
In case the matrix is not square, you first need to extract a square matrix. There are two cases:
You have more variables than equations: then you have either no solution, or an infinite number of them.
You have more equations than variables: in this case, find a square sub-matrix of non-null determinant. Solve for this matrix and check the solution. If the solution doesn't fit, it means you have no solution. If the solution fits, it means the extra equations were linearly-dependant on the extract ones.
In both case, before checking the dimension of the matrix, remove rows and columns with only 0s.
As for the gaussian elimination, it should work directly with non-square matrices. However, this time, you should check that the number of non-empty row (i.e. a row with some non-0 values) is equal to the number of variable. If it's less you have an infinite number of solution, and if it's more, you don't have any solutions.