I am using the polyeig command in Matlab to solve a polynomial eigenvalue problem of order 2 in Matlab. I know that the system has a single 0 eigenvalue (this is due to the form of the zero coefficient matrix where each diagonal element is -1 times the sum of the elements in he same row so the vector (1 1 1 ... 1) has 0 eigenvalue).
Size of the system is about 150 to 150.
When I use the polyeig command the lowest eigenvalue I get is of the order 1E-4 (which is supposed to be the 0 eigenvalue) and the second lowest is of the order 1E-1. As the system size decreases the lowest eigenvalue decreases to something of the order 1E-14 which is reasonable but 1E-4 is too much.
Is there anyway to achieve better accuracy or any other library you would suggest?
I could also turn the polynomial eigenvalue problem to generalized eigenvalue problem in higher dimensions (2 times the given dimension) but I am not sure how that affects speed and accuracy. I would like to see if there is a simpler solution before reformulating the problem. So I would welcome any suggestions on these matters.
EDIT: The problem is resolved it was actually about the precision of the INPUT file that I was using which was printed only up to 4 digits. Having found better ones the precision has increased. Thanks in any case.
The problem turned out to be with the input file I was using which was printed only up to 4 decimal points. Now even with matrices of 800x800 I only get accuracy problems up to e-11 which is good.
Related
My extended kalman filter (EKF) program works well: my estimated state vector is same as real state vector when I give any positive definite number to measurement noise R, even though I gives 10^ -14 to R.
But I want to make covariance analysis, and one part of covariance analysis I need to set zero measurement noise. When I do this, I get singularity warning from K= (H*P*H'+R)^-1 (kalman gain part of measurement correction part of EKF).
I checked eigenvalues and rank. When I get R=0, some eigenvalues becomes negative a few seconds later and rank is decrease from 15 to 1. When I get R>0, all eigenvalues are positive definite and rank goes to 15 to 7. How can I solve problem, I can not detect cause of this problem.
How could I go about this?
I meant initial covariance matrix and process noise matrix are given but measurement matrix is zero to observe effect of measurement noise on total error of estimated covariance matrix. Also, I solved my problem with 2 options. First one is measurement noise should be higher than zero, it can be close to zero. So, Gain is not goes to infinity. Second one is, If HPH' part of Gain calculation is close enough to zero, do not make correction on Gain because we do not need to correction on covariance matrix, they very close to each other (prior and posteri) if HPH' is zero. Briefly, I solved my problem. Thanks.
I am classifying a dataset with four classes using pretrained VGG19. To calculate accuracy, I used this formula:
accuracy = sum(predictedLabels==testLabels)/numel(predictedLabels) --Eq 1
Then I calculated the confusion matrix using:
confMat = confusionmat(testLabels, predictedLabels) **--Eq 2**
From which I got a matrix with 4 rows and 4 columns since I had 4 classes.
Now, we know that the accuracy formula is also:
Accuracy=TP+TN/(TP+TN+FP+FN) **Eq-3**
So I also calculated Accuracy from my confusion matrix formed through above Eq. 2. where
TP=value in (row==column),
FP=sum of column-TP,
FN=sum of row-TP,
TN=sum of the diagonal-TP
If I am doing above steps alright, then my confusion is that I am getting different accuracies from two methods Eq 1 and Eq 3. The accuracy I am getting with Eq. 1 is equivalent to the formula TP/(TP+TN). so, If this is the case, then Eq. 1 is the wrong formula for calculating accuracy. But, this formula has been used across all matlab deep learning codes.
So, MATLAB is doing something wrong (which has the probability 0, I know) or I am doing something wrong. But, unfortunately, I am unable to pinpoint my mistake.
Now, the question is,
Am I doing it wrong? Where am I missing the step? How to correct it? What is the logical explanation of this anomaly?
EDIT
This anomaly in accuracy calculation happens due to class imbalance problem. that is when, there are different number of samples in each class. therefore, the regular accuracy formula in Eq. 3 will not work in such cases.
The main issue is that negative and positive is for prediction (is this a cat or not), while you are doing classification with more than two categories. The classifier doesn't give you positive and negative (for is it a cat prediction), so it is not possible to relate to answers as true positive or false positive etc. Therefore equation 3 is meaningless, and so is the method for computing TP, TN etc. For example, if TP is row=column as you defined, then these are the accurate values in the diagonal of confMat. But what is TN? According to your definition it is TP (which is the diagonal) minus the diagonal. I hope this helps put things on the write track.
I use eigs to calculate the eigen vectors of sparse square matrices which are large (tens of thousands).
What I want is the smallest set of eigen vectors.
But
eigs(A, 10, 'sm') % Note: A is the matrix
runs very slow.
However, using eigs(A, 10, 'lm') gives me the answer relatively faster.
And as I tried, replacing 10 with A_width in eigs(A, 10, 'lm') so that this includes all the eigen vectors, doesn't solve this problem, 'cause this make it the as slow as using 'sm'.
So, I want to know why calculating the smallest vectors(using 'sm') is much slower than calculating the largest?
BTW, if you have any idea about how to use eigs with 'sm' as fast as with 'lm', please tell me that.
The algorithm used in pretty much any standard eigs function is (some variation of) the Lanczos algorithm. It is iterative and the first iterations give you the largest eigenvalues. This explains pretty much every observation you make:
Largest eigenvalues take the least amount of iterations,
Smallest eigenvalues take the maximum amount of iterations,
All eigenvalues also take the maximum amount of iterations.
There are tricks to "fool" eigs into calculating the smallest eigenvalues by actually making them the largest eigenvalues of another problem. This is usually accomplished by a shift parameter. Skimming over the Matlab documentation for eigs, I see that they have a sigma parameter, which might help you. Note the same documentation recommends proper eig if the matrix fits into memory, as eigs has its numerical quirks.
Since eigs is actually an m-file function, we can profile it. I have run a couple of basic tests, and it depends very much on the nature of the data in the matrix. If we run the profiler separately on the following two lines of code:
eigs(eye(1000), 10, 'lm'), and
eigs(eye(1000), 10, 'sm'),
then in the first instance it calls arpackc (the main function that does the work - according to the comments in eigs it's probably from here) a total of 22 times. In the second instance it is called 103 times.
On the other hand, trying it with
eigs(rand(1000), 10, 'lm'), and
eigs(rand(1000), 10, 'sm'),
I get results where the 'lm' option consistently calls arpackc many more times than the sm option.
I'm afraid I don't know the details of the algorithm, and so can't explain it in any deeper mathematical sense, but the page that I linked suggests ARPACK is best for matrices with some structure. Since matrices generated by rand have little structure, it is probably safe to assume the latter behaviour I described is not what you'd expect under normal operating conditions.
In short: it simply takes the algorithm more iterations to converge when you ask it for the smallest eigenvalues of a structured matrix. This being an iterative process, however, it very much depends on the actual data you give it.
Edit: There is a wealth of information and references about this method here, and the key to understanding exactly why this happens is surely contained somewhere therein.
The reason is actually much more simple and due to the basics of solving large sparse eigenvalue problems. These are all based on solving:
(1) A x = lam x
Most solution methods use some power law (e.g. a Krylov subspace spanned in both the Lanczos and Arnoldi methods)
The thing is that the a power series converge to the largest eigenvalue of (1). Therefore we have that the largest eigenvalues are found by the subspace spanned by: K^k = {A*r0,....,A^k*r0}, which requires only matrix vector multiplications (cheap).
To find the smallest, we have to reformulate (1) as follows:
(2) 1/lam x = A^(-1) x or A^(-1) x = invlam x
Now solving for the largest eigenvalue of (2) is equivalent to finding the smallest eigenvalue of (1). In this case the subspace is spanned by K^k = {A^(-1)*r0,....,A^(-k)*r0}, which requires solving several linear system (expensive!).
I'm running an optimization algorithm that requires calculation of the inverse of a matrix. The goal of the algorithm is to eliminate negative values from the matrix A and obtain the new matrix B. Basically, I start with known square matrices B and C of the same size.
I start by calculating the matrix A which is equal to:
A = B^-1 * C
Or in Matlab:
A = B\C;
I use this because Matlab told me B\C is more accurate than inv(B)*C.
The negative values in A are then divided by two and A is then normalised so that it's rows have length of 1. Using this new A, I calculate a new B with:
(1/N) * A * C' = B^-1
where N is just a scaling factor (# of columns in A). This new B would then be used again in the first step and these iterations continue until the negatives in A are gone.
My problem is I have to calculate B from the second equation and then normalise it.
invB = (1/N)*A*C';
B = inv(invB);
I've been calculating B using inv(B^-1) but after a few iterations I start getting messages that B^-1 is "close to singular or badly scaled."
This algorithm actually works for smaller matrices (around 70x70) but when it gets up to about 500x500 I start getting these messages.
Are there any better ways to calculate inv(B^-1)?
You should definitely head warnings about singular matrices. Results in numerical linear algebra tend to break down as you move toward matrices with high condition numbers. The underlying idea is if
A*b_1 = c
and we're actually solving the problem (because we are using approximate numbers when we use computers)
(A + matrix error)*b_2 = (c + vector error)
how close are b_1 and b_2 as a function of the matrix and vector errors? When A has small condition number b_1 and b_2 are close. When A has large condition number b_1 and b_2 are not close.
There is an informative piece of analysis you could do on your algorithm. At each iteration, after you've found B, find use Matlab to find the condition number of it. This is
cond(B)
You will likely see the number climb rapidly. This indicates that every time you iterate your algorithm, you should trust your result for B less and less.
Problems like this crop up all the time in numerical mathematics. If you'll be working with numerical algorithms frequently you should take some time to familiarize with the role of condition numbers in the field and preconditioning techniques as mentioned above. My preferred text for this is "Numerical Linear Algebra" by Lloyd Trefethen, but any text on Numerical Algebra should address some of these issues.
Best of luck,
Andrew
The main issue is that your matrix has a high condition number (i.e. really small rcond(B) in your case). This is due to the iterative structure in your algorithm, I guess. As you do each iteration your small singular values get smaller and smaller so your condition number grows exponentially. You should check preconditioning to avoid this kind of behavior.
from a simulation problem, I want to calculate complex square matrices on the order of 1000x1000 in MATLAB. Since the values refer to those of Bessel functions, the matrices are not at all sparse.
Since I am interested in the change of the determinant with respect to some parameter (the energy of a searched eigenfunction in my case), I overcome the problem at the moment by first searching a rescaling factor for the studied range and then calculate the determinants,
result(k) = det(pre_factor*Matrix{k});
Now this is a very awkward solution and only works for matrix dimensions of, say, maximum 500x500.
Does anybody know a nice solution to the problem? Interfacing to Mathematica might work in principle but I have my doubts concerning feasibility.
Thank you in advance
Robert
Edit: I did not find a convient solution to the calculation problem since this would require changing to a higher precision. Instead, I used that
ln det M = trace ln M
which is, when I derive it with respect to k
A = trace(inv(M(k))*dM/dk)
So I at least had the change of the logarithm of the determinant with respect to k. From the physical background of the problem I could derive constraints on A which in the end gave me a workaround valid for my problem. Unfortunately I do not know if such a workaround could be generalized.
You should realize that when you multiply a matrix by a constant k, then you scale the determinant of the matrix by k^n, where n is the dimension of the matrix. So for n = 1000, and k = 2, you scale the determinant by
>> 2^1000
ans =
1.07150860718627e+301
This is of course a huge number, so you might expect that it should fail, since in double precision, MATLAB will only represent floating point numbers as large as realmax.
>> realmax
ans =
1.79769313486232e+308
There is no need to do all the work of recomputing that determinant, not that computing the determinant of a huge matrix like that is a terribly well-posed problem anyway.
If speed is not a concern, you may want to use det(e^A) = e^(tr A) and take as A some scaling constant times your matrix (so that A - I has spectral radius less than one).
EDIT: In MatLab, the log of a matrix (logm) is calculated via trigonalization. So it is better for you to compute the eigenvalues of your matrix and multiply them (or better, add their logarithm). You did not specify whether your matrix was symmetric or not: if it is, finding eigenvalues are easier than if it is not.
You said the current value of the determinant is about 10^-300.
Are you trying to get the determinant at a certain value, say 1? If so, rescaling is awkward: the matrix you are considering is ill-conditioned, and, considering the precision of the machine, you should consider the output determinant to be zero. It is impossible to get a reliable inverse in other words.
I would suggest to modify the columns or lines of the matrix rather than rescale it.
I used R to make a small test with a random matrix (random normal values), it seems the determinant should be clearly non-zero.
> n=100
> M=matrix(rnorm(n**2),n,n)
> det(M)
[1] -1.977380e+77
> kappa(M)
[1] 2318.188
This is not strictly a matlab solution, but you might want to consider using Mahout. It's specifically designed for large-scale linear algebra. (1000x1000 is no problem for the scales it's used to.)
You would call into java to pass data to/from Mahout.