chol() returns error even the matrix is positive definite.
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I have a matrix , where I is a identity matrix and B is a matrix. It is clear that M is a positive definite matrix. However, I am facing a situation in which the Matlab Cholesky decomposition function
chol(M);
returns an error stating that M must be positive definite. I examine matrix B and realise that it's elements are pretty small, resulting in the elements in matrix are extreme large in magnitude (the magnitudes are about e+15 to e+17).
Any advice/suggestion how to handle this issue is much appreciated !
Update on 9 Dec 22:
To help explaning the issue more cleary, I attached the data containing the matrix. Please load the data 'example.mat' and run the code below
[p, r] = size(B);
B1 = d.\B;
z1 = z./d;
z2 = B1'*z1;
R = chol(eye(r) + B1'*B1);
2 Comments
Walter Roberson
on 6 Dec 2022
Taking B'*B cannot result in large values if the entries are all small... not unless there are a huge number of entries. Each element of B'*B is the dot product between a row and another row of B. The maximum possible value would be max(abs(B))^2 * number of columns
Perhaps you are looking at the pseudoinverse of the matrix instead of the transpose
Answers (2)
Bruno Luong
on 9 Dec 2022
Edited: Bruno Luong
on 9 Dec 2022
This statement
B1 = d.\B;
looks very suspecious to me (the .\ operator), if you remove the dot it will work.
Here you do element-wise of d that oscillate to 0, create a huge number in B1 (it can goes to infinity indeed s such noisy data)
[p, r] = size(B);
B1 = d\B;
R = chol(eye(r) + B1'*B1);
Now wht you trying to do and implement only you know, but I suspect your code is incorrect for whatever the goal you want to achieve.
2 Comments
Bruno Luong
on 9 Dec 2022
Edited: Bruno Luong
on 9 Dec 2022
Only you know what is correct, but your d looks like this
It looks like it contains noise close to 0 and it cross randomly 0 mny time. 1./ d.\1 is then randomy big.
It that make sense for you then OK, but don't expect getting anything meaningful from that.
B1'*B1 have condition number and large element of the order of probably 1e20. Add eye(r) to it is meaning less, doing cholesky doesn't make any sense numerically. Your code won't produce anything usuful but gardebage with such data.
Askic V
on 6 Dec 2022
Edited: Walter Roberson
on 7 Dec 2022
Without actual code it is very hard to know for sure, but using chol() function in Matlab is actually a preferred way to test if the matrix is positive definite.
For me, it is very hard to understand how values of B'*B can be extremly large if values of B are small. Something is wrong. Please have a look at the example:
B = randn(3,3)*1e-6;
C = B'*B
max(C(:))
min(C(:))
Regarding your problem, I suggest you to read these advices:
4 Comments
Bruno Luong
on 9 Dec 2022
Edited: Bruno Luong
on 9 Dec 2022
@Askic V "In this particular example, you do have symmetric but not positive definite matrix"
I claim is wrong
C = eye(r) + B1'*B1
is always spd theorically since
x'*C*x is equal to |x|^2+|B1*x|^ 2 >= |x|^2 > 0 for any vector x ~= 0. Which is the requirement of spd (symmetric is easy to see).
Even B1'*B1 is non negative, eig fails to estimate the smallest eigen value.
OP simply has numerical error due to bad scaling issue.
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