If we call LAPACK DGGEV or DGGEVX routines for two badly-conditioned matrices in a C++ code, will we get the same eigen-values & vectors as MATLAB's eig command with 'qz' algorithm like: eig(a,b,'qz')?
Is there any possibility that eigenvectors of MATLAB are different from LAPACK DGGEV or DGGEVX?
A few things to note:
By definition A·v = λ·v, eigenvectors are not unique. You can multiply by any constant and still get another valid eigenvector.
The convention in MATLAB is that for eig(A), the eigenvectors are scaled so that the norm of each is 1.0, and for eig(A,B), the eigenvectors are not normalized (see here for an example). Here is the relevant part in the documentation:
V: right eigenvectors, returned as a square matrix whose columns are the right eigenvectors ofAor generalized right eigenvectors of the pair(A,B). The form and normalization ofVdepends on the combination of input arguments:
[V,D] = eig(A)returns matrixV, whose columns are the right eigenvectors ofAsuch thatA*V = V*D. The eigenvectors inVare normalized so that the 2-norm of each is 1.
[V,D] = eig(A,'nobalance')also returns matrixV. However, the 2-norm of each eigenvector is not necessarily 1.
[V,D] = eig(A,B)and[V,D] = eig(A,B,algorithm)returnsVas a matrix whose columns are the generalized right eigenvectors that satisfyA*V = B*V*D. The 2-norm of each eigenvector is not necessarily 1. In this case,Dcontains the generalized eigenvalues of the pair(A,B), along the main diagonal.If
Ais symmetric andBis symmetric positive definite, then the eigenvectors inVare normalized so that the 2-norm of each is 1.
In addition, eigenvalues are not sorted. You are only guaranteed that the columns of V are the corresponding right eigenvectors to the eigenvalues in D. That's not the same as svd.
In fact, there is no total ordering of complex numbers. The convention in MATLAB is that the sort function sorts complex elements first by magnitude (i.e. abs(x)), then by phase angle on the [-pi,pi] interval (i.e. angle(x)) if magnitudes are equal.
Having considered the above, when solving the generalized eigenvalue problem, you should be getting similar results to MATLAB by using DGGEV routine in LAPACK.
B-norm is it the same as 2-norm?
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B-norm mean that Z**T*B*Z = I, mentioned here in DSYGV documentation?
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2-norm in my quote above. I'm not so sure now... Anyway see here to understand the naming convention of LAPACK routines: xyyzzz where x is the type (single, double, complex), yy is the type of the matrix (general, symmetric, diagonal, etc..), and zzz is the computation performed (e.g: EV, GV, SVX, etc..). So if there is a version of the algorithm specialized for the matrix shape you have you should use it instead (say symmetric positive-definite)
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The best way to find the answer to this question was to experiment:
We calculated the Eigen values & vectors of two badly-conditioned matrices with MATLAB R2013b by using eig command like eig(a,b,'qz'). The ill-conditioned matrices were 342x342 in size.
Based on the explanations on the following link, MATLAB R2009a implemented LAPACK DGGEV for doing eig(a,b,'qz'):
Therefore, we implemented DGGEV (and also DGGEVX) routines from LAPACK in our C++ code by using LAPACKE which is a "C" language wrapper for LAPACK developed by INTEL-MKL team.
The conclusion was that we got exactly the same Eigen values & vectors as MATLAB eig(a,b,'qz') precisely.
Please note that if you use DGGEVX (expert version of DGGEV), you should turn the balancing off by using 'N' for BALANC argument, to get exactly the same Eigen vectors as MATLAB 'qz' algorithm.
We also checked the correctness of Eigen values & vectors by checking if they satisfy A*V-B*V*D=0, and the coclusion was that Eigen values & vectors were correct, as the residual error was close enough to zero.